Singular Elliptic Problems: Bifurcation & Asymptotic Analysis (Math Applications Series)

OXIORD [tCTURS AND ITS ArrIICATIONS

IN tlAi111WAYIC.1 17

Singular Elliptic Problems Bifurcation and Asymptotic Analysis

Marius Ghergu Viceniiu RAdulescu

OXFORD LECTURE SERIES IN MATHEMATICS AND ITS APPLICATIONS For a full list of titles, please visit http://www.oup,co.uk/academic/science/maths/ series/OLSMA/ 10. P.L. Lions: Mathematical topics in fluid mechanics, Vol. 2: Compressible models 11. W.T. Tutte: Graph theory as I have known it 12. Andrea Braides and Anneliese Defranceschi: Homogenization of multiple integrals 13. Thierry Cazenave and Alain Haraux: An introduction to semilinear evolution equations 14. J.Y. Chemin: Perfect incompressible fluids 15. Giuseppe Buttazzo, Mariano Giaquinta and Stefan Hildebrandt: One-dimensional variational problems: an introduction 16. Alexander I. Bobenko and Ruedi Seiler: Discrete integrable geometry and physics 17. Doina Cioranescu and Patrizia Donato: An introduction to homogenization 18. E.J. Janse van Rensburg: The statistical mechanics of interacting walks, polygons, animals and vesicles 19. S. Kuksin: Hamiltonian partial differential equations 20. Alberto Bressan: Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem 21. B. Perthame: Kinetic formulation of conservation laws 22. A. Braides: Gamma-convergence for beginners 23. Robert Leese and Stephen Hurley: Methods and Algorithms for Radio Channel Assignment 24. Charles Semple and Mike Steel: Phylogenetics 25. Luigi Ambrosio and Paolo Tilli: Topics on Analysis in Metric Spaces 26. Eduard Feireisl: Dynamics of Viscous Compressible Fluids 27. Antonin Novotny and Ivan Straskraba: Introduction to the Mathematical Theory of Compressible Flow 28. Pavol Hell and Jarik Nesetril: Graphs and Homomorphisms 29. Pavel Etingof and Frederic Latour: The dynamical Yang-Baxter equation, representation theory, and quantum integrable .systems 30. Jorge Ramirez Alfonsin: The-Diophantine Frobenius Problem 31. Rolf Niedermeier: Invitation to Fixed Parameter Algorithms 32. Jean-Yves Chemin, Benoit Desjardins, Isabelle Gallagher and Emmanuel Grenier: Mathematical Geophysics: An introduction to rotating fluids and the Navier-Stokes equations 33. Juan Luis Vizquez: Smoothing and Decay Estimates for Nonlinear Diffusion Equations 34. Geoffrey Grimmett and Colin McDiarmid: Combinatorics, Complexity and Chance 35. Alessio Corti: Flips. for 3-colds and 4-folds 36. Kirsch and Grinberg: The Factorization Method for Inverse Problems 37. Ghergu and Radulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis

Singular Elliptic Problems: Bifurcation and Asymptotic Analysis Marius Ghergu Vicentiu D. RAdulescu

CLARENDON PRESS OXFORD 2008

OXFORD UNIVERSITY PRESS Oxford University Press, Inc., publishes works that further Oxford University's objective of excellence in research, scholarship, and education.

Oxford New York Auckland CapeTown Dares Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne MexicoCity Nairobi NewDelhi Shanghai Taipei Toronto With offices in

Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam

Copyright © 2008 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press

All rights reserved. No part of this publication may be reproduced. stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Ghergu. Marius Singular elliptic problems. bifurcation and asymptotic analysis I Marius Ghergu, Viccntiu D. RSdulescu. p. cm. - (Oxford lecture series in mathematics and its applications; 37) Includes bibliographical references and index. ISBN 978-0-19-533472-2

1. Differential equations, Elliptic-Asymptotic theory. 2. Differential equations, Nonlinear. 3. Bifurcation theory. 1. Radulescu, V. II. Title. QA377.G47 2008 515`.3533-dc22

2007060129

135798642 Printed in the United States of America on acid-free paper

To our families, for their patience and continuous support over the years

PREFACE The most incomprehensible thing

about the world is that it is comprehensible. Albert Einstein (1879-1955)

The development of nonlinear analysis during the last few decades has been profoundly influenced by attempts to understand various phenomena from mathematical physics. One of the beauties of the subject is the immense breadth of mathematics that has been applied in this pursuit. There is an enormous body of literature in nonlinear elliptic partial differential equations that stretches back half a century. However, we shall make almost no reference to this literature, and shall rely almost entirely upon personal re-

sults. These lecture notes are primarily intended to fill, in a substantial way, the absence of a book dealing with the qualitative analysis of some basic singular stationary processes arising in nonlinear sciences. This volume aims to offer an introduction to this subject, and also to present some research problems. The models that we analyze represent a compromise between the description of physical phenomena and analytical requirements; accordingly, our presentation is characterized by a strict interplay between mathematics and nonlinear sciences. The book is an outgrowth of our original research on the subject during the last few years, and much of the development is motivated by problems arising in applications. However, most of the proofs have been completely reworked and we are especially careful to explain where each chapter is going, why it matters, and what background material is required. Although the theory that we describe could have been carried out on differentiable manifolds even from the beginning,

we have chosen to develop it on domains on the Euclidean space. However, the techniques we develop can be extended to Laplace-Beltrami operators on Riemannian manifolds.

The major thrust of this book is the qualitative analysis of some classes of nonlinear stationary problems involving different types of singularities. Be aware,

this is definitely a research book. We are mainly concerned with the following types of problems. We first study singular solutions of the logistic equation, with a basic model that is described by the semilinear elliptic equation Du = uP, where p > 1. The research program around this equation flourished after the pioneering papers by Bieberbach and Rademacher, continued with the deep contributions of Loewner and Nirenberg in Riemannian geometry, and creating recently (because of the works by Dynkin and Le Gall) a nonlinear analogue of the classical relation between Brownian motion and potential theory. Equations of this type arise in astrophysics, genetics, meteorology, theory of atomic spec-

viii

Preface

tra, and the Yamabe problem in geometry. A first consequence of such types of nonlinearities is the possibility of the existence of a "large solution"-that is, a solution blowing up at the boundary. When the large solution is unique, it is a maximal solution and dominates any solution. In connection with the previously mentioned applications, the existence of the large solution in a ball was used by Iscoe to establish the compact support property of super-Brownian motion, demonstrating the importance of the relationship between properties of superdiffusion and the equation. Next, we are concerned with Lane-EmdenFowler equations and Gierer-Meinhardt systems with singular nonlinearity. The model problem in such cases is described by equations like --Au = u-°, where a is a positive real number. To our best knowledge, the first study in this direction is from Fulks and Maybee, who proved existence and uniqueness results by using a fixed point argument; moreover, they showed that solutions of the associated parabolic problem tend to the unique solution of the corresponding elliptic equation. Different approaches are the result of Coclite and Palmieri, respectively Crandall, Rabinowitz, and Tartar, who approximated the singular equation with regular problems, where the standard monotonicity techniques do work. Singular problems of this type arise in the context of chemical heterogeneous catalysts and chemical catalyst kinetics, in the theory of heat conduction in electrically conducting materials, singular minimal surfaces, as well as in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, glacial advance, transport of coal slurries down conveyor belts, and in several other geophysical and industrial contents. In both cases, because of the meaning of the unknowns (concentrations, populations, etc.), the positive solutions are relevant in most situations. We intend to give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear elliptic equations, as well as their applications to various processes arising in mathematical physics. Our approach leads not only to the basic results of existence, uniqueness, and multiplicity of solutions, but also to several qualitative properties, including bifurcation, asymptotic analysis, blow-up and so forth. Moreover, because the book is concerned primarily with classical solutions, the monotone iteration processes we apply for various classes of nonlinear singular problems are adaptable to numerical solutions of the corresponding discrete processes. To place the text in better perspective, each chapter is concluded with a section on historical notes that includes references to all important and relatively new results. In addition

to cited works, the list of references contains many other works related to the material developed in this volume. The organization of the book is briefly summarized as follows. The first chapter deals with preliminary material, such as the method of sub- and supersolution, several variants of the maximum principle (Stampacchia, Vazquez, Pucci, and Serrin), and various existence and uniqueness results for nonlinear elliptic boundary value problems.

Preface

ix

Part II is composed of two chapters, which are concerned with singular solutions of logistic-type equations or systems. There are studied both equations with blow-up boundary solutions and entire solutions blowing up at infinity for elliptic systems. In all these cases, the major role played by the Keller-Osserman condition is discussed. In the third part of this book we are concerned with elliptic problems involving singular nonlinearities, either in isotropic or in anisotropic media. Chapter 4 deals with sublinear elliptic problems that are affected by singular perturbations. We distinguish between equations on bounded domains or on the whole space and we are also concerned with a related bifurcation problem. Chapters 5 and 6 are devoted to the study of a bifurcation problem in the case of linear growth for the nonlinearity. Two different situations are distinguished and a complete discussion is developed in both circumstances. The superlinear case is studied in Chapter 7, by means of variational arguments, whereas Chapter 8 is concerned with stability properties of solutions. Chapter 9 is devoted to the study of the "competition" between various terms in a singular Lane-EmdenFowler equation with convection and variable (possible, singular) potential. In the last chapter of these lecture notes, the qualitative analysis of solutions is extended to the case of singular Gierer-Meinhardt systems. We refer to the works of J.M. Ball [11,12], V. Barbu [19], L. Beznea and N. Boboc [22], H. Brezis [30], and P.G. Ciarlet [46,47] for related results and various applications to concrete phenomena.

Four appendices illustrate some basic mathematical tools applied in this book: elements of spectral theory for differential operators, the implicit function theorem, Ekeland's variational principle, and the mountain pass theorem. These auxiliary chapters deal with some analytical methods used in this volume, but also include some complements. Each problem we develop in this book has its own difficulties. That is why we intend to develop some standard and appropriate methods that are useful and that can be extended to other problems. However, we do our best to restrict the prerequisites to the essential knowledge. We define as few concepts as possible and give only basic theorems that are useful for our topic. The only prerequisite for this volume is a standard graduate course in partial differential equations, drawing especially from linear elliptic equations to elementary variational methods, with a special emphasis on the maximum principle (weak and strong variants). This volume may be used for self-study by advanced graduate students and engineers, and as a valuable reference for researchers in pure and applied mathematics and physics. Our vision throughout this volume is closely inspired by the following words of Henri Poincare on the role of partial differential equations in the development of other fields of mathematics and in applications: Nevertheless, each time I can, I aim the absolute rigor for two reasons. In the first place, it is always hard for a geometer to consider a problem without resolving it completely. In the second place, these equations that I will study are susceptible, not only to

x

Preface

physical applications, but also to analytical applications. It is using the existence theory of the Dirichlet problem that Riemann founded his magnificent theory of Abelian functions. Since then, other geometers have made important applications of the same principle to the most fundamental parts of pure analysis. Is it still permitted to content oneself with a demi-rigor? And who will say that the other problems of mathematical physics will not, one day, be called to play in analysis a considerable role, as has been the case of the most elementary of them? (Henri Poincare 1164])

May 2007

ACKNOWLEDGMENTS We acknowledge, with unreserved gratitude, the crucial role of Professor Philippe G. Ciarlet, who encouraged writing a book on this subject, but also for his constant support over the years. We are greatly indebted to Sir John M. Ball for accepting our work in his prestigious series at Oxford University Press. We are also grateful to Dr. Michael

Penn, the Oxford book program editor, and to Paul Hobson, the production editor, for their efficient and enthusiastic help, as well as for numerous suggestions related to this volume.

We extend our warm thanks to Professors Catherine Bandle, Viorel Barbu, Lucian Beznea, Bernard Brighi, Olivier Goubet, Patrizia Pucci, and Michel Willem for their useful comments and remarks during the preparation of this work.

We thank Dr. Nicolae Constantinescu for the professional drawing of figures contained in this book. Both authors acknowledge the support of grants 2-CExO6-11-18/2006 and CNCSIS PNII-79/2007. They have been also supported by grants CNCSIS AT191/2007 (M. Chergu) and CNCSIS A-589/2007 (V. RAdulescu).

CONTENTS I 1

PRELIMINARIES

Basic methods 1.1 1.2 1.3

1.4

1.5 1.6

3

A fixed point result The method of sub- and supersolution Comparison principles 1.3.1 Weak and strong maximum principle 1.3.2 Maximum principle for weakly differentiable functions 1.3.3 Stampacchia's maximum principle 1.3.4 Vazquez's maximum principle 1.3.5 Pucci and Serrin's maximum principle 1.3.6 A comparison principle in the presence of singular nonlinearities Existence properties and related maximum principles 1.4.1 Dead core solutions of sublinear logistic equations 1.4.2 Singular solutions of the logistic equation Brezis-Oswald theorem Comments and historical notes -

3 5 9

9 10 11

12 15 17 20

23 24 26 32

II BLOW-UP SOLUTIONS 2

Blow-up solutions for semilinear elliptic equations 2.1 2.2

2.3

2.4 2.5

Introduction Blow-up solution for elliptic equations with vanishing potential 2.2.1 Existence results in bounded domains 2.2.2 Existence results in the whole space Blow-up solutions for logistic equations 2.3.1 The case of positive potentials 2.3.2 The case of vanishing potentials An equivalent criterion to the Keller-Osserman condition Singular solutions of the logistic equation on domains with holes

2.6 2.7

2.8

3

Uniqueness of blow-up solution A Karamata theory approach for uniqueness of blow-up solution Comments and historical notes

Entire solutions blowing up at infinity for elliptic systems 3.1

Introduction

37 37 40 40 42 45 45 46

49 52 58 64 71

75 75

Contents

xiv

3.2

3.3

Characterization of the central value set 3.2.1 Bounded or unbounded entire solutions 3.2.2 Role of the Keller-Osserman condition Comments and historical notes

76 76 81

89

III ELLIPTIC PROBLEMS WITH SINGULAR NONLINEARITIES

4

Sublinear perturbations of singular elliptic problems 4.1

4.2 4.3 4.4 4.5 4.6

4.7

4.8 5

5.4 5.5

5.6 5.7

112

Singular elliptic problems in the whole space 4.7.1 Existence of entire solutions 4.7.2 Uniqueness of radially symmetric solutions Comments and historical notes

113 113 117

Introduction A general bifurcation result Existence and bifurcation results Asymptotic behavior of the solution with respect to parameters Examples 5.5.1 First example 5.5.2 Second example The case of singular nonlinearities Comments and historical notes

121

125 126 127 129 134 137

137 138 140 142

Bifurcation and asymptotic analysis: The nonmonotone case 6.1 6.2 6.3

6.4 6.5 7

93 93 98 99 104 107

case

Bifurcation and asymptotic analysis: The monotone case 5.1 5.2 5.3

6

Introduction An ODE with mixed nonlinearities A complete description for positive potentials An example Bifurcation for negative potentials Existence for large values of parameters in the sign-changing

143

Introduction Auxiliary results Existence and bifurcation results in the nonmonotone case An example Comments and historical notes

Superlinear perturbations of singular elliptic problems 7.1 7.2 7.3 7.4

Introduction The weak sub- and supersolution method H1 local minimizers Existence of the first solution

143 144 147 154 154

157 157 158 161 163

7.5

7.6 7.7 7.8 8

xv

Existence of the second solution 7.5.1 First case 7.5.2 Second case C1 regularity of solution Asymptotic behavior of solutions Comments and historical notes

169

Stability of the solution of a singular problem 8.1

8.2 8.3 8.4 8.5

9

Contents

Stability of the solution in a general singular setting A min-max characterization of the first eigenvalue for the linearized problem Differentiability of some singular nonlinear problems Examples Comments and historical notes

The influence of a nonlinear convection term in singular elliptic problems 9.1 9.2 9.3 9.4 9.5 9.6 9.7

9.8 9.9

Introduction A general nonexistence result A singular elliptic problem in one dimension Existence results in the sublinear case Existence results in the linear case Boundary estimates of the solution The case of a negative singular potential 9.7.1 A nonexistence result 9.7.2 Existence result Ground-state solutions of singular elliptic problems with gradient term Comments and historical notes

10 Singular Gierer-Meinhardt systems 10.1 Introduction 10.2 A nonexistence result 10.3 Existence results 10.4 Uniqueness of the solution in one dimension 10.5 Comments and historical notes

Appendix A Spectral theory for differential operators A.1 Eigenvalues and eigenfunctions for the Laplace operator

A.2 Krein-Rutman theorem

Appendix B Implicit function theorem Appendix C Ekeland's variational principle Minimization of weak lower semicontinuous coercive functionals C.2 Ekeland's variational principle

170

174 182

184

189

191 191

199 201 204 205

207 207 208 210 215 226 228 231 232 233

238 241 243 243 244 248 255 261

265 265 266 269

273

C.1

273 273

xvi

Contents

Appendix D Mountain pass theorem D.1 Ambrosetti-Rabinowitz theorem

D.2 Application to the Emden-Fowler equation D.3 Mountains of zero altitude

277 277 281 282

References

283

Index

295

PART I

PRELIMINARIES

1

BASIC METHODS If I have seen further than others, it is by standing upon the shoulders of giants. Sir Isaac Newton (1642-1727), Letter to Robert Hooke, 1675

The main purpose of nonlinear analysis is to study and describe various material systems arising in mathematical physics and other applied sciences. The language of nonlinear analysis is very close to that of functional analysis and partial differential equations. This chapter deals with some general features about the qualitative analysis of nonlinear elliptic partial differential equations. The main issues we raise in what follows concern the existence and uniqueness of the solution. The existence of solutions is essential for a prescribed model to make sense. The uniqueness of the solution is a natural requirement for many problems. In some cases, uniqueness becomes a difficult question, but related interesting questions concern multiplicity of solutions, as well as the existence of maximal or minimal solutions. That is why monotonicity properties, described by means of comparison principles, play a central role in this chapter and throughout the book. 1.1

A fixed point result

This type of result applies successfully to classical ordinary differential equations (ODE) boundary value problems. Given a second order equation y"(t) = f (t, y(t), y'(t)) subject to the Dirichlet boundary condition, we look for a solution as a fixed point of a compact map F (i.e., F(y) = y) defined on a suitable closed subset of a normed vector space. There are many results in this direction, but we restrict our attention to the following basic property, which is due to Leray and Schauder [131].

Theorem 1.1.1 (Leray-Schauder) Let C be a convex set in a normed vector space E and let U C C be an open set that contains the origin. Then each compact map F : U -* E has at least one of the following properties: (i) F has a fixed point. (ii) There exist x E 8U and 0 < A < 1 such that x = AF(x). For the proof we refer the reader to Granas and Dugundji [98], O'Regan [152], and O'Regan and Precup [153]. We next give an application of this result to a singular ODE.

Basic methods

4

Consider the Sturm-Liouville problem

0
f y"(t) + f {t, y(t)) = 0 y(O) = a, y(l) = b,

where f : (0,1) x R -i R is a continuous function such that (Fl) for all r > 0 there exists ?P,. E L 10'C(0, 1) satisfying If (t, y) I < ?Pr(t)

for all (t, y) E (0, 1) x [-r, r],

(1.2)

and

fJ

1

t(1 - t)i,br(t)dt < oo.

(1.3)

0

We also consider the problem 0 < t < 1,

J y"(t) + Af (t, y(t)) = 0

{1.4) It

y(0) = y(1) = 0.

Theorem 1.1.2 Assume that f fulfills (Fl) and there exists M > 0 such that for all 0 < A < 1 and for any solution y E C2(0,1) fl C[0,1] of (1.4), we have

W. < M. Then problem (1.1) has a solution y E C2(0,1) nCIO, 1] such that jyj,' < M.

Proof We first remark that it suffices to argue if a = b = 0. Indeed, if this is not the case, we define z(t) = y(t) - all - t) - bt, for any 0 < t < 1. Thus, z verifies

0 < t < 1,

z"(t) + g(t, z(t)) = 0 z(0) = z(1) = 0,

where g(t, x) = f (t, x + a(l - t) + bt), (t, x) E (0, 1) x R. We also observe that g satisfies the hypothesis (Fl). Thus, we can assume a = b = 0. By virtue of (1.2) and (1.3), problem (1.1) is equivalent to Y(t) = (1 - t) fo sf t (s, y(s))ds + At t (1 - s) f (s, y(s))ds.

it

(1.5)

Set U:= {y E C[0,1] : ]yj,, < M} and define the map F : U -+ C[0,1] by I It Fy(t) = all - t) + bt + (1 - t) l s f (s, y(s))ds + t I ( 1 - s) f (s, y(s))ds. JJJo

Then F is a compact operator. We claim that property (ii) in Theorem 1.1.1 cannot be true, which implies immediately that F has a fixed point. Indeed, if there exist y e W and 0 < A < 1 such that y = AFy, then y is a solution of (1.4). Thus, by our hypothesis, IyL,, < M-that is, y W. This is clearly a contradiction. So, by Theorem 1.1.1, we deduce that F has a fixed pointy e U. In view of the equivalent formulation given in (1.5), this means that y is a solution of (1.1). This ends our proof.

The method of sub- and supersolution

1.2

5

The method of sub- and supersolution

Let S2 C RN(N > 1) be a bounded domain with a smooth boundary 8Sl (for instance, we can assume that aft is of class C3) and F = fi(x, t, ) : St x R x RN R be a Holder continuous function with exponent y E (0, 1) and continuously differentiable with respect to the variables t, t; and such that

(Al) for any 11o cc St and any 0 < a < b < oo, there exists a constant C > 0 such that

,p(x,t,Z:)l
Au = 4'(x, u, Vu)

u=0

in n, on 60.

By classical solution of (1.6) we mean a function u E C2'y(Q)nC(S2) that satisfies (1.6).

Definition 1.2.1 A function u E C2(Q) n C(Si) is said to be a subsolution of the problem (1.6) if u < 0 on aQ and

-Au <$(x,u,Vu)

in Q.

(1.7)

Similarly, is E C2 (Q) fl C(St) is a supersolution of (1.6) if `u > 0 on a g and

-Au >

u, Vu)

in Q.

The method of sub- and supersolution in its classical form was developed by H. Amann [5], [6] in the framework of nonlinear elliptic partial differential equations. Roughly speaking, this method establishes the existence of a solution, provided suitable sub- and supersolutions do exist. This result is a crucial property for showing the existence of solutions to wide classes of nonlinear elliptic problems.

Theorem 1.2.2 Let u and u be a sub- and a supersolution of problem (1.6) such that u < u in Q. Then the following properties hold true:

(i) There exists a solution u of (1.6) that satisfies u < u < U. (ii) There exists a minimal and a maximal solution U and U of problem (1.6) with respect to the interval [u, u]. Let us assume that (D is defined only on the set I x (0, oo) x RN, so that 4) may by singular in 80 x {0} x W'. In this case we are looking for a solution of the problem -Du = (D(x, U, Du) in S2, u > 0

in St,

u=0

onaQ.

(1.8)

Basic methods

6

A subsolution u of (1.8) is a function u E C2(9) fl C(?l), which is positive in 52, u = 0 on 852, and fulfills (1.7)_ Accordingly, u E C2(Sl) f1C(S2) is a supersolution

of (1.8) if u is positive in 0, u = 0 on 85l, and -i

> 4D(x, U, Vu) in 0.

Theorem 1.2.3 If u and u are respectively sub- and supersolutions of (1.8) such that u < u in Sl, then (1.8) has at least one classical solution.

Proof The proof relies on the domain approximation method. In this way we avoid the possible singularities of qb(x,u,Vu) at the boundary. Let (S2k)k>1 be a sequence of subdomains of H having smooth boundaries and such that 11 CC and

S2 = U 52k. k>1

For each k > 1, consider the problem -Au = 4D(x, U, Vu)

u>0

in Q k,

in52k, on 8S2k.

u=u

Obviously, the restriction of u and u on S2k are sub- and supersolutions of problem (1.9). Thus, by Theorem 1.2.2, there exists a minimal solution Uk E C2'-'(S2k) f1 C(S2k) of (1.9) such that u < uk < u in Stk. We extend Uk to the whole S2 by taking Uk = u in S2 \ Stk. Hence, (uk)k>1 is a sequence of continuous

functions such that

u
in S2k,

in S2,

for all k > 1.

(1.10)

(1.11)

For all x E 52, define u(x) := limky, uk(x). To obtain u E C2,"(52) fl C(11) and u is a solution of (1.8), we need the following result.

Lemma 1.2.4 For each j > 1 there exists a corresponding constant Cj > 0 such that for all k > j + 1. (1.12) IIukIIC2.,( ) < Cj

Proof Let j > 1 be fixed and take two subdomains Q1 and Q2 such that

S2jCCQ1CCQ2CCS2j+1 Then, for all k > j + 1 we have -Auk ='D (x, uk, VUk)

in ?2j+1.

If we denote ¢k := fi(x, Uk, VUk), k > j + 1, the previous equality reads

The method of sub- and supersolution

-AUk = t5k

7

in ?ij+1

Because u < uk < u in 12j+1, for all k > j + 1, we deduce that (uk)k>j+1 is uniformly bounded in Qj+1. On the other hand, by the interior gradient estimate 1123, Theorem 3.1, p. 266], there exists C1 > 0 independent of k such that Max IVuk(x)I < C1 max uk(x) < C1IIUIIoo. xEQ2

xES3j},

Hence, the sequence (Vuk)k>j±1 is uniformly bounded in Q2. Therefore, the sequence ('k)k>j+1 is bounded in Q2. Next, by the interior L" estimate [95, Theorem 9.11, p. 2351, we conclude that for any p > 1 there exists C2 > 1) that does not depend on k such that IIukIIW2., (Q,) < C2(II0kIILr(Q2) + IIukIILr(Q2)) < C2[Q2I1/P(max xEQ2

I0k(x)I + Max Iuk(x)I). xEQ2

Thus, the sequence (uk)k>j+1 is bounded in W2'P(Q1). Letting p = N/(1 -y), it follows by Sobolev-Morrey's inequality that the sequence (Vuk)k>j+1 is bounded in C1(Q1). Finally, by Holder interior estimates [95, Theorem 6.2, p. 90], there exists a constant C3 > 0 independent of k such that IIukIIC2.-,(-aj) <- C3(II0kIIC,(Q,) + max Iuk(x)I). xE Q,

This proves the inequality (1.12). By Lemma 1.2.4, the sequence (uk)k>1 is bounded in C2.7(lzj), for all j > 1.

Because the embedding C2'-'(Stj) - C2(S2j) is compact, there exists a subsequence of (uk)k>j that converges in C2(SZj). This yields u E C2(Qj), for all j > 1-that is, u E C2 (Q). Moreover, we have

-Auk = Vx, uk, Vuk)

in S2j, for all k > j + 1.

Thus, taking the limit of the subsequence converging in C2(-Qj), we conclude that

u satisfies -Au = 4)(x, u, Vu) in the whole Q. Furthermore, from u < u < u in S2 we obtain u(x) = 0, for all x0 E 0Q, so u E C(! ). It remains to prove u E C2°^'(52), which follows easily from the interior regularity theory of elliptic equations. This finishes the proof of Theorem 1.2.3. Consider the problem

-©u = F(x, u)

in (2,

u>0

in n,

u=0

on t9Q.

(1.13)

Assume that gi : 11 x (0, oo) -> (0, oo) is a Holder continuous function with exponent y (0 < -y < 1) on each compact subset of Sl x (0, oo) and it satisfies the following assumptions:

Basic methods

8

(A2) lim supt_ 0 ±( 't) < Al := A1(-A, 52) uniformly for x E 52. (A3) limt\o ±42!I = co uniformly for x E Q. Note that may be singular at the origin with respect to the second variable. The existence of a classical solution to (1.13) is given by Theorem 1.2.5.

Theorem 1.2.5 Assume 4) satisfies hypotheses (Al), (A2), and (A3). Then -problem (1.13) has at least one positive solution u E C2.7(52) fl C(S2).

Proof For any positive integer k, consider the approximated problem

-Au = 4i(x,u)

in 52,

u>0

in Q,

u=

on 852.

(1.14)

Let co be the normalized positive eigenfunction corresponding to the first eigenvalue al of the problem

in c,

Au = Au u=0

on 852.

(1.15)

Using (A3), there exists c > 0, which is small enough and ko > 1, which is large enough, such that Vk._

-CW1+

1

is a subsolution of (1.14) for all k > ko. Without loss of generality, we assume that ko = 1. To provide a supersolution, let us fix 0 < A < Al such that lim

t-oo Then there exists

{x, t)

t

< A < A,,

uniformly for x E Q.

E C2(S2) such that

Ae > . >0

in 52,

on 852.

We now choose M > 0 large enough such that S := MC is a supersolution of (1.14) for all k > 1 and vk < C in Q. There exists ul E C2(52) a solution of (1.14) in Q. Now, ul is a supersolution of (1.14) with k = 2 and v2 < U1 in Q. Hence, there exists u2 E C2 (E2) a solution of (1.14) (with k = 2) such that v2 < u2 < u1 in Q. Repeating the previous process, we obtain a sequence (uk)k>1 such that

with k = 1 such that v1 < ul <

(i) Uk is a solution of (1.14); (ii) 0 < cvl < Vk+1 < Uk+1 < uk in 52.

Comparison principles

9

We set u(x) := limk-,,, uk(x), for all x E S2. From (ii) we have

0
in R, for all k>1,

which yields it = 0 on BSl. Standard elliptic regularity arguments imply that C2"-y (9) n C(1l) is a solution of (1.13).

uE

11

Comparison principles

1.3

Weak and strong maximum principle Consider the linear differential operator 1.3.1

Cu:=

- > aij (x) 1
82u + exiaxj

bi(x) i_1N

au + c(x)u, axi

it E C2(Il) n C(Si)

where E2 is a bounded domain in RN, N > 1, and aij, bi, c E C(SZ), for all

1
Definition 1.3.1 We say that G is uniformly elliptic if there exists cl, c2 > 0 such that c1 1

aij(x)Sij < c2 1

12 <

12,

(1.16)

1
for all 6 = ('ij)1
RN.

Theorem 1.3.2 (Weak maximum principle) Assume that c > 0 in 0 and let u E C2(Q) n C(S2) be such that

r£u>0 St

u>0

in 0, on 852.

Thenu>0 in 92. In particular, the weak maximum principle states that if it E C2(1) n C(S2)

is a superharmonic function (that is, -Au > 0 in 0) such that it > 0 on aSt, then it > 0 in Q. Accordingly, if it (=- C2(S2) n C(1) is a subharmonic function

(that is, -Du < 0 in Q) such that it < 0 on 69, then it < 0 in P. We point out that, quoting J. L. Doob [69], "F. Riesz inaugurated the systematic study of superharmonic and subharmonic functions" (see Riesz [177]). A refined result that provides better estimates in a more general case is the following theorem.

Theorem 1.3.3 Assume that c > 0 in 1, f c C(fl), and let it E C2(fl) n C(S2) be such that Cu > f in S2. Then there exists C > 0 depending only on diam!l and the coefficients of G such that

supu<supuf+CIIfI00. n

an

Basic methods

10

The strong maximum principle is mainly needed to exclude the existence of nontrivial interior maximum points. The domain St is required to be smooth in the sense of the definition stated next. Definition 1.3.4 We say that a domain SZ satisfies the interior sphere condition at a point xo E 8S2 if there exists an open ball B C SZ such that B n St = {xo}. Accordingly, if B C R' \ S2, then we obtain the exterior sphere condition.

Theorem 1.3.5 (Strong maximum principle) Assume that Sl is a bounded domain satisfying the interior sphere condition and let G be an elliptic operator as noted earlier with c > 0. Let u E C2(Q) n C(ci) be such that

5 Lu>0

in S2,

u>0

on &Q.

Then the following properties hold true: (1) If u achieves its minimum in 9, then u as constant.

(ii) If xo E 8Q is such that u > u(xo) in 0 and 11 satisfies the interior sphere condition at xo, then the outer normal derivative of u at xo, if it exists, satisfies an(xo) < 0. We refer to the monograph by Gilbarg and Trudinger [95] for the proofs of the results' contained in this section. 1.3.2

Maximum principle for weakly differentiable functions

Let

8

Lu := 1
au ! + 8x- {\ ak, axi// a_11V

bi(x) "u + c(x)u, i9xi

u E H'(i)

(1.17)

be a linear uniformly elliptic operator (in the sense that the matrix (ate) satisfies condition (1.16) in Il) in divergence form. We assume for simplicity that ai by, c E C(S2). Let a : H'(I2) x H1(c) -+ IR be the associated bilinear form of L, defined by

a(u,v)

f1 a=i axz axe +

f(N1b1 axi

+ Cu

dx,

for all u, v E H' (Q).

Definition 1.3.6 We say that u E H1(0) satisfies Lu > 0 in SZ if a(u, v) > 0 for all v c Co (0) with v > 0. The following is an analogous result to Theorem 1.3.2 for weakly differentiable functions.

it

Comparison principles

Theorem 1.3.7 Assume that c > 0 in SZ and let u E H1(1l) be such that

Cu>0

l u>0

in n, ona1Z.

Then u>0 in 0. Corollary 1.3.8 Let k > 0 and u E H1(11) be such that

-Au + ku > 0

in 12,

u>0

on *Q.

Then either u = 0 or there exists y > 0 such that u(x) > ydist (x, alt), for any

xE1t. Proof Assume that u 0 0. Then, there exists m > 0 and a ball B cc 12 with the property u > m in B. Let v E C2(St \ B) be a classical solution of

in9\B,

1-Av+kv=0 U=M

on aB, on a9.

u=0

By the strong maximum principle we find y > 0 such that v(x) > y dist (x, alt), for all x E 0 \ B. Also note that

-A(u - v) + k(u - v) > 0 u-v>0

in St \ B,

on a(c \ B).

By Theorem 1.3.7 we obtain u > v in St, hence the conclusion. 1.3.3 Stampacchia's maximum principle Another comparison result for weakly differentiable functions is Stampacchia's maximum principle. Definition 1.3.9 The bilinear form a associated with the differential operator C defined in (1.17) is called coercive if there exists a > 0 such that

a(u, u) > a hull',

)

for all u E H' (0).

Theorem 1.3.10 (Stampacchia's maximum principle) Assume that the bilinear form a associated with the linear operator L is coercive. Let u E H1(Q) be such that a(u, v) > 0 for all v E H1(11), v > 0. Then u > 0 in Q.

Proof Let u^ = max{-u,0}. Then u- E H'(Sl) and Vu = -Vu on the set where u < 0 and Vu- = 0 on the set where u > 0. Because Lu > 0, it follows

that a(u, u-) > 0. On the other hand, a(u, u ) = -a(u-,u-) and, from the coercivity of a, we deduce that

a11u 11H1(-) 0 in Q.

Basic methods

12

More generally, Stampacchia's maximum principle states that if the bilinear

form a is coercive and u, v E H' (Q) are such that Lu > 0 and Lv > 0 in i, then w = min{u, v} also verifies Lw > 0 in Q. We refer to the book of Stampacchia [183] for further details. 1.3.4

Vazquezs maximum principle

We recall that the standard version of the maximum principle asserts that if u E C2(1l) n C(Ti) is a superharmonic function such that u > 0 on aI, then the following alternative holds: Either u > 0 in 1 or u = 0 in Q. Stampacchia extended this result for linear perturbations of the Laplace operator and showed

that the same conclusion holds if u > 0 on asl and -Au + an > 0 in Q, provided -A + aI is coercive in H'(Sl), where a E LO°(Sl). A natural question is to determine whether a similar result holds true when the Laplace operator is affected by a suitable nonlinear perturbation. In [190], J. L. Vazquez established an important extension of the maximum principle for semilinear and quasilinear problems, in close relationship with the behavior of the nonlinearity around the origin.

Theorem 1.3.11 (Vazquez's maximum principle) Let Sl be a bounded domain in RN, N > 3, and let u E C2(c) n C(i) be such that u > 0 in 11 and Au < f (u)

in Sl,

(1.18)

where f : [0, oo) -+ R is a continuous and increasing function such that f (0) = 0 and f satisfies the integral condition dt

F(t)

Jo

=00 >

with F(t) :=

J0

t f (s)ds.

(1.19)

Then the following alternative holds: Either u > 0 in 52 or u =_ 0 in D.

Proof We start with the following auxiliary result. Lemma 1.3.12 Consider the Sturm-Liouville problem

f v"=Klv'+K2f(v)

j

v(0) = 0,

in (0,ri),

(1.20)

v(rl) = vl,

where K1i K2, rl, and vI are positive real numbers. If f (0) = 0 and f is increasing, then problem (1.20) has a unique solution. If, moreover, f satisfies condition (1.19), then the unique solution v of (1.20) satisfies v'(0) > 0 and 0 < v < vi in (0, r1)

Proof We first observe that v := 0 is a subsolution and v := C is a supersolution of (1.20), provided C is large enough. Thus, problem (1.20) has at least a

Comparison principles

13

solution. To argue that this solution is unique, let us assume that V1 and V2 are solutions of (1.20) and set v := V1 - V2. Then

f v" = Kl v' + K2 (f (Vi) - f(V2)) v(0) = v(ri) = 0.

in (0, r1)

(1.21)

We claim that V1 < V2 in (0, ri). Indeed, if not, there exists xo E (0, r1) such that v(xo) = supZE(o,ri) v(x) > 0. Hence, v'(xo) = 0 and v"(xo)
0 > v"(xo) = Klv'(xo) + K2 (f (Vi (xo)) - f(V2(xo))) > 0, which is a contradiction. Thus, V1 < V2 in (0,ri) and, after changing the roles of V1 and V2, we deduce that Vl = V2, which shows that problem (1.20) has a unique solution. Let us now assume that f satisfies condition (1.19) and let v be the unique solution of (1.20). Set

ro := sup{O < r < ri : v(r) = 01. Then 0 < ro < ri and v(ro) = 0. We show in what follows that ro = 0, which is enough to conclude the proof of the lemma. Arguing by contradiction, we assume

that ro > 0. Then v'(ro) = 0 and v' > 0 in (ro,ri). Because v cannot have local maxima in (ro, ri), it follows that the mapping v : [ro, ri] -+ 10, vl] is a bijection. Multiplying (1.20) by v' and integrating, we find

w' = 2K1w + 2K2(F(v))' ,

(1.22)

where w := vi2. After multiplication in (1.22) by e-2K1r and integration on [ro, r] we deduce that

e-2K1rw(r)

- e-2K1T0w(ro) = 2K2

rr e-2K1s

(F(v(s)))'ds.

Tp

Because w(ro) = 0, this relation yields e-2K1Tw(r) = 2K2

rr

e-2K1s

Jr,

(F(v(s)))'ds

< 2K2 f C 21,r0 (F(v(s)))'ds = 2K2 er02K1r,, (F(v(r)) - F(v(ro) )) . Hence,

v'(r) F(v(r))

<_

2I{2eK,(r1-r))(ri

- ro) < C for any 0 < r < r1.

(1.23)

Because v : [ro, ri] -} [0, vi] is a bijection, after integration in (1.23) we obtain

fo"i which contradicts our assumption (1.19).

C]

Basic methods

14

We are now in position to prove Vazquez's maximum principle. We first denote by v(r; K1, K2, rl, vI) the unique solution of problem (1.20).

Denote go :_ {x E SZ u(x) = 0}, where u > 0 solves the differential inequality (1.18). Let xl E 0 \ S20 be such that dist (xl, Sto) < dist (x1, 8Q). Assuming that 00 is a proper subset of St, set R := dist (xl, Slo) and fix x0 E :

go 11 B(xi, R). Then u(xo) = 0. Next, we apply Lemma 1.3.12 for rl = R/2, vi = inf{u(x) : Ix - xl I = R/21,

Kl = 2(N - 1)/R, and K2 = 1. Define u(x) := v(R - Ix - xi I : Kl, K2, rl, vi), where v is given by Lemma 1.3.12. Then w(R) = v(O) = 0 and %R/2) = v(R/2). Using the equation fulfilled by v we obtain

2(N-'1)v,=f(v) r

r

(1.24)

inw:={xES?:R/2 0, in W v,(x) < u(x) + E (1 + Ixlz)-1/2 (1.25) Assuming by contradiction that (1.25) is not true, there exists xM E w such that IxI2)-1/2 the mapping 11(x) - u(x) - E (1 + attains its maximum in xM and that this value is positive. Therefore 0 > A (u(x) - u(x) - E(1 +

Ixl2)-i/2)

jx=xHr

= Du(xM) - Au(xM) + E (N - 3)(1 + IxM I2)-3/2 + 3E (1 + IxM l2)-5/2 > D1(xM) - f(u(xM)) + 3E (1 + IXMl2)-5/2 . So, by relation (1.24), 0 > 3e(1 + IxM12)-5/2 > 0.

This contradiction shows that relation (1.25) is true. Passing, now, at the limit

as E -> 0 we conclude that it < u in w. Therefore u(xo) = u(zo) = 0, where xo E aw. Because u > 0 in St, we deduce that x0 is an interior minimum point of u in Q. Thus, Vu(xo) = 0 and hence au(xo)/can = 0. On the other hand, we compute the normal derivative of u at xo e 8w. We have Ou(x0) = lim

u(xo - t(x - xo))

t\O t u(xo - t(x - x0)) = lim v(tR) > lim = Rv'(0) > 0. t\,o t t\O t This contradiction shows that our assumption that go is a proper subset of S2 is false, hence either go = Q (and, in this case, u =- 0 in S2) or SZ0 = 0 (which corresponds to u > 0 in 1). This completes the proof of Vazquez's maximum

an

principle.

We refer to Vazquez [190] for a corresponding result in the quasilinear case.

Comparison principles

15

Pucci and Serrin's maximum principle The growth assumption (1.19) in the statement of Theorem 1.3.11 is a KellerOsserman-type condition around the origin, which establishes that, to guarantee the maximum principle for the nonlinear operator -Au + f (u), it suffices that f has at least a linear decay rate near the origin. A straightforward computation shows that all functions f (u) = up (where p > 1) satisfy relation (1.19). The particular case p = 1 corresponds to the linear setting, as described in Stampacchia's maximum principle. It has been argued by Pucci and Serrin [169-172] that the maximum principle may fail if assumption (1.19) is not fulfilled. Moreover, Pucci and Serrin have extended the maximum principle to nonhomogeneous differential operators. They have considered the inequality problem 1.3.5

f div{A(IVuI)Vu} < f(u)

u>0

in 1,

infl,

(1.26)

where Sl is a bounded in 1R^'. The main assumptions on A and f are the following:

(Al) A : (0, oo) - (0, oo) is a continuous function. (A2) The mapping -P(t) := tA(t) increases in (0, oo) and limt\o (D(t) = 0. (Fl) f : [0, oo) -+ [0, oo) is continuous and f (0) = 0. (F2) f is nondecreasing on some interval [0, d], 6 > 0. Some important particular cases are the following: (i) If A(t) = 1, we obtain the Laplace operator. (ii) If A(t) = tp-2 (p > 1), we obtain the p-Laplace operator. +t2)-1/2, (iii) If A(t) _ (1 we obtain the mean curvature operator. (iv) If A(t) _ (1 + t2)-p/2 (p < 1), we obtain the generalized mean curvature operator.

(v) If A(t) = tp-2 + tq-2 (1' < p < q), we obtain an important operator occurring in quantum physics (see, for example, [20]).

Definition 1.3.13 A function u E C' (Q) is a solution to the problem (1.26) if for any 0 E Ca (11), 0 > 0,

fn [A(]Vul)Vu Vq+ f(u)o] dx > 0. Definition 1.3.14 We say that the strong maximum principle holds for problem (1.26) if u is an arbitrary solution of (1.26) with u(xo) = 0 for some xo E f ,

then u=0 in Q. An important role in the statement of the maximum principle of Pucci and Serrin is played by the mapping

H(t) := t4>(t) -

J0

t P(s)ds.

Basic methods

16

Theorem 1.3.15 (Pucci-Serrin maximum principle) Assume conditions (Al), (A2), (Fl), and (F2) are fulfilled. Then the strong maximum principle is valid for the inequality problem (1.26) provided that either f (t) - 0 for t E [0, µ), p. > 0, or f (t) > O for t E (0,6) with f >O and

frl o

dt

H-1(F(t))

where F(t) := / t f (s)ds.

= oo,

(1.27)

o

Conversely, if f (t) > 0 fort E (0, S), then condition (1.27) is also necessary for the validity of the strong maximum principle. We observe that assumption (1.27) reduces to the Keller-Osserman condition (1.19) in the case of the Laplace operator. A related phenomenon concerns the existence of dead cores. More precisely, an elliptic equation or inequality is said to have a dead core solution u in some domain 9 C RN provided that there exists an open subset w strictly contained in 1 (called the dead core of u) such that

u=_0 inil

and

u>Oinft\w.

Such singular phenomena arise in chemical models, when the solution represents the density of a reactant. The case when the solution vanishes corresponds to some region (dead cone) where no reactant is present (see [18], [171]). The boundary of such subdomains is then called the free boundary. Two such situations are chemical reactions and nonlinear diffusion models in population dynamics. In these situations the dead core region is the subdomain where the

concentration of reactant is zero or the population dies. More details on free boundary problems can be found in the book by Diaz [66] in which existence and regularity of such problems is studied. By Theorem 1.3.15, problem (1.26) can have a dead core only if condition (1.27) fails-that is, if f > 0 in (0, oo) and 1

1

dt

H-1 (F(t))

< oo .

(1,28)

Actually, condition (1.28) is not only necessary, but also sufficient for the existence of dead core solutions. The following result is from Pucci and Serrin [171], (1721.

Theorem 1.3.16 Assume hypotheses (Al) and (A2) are fulfilled. Let f : R -r R be a continuous nondecreasing function such that f (0) = 0 and satisfies condition

(1.28). Suppose b(oo) = H(oo) = oo. Let u be a solution of the inequality problem (1.26), with 0 < u(x) < m on 91 for some constant m > 0. Then the following properties are valid:

(i) 0
Comparison principles

17

(ii) Assume that r°°

Ra :=

J

dt

H_1(F(t)/N)

< oo'

and let BR be a ball of radius R > Ra compactly contained in Q. Then u has a dead core in f for all m > 0. (iii) If B is any ball compactly contained in 9, then u - 0 in B provided that m > 0 is sufficiently small.

The previous remarks show that the equation Au = [uIp-lu allows a dead core only if 0 < p < 1. We will illustrate this phenomenon in one of the following sections of this chapter.

A comparison principle in the presence of singular nonlinearities The result in this section concerns problem (1.13). By the maximum principle we deduce that if t 6--i 4D (x, t) is decreasing at each x E St, then problem (1.13) has at most one solution. The following theorem is a comparison principle that holds in a more general setting. 1.3.6

Theorem 1.3.17 Let t : S2 x (0, oo) -> R be a continuous function such that (H1) the mapping (0, oo) 3 t'--i fi(x, t)/t is strictly decreasing at each x E S2. Assume that there exist v, w E C2 (1) fl C(Q) such that

(i) Ow+4(x,w) <0 <1 v+4)(x;v) in 9; (ii) v, w > 0 in S2 and v < w on ice; (iii) Ov E L1(52) or Aw E L1(St). Then v < w in Q. Proof We follow an idea of Brezis and Kamin [32] (see also [87], [182]).

Suppose by contradiction that v < w is not true in Q. Then, we can find

co,60>0andaball Bcc 0 such that

v-w>co inB, f JBvw f P(x'W

w)

_

4D(x V

v)

(1.29)

dx > 80,

{1.30}

Let us assume that Aw E L1(c2) and set M = max{1, IIOw[ILl(sl)}, e = min {1,

601

1 .

4

Consider 0 E C1(R) a nondecreasing function such that 0 _< 0 < 1, 0(t) = 0, if t < 1/2 and 0(t) = 1 for all t 1. Define

9E(t)=0(.), tER.

Basic methods

18

Because w > v on 852, we can find a smooth subdomain 52" cc SZ such that B C Q*

and v - w < 2 in 52 `S2".

Using hypotheses (i) and (ii) we deduce

fn -

(wdv - vLw)9e(v - w)dx >

J.

vw

\

(X, W)

w

- 'D (x'v) 9E(v - w)dx. / V

(1.31)

By relation (1.30), we have

.

vw

(,D(x, W) w

- 4(x,v)) 9e(v -w)dx I

v

r > I vw a

{x,v}`

\ = J vw [ a \

w

V

0f(v-w)dx

`J

(x' w) -'F(x, v)1 dx v

w

ll

> So.

To raise a contradiction, we need only to prove that the left-hand side in (1.31) is smaller than 6o. For this purpose, define t

OE(t) :_

s9E(s)ds,

t E lit

a

It is easy to see that OE(t) = 0, if t < 2

and 0 < Oe(t) < 2e, for all t E R.

(1.32)

Now, using Green's first formula, we evaluate the left side of (1.31):

Js

(wav - vOw)9e(v - w)dx

f

= Jaj wBE(v - w) av do (x) -

-

©v)B(v - w)dx f(Vw. .

wO (v - w)Ov V(v - w)dx z

+f

p

j

v8E(v - w) wdo(x) si

c9n

Ov)9e(v - w)dx + J vOE(v - w)©w V(v - w)dx

= Jst 9E(v - w)(vVw - wVv) - 0(v - w)dx. The previous relation can be rewritten as

Comparison principles

19

f(wIv - viw)O (v - w)dx = J w9E(v - w)V(w - v) - V (v - w)dz + J (v - w)O (v - w)Vw V(v - w)dx. sa

Because

J

w8e(v - w)V(w - v) - V(v - w)dx < 0, the last equality yields

L Therefore,

j(wLv - v©w)O (v - w)dx f Vw V((4(v - w))dx.

s

Again by Green's first formula, and by (1.32), we have r Q.

(WAv - vAw)8E(v - w)dx < f ®£(v - w) an do (x) n

n $ (v - w)Awdx

J

<-J ee(v-w)Awdx<2E J

Idwjdx

< 2EM < 2° Thus, we have obtained a contradiction. Hence v < w in Sl, which completes the

0

proof.

Theorem 1.3.17 states that if v and w are respectively sub- and supersolutions of problem (1.13) such that either Av E L1(cl) or Aw E L'(SZ), then we always have v < to in 0. As a consequence, problem (1.13) has at most one solution, u E C2(Il) n C(Sl), such that ©u E Ll(St). The proof of Theorem 1.3.17 was given in [87] (see also [182] for a particular case), following an idea that goes back to the method of Brezis and Kamin [32]. Theorems 1.2.5 and 1.3.17 are suitable for semilinear elliptic problems involving singular nonlinearities. In both these results, the nonlinearity 1 is defined only on (0, oo) with respect to its second variable, so 4P may be singular at the origin. Some simple examples of functions that fulfill the hypotheses in Theorems 1.2.5 and 1.3.17 are

(i) P(x, t) = t-a, a> 0; (ii)

t) = a(x)t-' +tp, where a E C^1 (-Q), a > 0 in SZ, 0 < p < 1, and a > 0.

Basic methods

20

1.4

Existence properties and related maximum principles

In this section we establish a general property that asserts the existence of a unique positive subharmonic function satisfying a nonlinear boundary value problem with nontrivial Dirichlet boundary value data. We illustrate this result with two maximum principles corresponding to nonlinear perturbations of the Laplace operator.

Theorem 1.4.1 Let D c RN be a bounded smooth domain. Assume 0 # p E Ca'"r(Il) is nonnegative, and f E C' [0, oo) is positive and nondecreasing on (0, oo) such that f (0) = 0. If 0 0 0 E C°,ry(O l) is nonnegative, then the boundary value problem in S2, Du = p(x) f (u)

u>0

in 1,

u=O

on 0`9

(1.33)

has a unique classical solution, which is positive in Q.

Proof We first observe that u = 0 is a subsolution of (1.33), whereas u = n is a supersolution of (1.33) if n is large enough. Hence problem (1.33) has at least one nonnegative solution u. Then, taking into account the regularity of p and f, a standard bootstrap argument based on Schauder and Holder regularity shows that u E C2(S2). To prove the uniqueness, let ul and u2 be two solutions of (1.33); we show

that ul > u2 in Q. Denote w := {x E SZ

U1(x) < u2(x)} and suppose that w # 0. Then the function v = ul - u2 satisfies :

Lv = p(x)(f(ul) - f(u2)) in w, and v = 0 on 8w. Because f is nondecreasing and p > 0, it follows that v is a superharmonic function in w that vanishes on ow. Thus, by the maximum principle, either v = 0 or v > 0 in w, which yields a contradiction. Thus ul < u2 in Q. Similarly we get u2 < u1 in 0, thus ul = u2. We state in what follows two proofs for the positivity of u. The first one relies essentially on Vazquez's maximum principle, whereas the second proof offers a direct approach. FIRST PROOF: Let w be the unique nonnegative classical solution of the problem

r Du = IIpII.f(w)

1 w=

in 0, on ag.

To conclude that u > 0 in 1 it is enough to show that u > w > 0 in Q. Because f E C1 [0, oo), we have 2

t\O F(t)

2t

t\o f (t)

2

f'(0) >

0,

(1.34)

which implies immediately that fo dt/ F(t) = oo. By Vazquez's maximum principle (Theorem 1.3.11) we conclude that w > 0 in Q.

Existence properties and related maximum principles

21

We now prove that u > w in Q. To this aim, fix e > 0. We claim that

w(x) < u(x) + e(1 +

1x12)-1/2

for any x E Q.

(1.35)

Assume the contrary. Because u = w on t9S2, we deduce that max{w(x) - u(x) - e(1 +

1x12)-1/2}

zEn

is achieved in Q. At this point we have 1x12)-1/2)

O >A (w(x) - u(x) - e(1 +

=11p11-f(w(x)) - p(x)f (u(x)) - c (1 + IxI2)-1/2 1x12)-3/2 ?p(x) (f (w(x)) - f (u(x))) + e(N - 3)(1 + +3-,(l

IX12)-5/2 > 0,

which is a contradiction. Because e > 0 is chosen arbitrarily, inequality (1.35) implies u > w in IL. SECOND PROOF: Because 0 is not identically zero, there exists x0 E S2 such that

u(xo) > 0. To conclude that u > 0 in IL, it is sufficient to prove that u > 0 in B(xo,T), where F = dist (xo, aQ). Without loss of generality we can assume xo = 0. By the continuity of u, there exists r E (0, f) such that u(x) > 0 for all x with Ixl < r. So, p := mini.=, u(x) > 0. We define

M :=

P+1

n := fp

dt

f(t)

_

rP+1

and E

dt

f(t) for 0 < s < p.

It remains to show that u > 0 in A(r,T), where A(r,T) :_ {x E RN : r < I xl < T}. For this purpose, we need the following auxiliary result.

Lemma 1.4.2 Fore > 0 small enough, the problem 1-AV = M

in A(r,T),

v =

aslxl = r,

V = v(e)

aslxl = T

(1.36)

has a unique solution, which is increasing in A(r,T). Proof By the maximum principle, problem (1.36) has a unique solution. More-

over, v is radially symmetric in A(r,f)-namely, v(x) = v(r), r = 1x1. The function v satisfies

r
Basic methods

22

Integrating this relation twice we find Cl

2-N

2-1V ?7(r) _ - M r2 N- r + C2, r < r < r, 2N where Cl and C2 are real constants. The boundary conditions v(r) = 77 and v(r) = v(e) imply

M 2 - r) 2)

Cl = v(e} - 1l + ZN (T

N-2

r2-N

- r2-N .

From (1.34) we deduce that v(e) - oo as a --> 0. Thus, taking e > 0 sufficiently small, Cl becomes large enough to ensure that v'(r) > 0 for all r E (r, r). This concludes the proof. Set E > 0 sufficiently small such that the conclusion of Lemma 1.4.2 holds. Let u be the function defined implicitly as follows: a+i

dt = v(x) u(x)+e f (t)

for all x E A(r, r).

(1.37)

It is easy to check that Du > M f (u + E) > P(X)f(u)

in A(r, r),

u(x) = p - e < u(x)

as JxI =

u(x) = 0 < u(x)

as ixj =

Using the maximum principle, as in the proof of (1.35), we deduce that u < u in A(r, r). By (1.37) and Lemma 1.4.2 we deduce that u decreases in A(r, F). Thus, u > 0 in A(r, r). This completes the proof. Under the same assumptions on p and f as in the statement of Theorem 1.4.1 we have the following corollary.

Corollary 1.4.3 (Strong maximum principle) Let S C RN be a domain. If u is a nonnegative classical solution of the equation Au = p(x) f (u) in 0, then the following alternative holds: Either u - 0 in l or u is positive in Q.

Proof Assume that u # 0 in ft. Hence, there exists xo E Il such that u(xo) > 0.

We claim that u > 0 in Q. To this aim, let us fix xl E St and let w cc Q be a smooth bounded domain such that xl E w and xo E oaw. By Theorem 1.4.1, there exists a unique v E C2(SZ) such that Av = po f (v)

in w,

v>0

in W,

v=u$0

onaw,

(1.38)

where po := 1 + sups, p > 0. Because f is nondecreasing, we easily deduce u > v > 0 in w. Thus, u(xl) > 0, which yields u > 0 in ft

Existence properties and related maximum principles

23

Corollary 1.4.4 Let 0 C RN be a smooth bounded domain. If U E C2(9)nC(Si) satisfies

©u + au = p(x) f (u)

in Q, in fl, onaS2,

u>0 u0- 0 then u > 0 in Q.

Proof Let 0 E C°,"(aSZ) be a nonnegative function such that q 0 and q5 < u on act. By Theorem 1.4.1, there exists a unique u E C2(0) fl C(S2) such that f dv = JAIv + Ilpl1oo f (v)

2v=O

in SZ,

inc,

v>0

(1.39)

on 09.

Moreover, v > 0 in Q. Note that u is a supersolution for (1.39) and a standard D argument shows that u > v > 0 in Q. Dead core solutions of sublinear logistic equations The positiveness of the solution in Theorem 1.4.1 follows essentially by the assumption f E C'[0, oo). We show in what follows that if f is not differentiable at the origin, then problem (1.33) has a unique solution that is not necessarily positive in Q. However, in this case, the positiveness of the solution may depend on c and on the geometry of S2. Indeed, let us consider the problem 1.4.1

r AU =

in 11,

u>0

in!Q,

u=c

on ad,

(1.40)

where c > 0 is a constant. The existence of a solution follows after observing that u = 0 and u = c are respectively sub- and supersolutions for our problem, whereas the uniqueness follows from Theorem 1.3.17. The following example illustrates that in certain situations, the unique solution of problem (1.40) may vanish.

Example 1.1 Let Q = B(0,1) C R', w(x) = alx14 with c < a < (4N + 8) -2. We have

r Ow = (4N + 8)alxI2 < vjx12 = v Sl

w=a>c

in d, on alt.

This means that w is a supersolution of (1.40). Hence, problem (1.40) has a

-

solution u such that 0 <- u < w in Q. Because w(0) = 0 then, necessarily,

u(0) = 0. The next example shows that in some cases, depending on c and on diam Q, the unique solution of (1.40) is positive.

Basic methods

24

Example 1.2 Suppose that St can be included in a ball B(xo, R) with R < R, := 2 ° c(N + 2)2. Define w(x) := aix - x014, where a is chosen so that c

R4 > a >

1

(4N + 8)2

Then w satisfies

1'Lw=(4N+8)alx-xoI2> Sl

in St,

/ix-xoI2=V,,W-

w=aIx-xoI4
on81,

which shows that w is a subsolution of (1.40). We conclude that u(x) > w(x) > 0, for any x E St`{xo}. If diam 9 < 2R < 2R,, then there exists two points, xo and x1, such that 11 can be included in each of the balls B(xo, R) and B(xi, R). Using the previous conclusion we have u(x) > amax{Ix - x014, Ix - x114} > a

xl - xe

4

2

>0.

Choosing a = c/R4, Ixl - xo I = 2R - diam S2, and R = R, we find

(2R- diam U)4 =

di2mRS2)4

R4

>0

in 0.

Hence, u is positive solution of (1.40). Singular solutions of the logistic equation We now illustrate how Theorem 1.4.1 can be applied in the qualitative analysis of a class of singular solutions for the logistic equation. We are first concerned with the nonlinear problem 1.4.2

I Au = up

u>0 U=00

in B1, in B1, on 8B1,

(1.41)

where p > 1 and B1 C RN is the unit ball centered at the origin. The notation u = oo on 8B1 signifies that limy,/l u(x) = oo. A function u E C2(B1) satisfying (1.41) is called blow-up boundary (explosive) solution of problem (1.41).

Theorem 1.4.5 Assume that p > 1. Then problem (1.41) has a unique solution u. Moreover, u is infinitely differentiable in B1, u is radially symmetric, and

- r 2(p + 1 */'1

(p - 1)2

1/(p-ll

Existence properties and related maximum principles

25

Proof Using Theorem 1.4.1 we deduce that for any positive integer n, there exists a unique function u,z E C2(B1) such that

f Dun = un un > 0

in B1, in B1,

un=n

(1.42)

onrB1.

By uniqueness we deduce that un has radial symmetry. Standard elliptic regularity implies that un E C°O(B1). Next, by the maximum principle, un < U"+1 in B1, for any n > 1. We now observe that u(x) := C(1 - r2)-2/(p-1) is a supersolution of problem (1.42), provided C is large enough. This shows that for any n > 1, un < u, hence (un) is locally bounded in B1. Thus, for any x E B1, there exists u(x) := llmn un(x) E R, which is a solution of (1.42). Schauder and Holder regularity arguments imply that u E C°O(B1). To deduce the blow-up rate of u near the boundary, we observe that the differential equation fulfilled by the solution is

N-1 u, = up

in (0, 1).

(rN-lu') = rN-lup

in (0, 1).

Uit + Therefore

r

(1.43)

After multiplication by rN-lu' in (1.43) and integration, we find p+1

ui2(r) = 2

P+1()

2N

-2

+'

t2N-1,up+1(t)dt

(p + 1)r2 N Jo

I

(1.44)

At this stage, it is enough to argue that

B(r) = o(A(r))

`

as r /1,

where

B(r) := r-2N f rt2N-1up+1(t)dt.

A(r) = uP+i (r) A(r)

Jo

Indeed, relations (1.44) and (1.45) yield

1-

p+ l u'(r) 2uP+1(r)

= 1 - 1 - B(r)

A(r) )

1/2

ti B(r) = o(1) as r / 1. 2A(r) -

Integrating this relation between r and 1 we deduce that (1-r)-

roo Ju(r)

p+ldt=o(1) 2tP+1

which implies the desired asymptotic estimate.

asr/1,

(1.45)

Basic methods

26

To prove (1.45), let us fix E E (0, 1/2). Thus, for any r E (1 - e, 1),

B(r) Ar

1

r

r

( ts

2N-1

,u t

P+1

u(r)

dt

(!q,,) p+1 +

= J 1_E +1( ) <2uuP+1(r)E +E.

r Jarr

\ s / 2N-1

U(r)

dt

Letting first r / 1 and then E \ 0, we obtain (1.45). In fact, all solutions of (1.41) have the same blow-up rate near the boundary, as described by (1.45). This implies, by the maximum principle, that problem (1.41) has a unique solution. 11 We refer to [71] for details and many related results.

With similar arguments as in the proof of Theorem 1.4.5, we can show that the nonlinear problem

AU=up

if 1 < jxj < 2,

if1
u=0

iflxj =2

has a unique solution, which is of class CO° and which has the same blow-up rate near jxi = 1, as in the statement of Theorem 1.4.5. We leave it to the reader to determine the details of the proof.

1.5 Brezis-Oswald theorem Consider the problem

1-Au = f (x, u) u > 0, u 0 0 U=0

in 0, in St,

(1.46)

on an,

where 11 C W' is a smooth bounded domain and f : Sl x [0, oo) -4R satisfies (f 1) f is a Holder continuous function and the mapping (0, oo) E) t H L-12) is decreasing; (f 2) for all t > 0, x E--* f (x, t) belongs to LO° (d);

(f 3) there exists a positive constant c > 0 such that f (x, t) < c(1 + t) for all (x, t) E 9 x [0, oo).

By a solution of problem (1.46), we mean a function it E C2(d) that satisfies (1.46). In view of (f 1), for all x E 9 there exist ao(x) := lim f M t\0 t

At). and a,,(x) := lim t-.oo t

Brezis-Oswald theorem

27

We have

-oo < ao(x) < oo

and

- oo < a... (x) < oo

in Q.

It follows that any solution u E Ho (0) n L°°(1l) of (1.46) verifies

-f (x, Ilull.) < f (x, u(x)) < c(Ilull,, + 1) in fl.

This yields f(x,u) E L°°(1l). Therefore, any solution u of (1.46) belongs to W2'p(St) for every 1 < p < oo. Denote by A0 and A,, the first eigenvalue of the operators -A - ao(x) and -0 - a00(x) respectively, with a zero Dirichlet boundary condition. Because ao and a.. ) may be ±oo, the precise meaning of this first eigenvalue must be understood in the following sense: For any measurable function a(x), we define

al(-0 - a(x)) :=

inf

fin IVnI2dx - f u 9&0} a(x)udx I . in Sl and au- < 0 on

Indeed, from (f 1) we have

f(x,u) > f(x,llull.) u Ilull. Hence f (x, u) > -Mu in SZ for some positive constant M. Thus, u satisfies -Au + Mu > 0 in fl. By the maximum principle (Theorem 1.3.5) we obtain the conclusion. Brezis and Oswald [37] established the following general result.

Theorem 1.5.2 Assume that f satisfies (f 1) - (f3). Then, problem (1.46) has solutions if and only if ao < 0 < a00. Moreover, in this case problem (1.46) has a unique solution.

Proof We divide the proof of Theorem 1.5.2 into three steps. Step 1: Necessary condition. Let u E HH(Q) n L0(Q) be a solution of (1.46). Multiplying by u in (1.46) and then integrating by parts we obtain J IVul2dx = j f(x,u)udx < J ao(x)udx.

s

n

This implies that A0 < 0. Set

b(x):= f(x, lull. + 1) E L°O S) Hull. + 1

Basic methods

28

and let (µ, ,O) be the first eigenvalue and eigenfunction of the linear operator

-A - b(x). That is, in 0,

1-AO - b(x)O = AO

0 >0

in Q,

1i = 0

on aa.

Because ac,, < b in 0, it follows that p < ate. Multiplying by

in (1.46) and

integrating over Il we find

f(b(x) + u)'(iudx = l

J sl

f (x, u)dx.

Note that f (x, u) > b(x)u by virtue of (f 1), which also yields u fn uV;dx > 0. Thus, ) > it > 0. Step 2: Sufficient condition. Consider the energy functional E : HO'(Sl) - (-oo,oo], defined by

E(u) = 2 jIVuI2ds_jF(x,u)dx

for all u E

1

where F(x, t) = ff f (x, s) ds and f (x, s) is extended to be f (x, 0) for s < 0. Remark that E E C' because F(x, t.) < c(t2 + Iti)

for all (x, t) E St x IR,

(1.47)

by virtue of (f3). In the statement of the next auxiliary result, we need the notion of lower semicontinuity. We recall that if S is a topological space, then a functional E : S -> R U {+oo} is lower semi continuous if and only if for any a E R, the sublevel set Sa := {u E S : E(u) _< a} is closed. If S is a metric space, this condition is

equivalent to E(u) < lim infra-a, E(un) whenever un - u. Lemma 1.5.3 The following properties hold true: (i) E(u) - oo as l1u11rii -+ oo. (ii) E is lower semi continuous. (iii) E(w) < 0 for some w E Ho (1).

Proof (i) Assume by contradiction that there exists (un)n>1 C HH(Sl) such that IIun11H,, -F oo and E(un) < C. From the definition of E and (1.47) we obtain 2

J

IVun12dx

< M j(un + 1)dx.

(1.48)

Brezis-Oswald theorem

29

Set to = 1Iun112 and vn = un/tn. From our assumption and (1.48) we have to -> oo,

and (vn)n>1 bounded in Ho(1l).

11Vn112 = 1

Therefore, one can assume that

vn-v vn -> v

weakly in Ha(Q)as n->oo, strongly in L2(Q) as n - oo,

vn - v

almost everywhere in Sl and 11v112 =1.

We claim that F(x, tnyn)

lim sup / n- oo

t2 S2

ao0u2dx .

dx < 1 2

to

(1.49)

{v>0}

Indeed, we have

F(x, tnvn)dx =

J J

F(x, tnvn )dx (1.50)

F(x,tnvn)dx. {v
Using (f 3) we can evaluate the first integral in the right-hand side of (1.50) as

f

F(x, tnvn )dx < o

f

[(tnT1n )2 + 1] dx,

{v
v
which leads to

F(x, tnyn)

f

to

v
dx < o(1)

asn -+oo.

(1.51)

In the same manner we have

J

F(x,tnvn)dx < c J tnivnldx

and so

F(x, tnvn)

f

t2n

dx < 0(1)

as n -> oo.

(1.52)

To evaluate the first integral of the right-hand side in (1.50) we first remark that

't) <

2

2

in 52,

which implies limo ri

F(x t2nv") < aoo(x)v n

2

on the set {v > 0}.

(1.53)

Basic methods

30

By (1.47) we also have

F(x tnnyn)

< C {(vn )2

in 9.

+ to ]

Because vn, --f v in L2(ul), by Fatou's lemma we deduce

r

F(x, 2,v"+)

lim sup J

noo

{u>0}

dx <

1

f

v>0}

o

a v2dx.

(1.54)

Combining (1.50) through (1.53) we obtain the claim. Furthermore, from E(un) C we deduce that

2 JS2 f IVunl2dx
2

s

J{v>0}

On the other hand, by the definition of a,, we have

J Ivv+I2 -

v2dx > A0IIv+IIz.

(1.56)

v>0}

St

From (1.55) and (1.56) we deduce v = 0, which is a contradiction because IIvII2 = 1. This proves W. (ii) The proof follows by Fatou's lemma and relation (1.47), which establishes

that F has a quadratic growth. (iii) Because ao < 0, there exists w E Ha (S2) such that

IOwI2dx-J

in

{v,o}

aowdx<0.

Moreover, replacing w with w+ and then truncating w+, we may assume that w E L°°(i) and w > 0 in Q. Notice that lim

F(x, t) > I

t-'o Hence lim

F(x, ew)

t2

2

ao

> 1ao(x)w2(x)

(x)

in Q.

on the set {w # Q.

Let co > 0 be such that f (x, s) > -cos for all 0 < s < IIwII(,, This yields F(x, ew) E22

>-cowz >-co

in 0.

Brezis-Oswald theorem

31

By Fatou's lemma we deduce iim inf £-i0

IWOO)

F(x 2 w) dx > I a°w2dx. e 2 (,wool

Therefore lim i°nf

f F(, w) dx >_

fn

2

f

a°w2dx. w ho}

The last inequality combined with the fact that a° < 0 shows that 1 2

J

IvwI -

in

F(x'2 w) dx < 0,

for e > 0 that is small enough. Hence v := ew satisfies the claim of (iii). We now come back to the proof of Theorem 1.5.2.

By Lemma 1.5.3 we see that inf {E(u) : u E H(1)} is achieved by some u E Ho (1l). We can assume that u _> 0 in Il; otherwise, we replace u with u+

and use the fact that F(x, u) < F(x, u+) because F(x, t) = f (x, 0)t < 0 for all t < 0. This implies that u E Ho (S2) is a weak solution of (1.46). Standard regularity arguments imply that u is actually a C2(S2) solution of (1.46). Step 3: Uniqueness. Let ul and u2 be two solutions of (1.46). Remark that

au2 U2

- AU1

= f (ul)

ul

- f (u2)

in Q.

u2

ul

Multiplying by ui - u2 in the previous relation and then integrating over St we find AU2

Dul

u2

ul

S2

2 2 (u1 - u2)dx

= fn

f (ul) ui

From Remark 1.5.1 we obtain that u1/u2, u2/ul formula, the left-hand side reads

r 10u2 u2

- Qul ul

f (U2)

2

2

(ul - u2)dx.

{1.57)

U2

LO°(9), and by Green's first

r r I Vul - ul Du2 + Vul - ul u2 2 U2 u2 2

(ul - u2)dx = J

ff S2

[

\`

dx.

I

Thus, the left-hand side in (1.57) is nonnegative. This implies f (U2) 1 ul `f (ul) -

u2

(ui -

u2)dx = 0,

which yields ul = u2. Hence, problem (1.46) has a unique solution. This concludes the proof. Remark 1.5.4 In [37], the existence of a solution it E Ho (S2) nL°°(S0) of problem (1.46) is obtained under weaker assumptions on f, -namely,

Basic methods

32

(i) for all t > 0, x'-- f f (x, t) belongs to LOO(Q); (ii) for almost everywhere x E Il the mapping [0, oo) i) t ' ---> f (x, t) is continuous;

(iii) for each b > 0 there exists a constant cb > 0 such that f (x, t) > -cat for all 0 < t < b and almost everywhere x E 11.

If f

satisfies

ao (x) := lim inf t\O

LLX,

t

t) < 0 < a... (x) := lim sup f (t) t--,oo

t

in Q,

then (1.46) has at least one solution in u E Ha (1) 11 LOO(Q)

Remark 1.5.5 If, instead of condition (f 1), we assume (f 1)' the mapping (0, oo) E) t --i LUtt is nonincreasing,

then the uniqueness does not necessarily hold. In this case we only obtain

Dul _ Vul ul

ul

and f (x, ui)

- f(x, u2)

ul

in S2.

u2

In particular, this implies that the quotient ul/u2 is constant. 1.6 Comments and historical notes The premises of the sub- and supersolution method can be traced back to E. Picard. He applied, in the early 1880s, the method of successive approximations to argue the existence of solutions for nonlinear elliptic equations that are suitable perturbations of uniquely solvable linear problems. This is the starting point of the use of sub- and supersolutions in connection with monotone methods. Picard's techniques were applied later by Poincare [166] in connection with problems arising in astrophysics. In particular, Poincare established that the nonlinear equation Du = e" has at least one solution. Motivated by this equation, which arises in the theory of Fuchsian functions as well as by many nonlinear problems describing various phenomena in mathematical physics, Poincare formulated and developed the continuity method, which consists of solving nonlinear equations by embedding them in a one-parameter family of problems. In a first step, this tool consists of solving a simple problem and then extending the solvability by a step-by-step change in the parameter. Because of the major contributions of Lyapunov (1906), Schmidt (1908), and Leray and Schauder (1934), this method became a powerful tool in the bifurcation theory. A major breakthrough in the understanding of monotone methods in connection with nonlinear differential equations is the result of the work by Scorza Dragoni [179], who considered the solvability of Sturm-Liouville problems of the type y"(x) + f (x, y(x), y'(x)) = 0

x(a) = A, x(b) = B,

in (a, b),

(1.58)

Comments and historical notes

33

by using comparison principles. The concepts of sub- and supersolution were introduced by Nagumo [146] in 1937 who proved, using the shooting method, the existence of at least one solution for problem (1.58). A direct consequence of the mean value formula for harmonic functions u : 0 -> Illi (where S2 is a domain in IRN) is that u cannot achieve its maximum value at an interior point of S2, unless it is constant. This result was extended by Picard and Lichtenstein to general linear uniformly elliptic operators. A pioneering contribution is a result of the work by Hopf [108], who proved that if u e C2(Sl) fl C(SZ) is a solution of the linear problem

Au=f(x) > 0 u = g(x)

in S2,

on asp,

(1.59)

and if u attains a nonnegative minimum in an interior point of S2, then u is constant. In particular, if g > 0 on On, then u >_ 0 in Q. This is the first form of the maximum principle for superharmonic functions. An important consequence of this result is that problem (1.59) has at most one solution. Another property related to the maximum principle is a result of the work by Perron [159], who established the existence of solutions to the linear Dirichlet problem (1.59) if

f =0. An immediate consequence of the maximum principle is that any superharmonic function on some domain Sl attains its minimum value on asI. An additional qualitative result states that if u E C2(92)nC1(ac) is superharmonic in i and xo E au such that u(xo) = infXEo u(x), then au(ro)/an < 0, provided On has the interior sphere condition in xo. According to Stampacchia's maximum principle, the same conclusion holds if u is a solution of the problem

-Au>au inn, where A < .11 (a1 stands for the least eigenvalue of the Laplace operator (-A) in Ho (S2)). Clement and Peletier [54] (see also Hess [105]) studied the same problem in the case when A is larger than the first eigenvalue of the Laplace operator. Their result is stated in the following antimaximum principle.

Theorem 1.6.1 Let i C RN be a bounded domain with a smooth boundary and assume that f is a positive and continuous function on U. Then there ensts 8 = 5(f) > 0 such that if Al < A < Al + b and if u c C2(Q) n C1(aIl) is such that

-Au - Au = f inn, then u < 0 in S2 and an > 0 on MI. We refer to Brezis and Browder [31] for further historical comments on the role of the maximum principle in the qualitative analysis of nonlinear partial differential equations.

Basic methods

34

The Keller-Osserman condition around the origin (1.19) in the statement of Vazquez's maximum principle may be reformulated, equivalently, as

1 Jo

dt t f (t)

= 00,

(1.60)

In particular, this hypothesis asserts that the nonlinearity f must not be too large near the origin. Indeed, condition (1.60) is fulfilled provided f (t)/t = 0(1) as t \ 0. The linear case corresponds to Stampacchia's maximum principle, but Vazquez's result also covers cases when f is not necessarily bounded around the

origin, for instance if f (t)/ (t loge t) = 0(1) as t \ 0. We have already seen in this chapter that dead core solutions appear, provided that condition (1.60) is not fulfilled, for instance if f (t) = tP, with p E (0, 1). The existence result established in Theorem 1.5.2 corresponding to sublinear Dirichlet boundary value problems on bounded domains was extended to the whole space by Brezis and Kamin (32]. They considered the problem

-Au = V(x)u'

in 1RN, N > 3,

(1.61)

where 0 < p < 1 and V E L' (R N) is a nonnegative and nontrivial potential. Under these assumptions, Brezis and Kamin [32] proved that the nonlinear problem (1.61) has a bounded positive solution if and only if the linear problem

-Av = V(x) in RN has a bounded solution. Moreover, in such a case, problem (1.61) has a minimal positive solution. An interesting conjecture raised by Sattinger [178] concerns the existence of a solution for the problem

Du= f(x,u)

u=0

in1, on a)

in the presence of sub- and supersolutions that do not satisfy the natural relation

u < u in Q. In such a case, Cossez and Omari [97] proved the existence of a solution in the case when f (x, t)/t remains, within small perturbations, between the first two eigenvalues of the linearized operator. Habets and Omari [100] extended this work by providing a general nonresonance condition with respect to the second curve of the Fu ik spectrum. Most of the results developed in this chapter can be extended to quasilinear elliptic equations described by the p-Laplace operator Opu := div (IVu]P-'VU), where 1 < p < oo. If p = 2, then L\ becomes the linear Laplace operator, whereas OP is nonlinear and degenerates at the zeros of Vu, provided p # 2. That is why in such a case the solutions need not to be smooth, nor even C1, and the corresponding equations should be understood in a weak sense.

PART II BLOW-UP SOLUTIONS

2

BLOW-UP SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS Truth lies neither in the narrow frame of doctrines nor outside them. Attempt to find it must carry us beyond all boundaries and dogmatic limitations. Stelian Mihalaq, 1975

In this chapter we are concerned with a class of singular solutions for logistictype equations. We focus on positive solutions with blow-up boundary behavior and we formulate several sufficient conditions for the existence of such solutions. One of the main results we describe here establishes a necessary and sufficient condition for the existence of blow-up boundary solutions in the singular case of a potential that vanishes on the boundary. The exact blow-up rate of solutions near the boundary is essential to state uniqueness results. The approach we develop to study the asymptotic behavior of solutions is original and is based on Karamata's regular variation theory.

A key role in all arguments in this chapter is played by the Keller-Osserman condition, which describes the growth of the nonlinearity at infinity. Our asymp-

totic analysis is based on the observation that the nonlinearity f fulfills the Keller-Osserman condition, provided f has a positive Karamata regular variation. Several recent works illustrate the importance of the Karamata theory, which is a tool for the interplay between classical mathematical analysis and modern nonlinear analysis. 2.1

Introduction

Consider the equation

i u = f(u)

in Q,

(2.1)

where 1 C RN (N > 1) is a smooth domain and f E C' [0, oo) is a nonnegative function that satisfies (fl) f(O)=O, f'> 0 and f > O on (O, oo). Definition 2.1.1 A nonnegative solution u E C2(1l) of equation (2.1) is called a blow-up (large, explosive) solution if u(x) --+ oo as dist(x, nSl) - 0 in case Sl bounded and if u(x) -+ co as Ixl -> oo in case Sl is an exterior domain or 0 = RN If0 = RN, such a solution is called an entire blow-up solution.

Blow-up solutions for semilinear elliptic equations

38

In view of Corollary 1.4.4, if f satisfies (f1), then any blow-up solution of (2.1) is positive. The first contributions in this direction are the result of the work by Keller [118] and Osserman (154]. They supplied the following result for the existence of blow-up solutions to (2.1) in bounded domains.

Theorem 2.1.2 Assume that Il is bounded, f satisfies (f 1), and (f2)

dt < oo, where F(t) := t f (s) ds. VF (t) J0 Then, equation (2.1) has a positive blow-up solution. Proof We first need the following auxiliary result.

J

Lemma 2.1.3 Assume that f satisfies (f 1) and (f 2). Then (i)

limf -(t) = oo; t

t-.oo

ds

=0. F(s) - F(t) Proof (i) Let g(t) = F(t)-1/z, t > 1. Then (ii)

tllmO°

Jc

t

t

t g(t) _ / g(s)ds + 1

I

sg'(s)ds + g(1),

for all t >1.

Because g is positive and g' _< 0, both integrals in the right-hand side of the previous equality define monotone functions. Thus, there exists limt-,, tg(t). This means that limt-,,,, F(t) /t2 exists in IRU{oo}. By (f2) we finally deduce that

limt, F(t) /t2 = oo. Because F(t) < tf (t) for all t > 0, we find limt, f (t) It = 00.

(ii) Set G(t) := f °° ds/ F(s) - F(t), t > 1. Notice that G is well defined because, by virtue of (f2), we have ds

G(t) _

F(s -+t - 1) - F(t) <

00.

Using (f 1) and the mean value theorem we deduce that

Hence,

F(s) - F(t) > F(s - t) for ails>t>1, F(s) - F(t) > (s - t) f (t) for alls>t>1. for allt>Iwehave zt ds °O ds G(t)

- Jt

F{s) - F(t) + Jzt 2t 1 ds foo s-t + t f (t) t 1°D

2 F7_(tt_)

+

d-r

F(,r)

F(s) - F(t) ds

F(s - t)

Introduction

39

Now the assumption (f 2) together with (i) allows us to conclude.

0

By virtue of Theorem 1.4.1, for all n > 1 there exists a unique solution un E C2(S2) of the problem

Au = f (u)

in 12,

u>0 u=n

in S2,

on 8c.

By the weak maximum principle we also have un < un+1 in S2, for all n > 1. We claim that (un)n>1 is uniformly bounded in any compact subset of Q. For this purpose, we show that for any xo E 1 there exists an open set w C fI such that xo E w, and (un)n>1 is uniformly bounded on w.

Fix xo E 0 and let B CC 52 be an open ball centered at xo with a radius R > 0. Consider the problem in B, in B,

Av = f (V)

v>0 v=n

I

on BB.

According to Theorem 1.4.1, problem (2.2) has a unique positive solution vn E C2(-R). Moreover, because un is subharmonic, we obtain un < vn on 8B and, by the weak maximum principle, it follows that

un < vn

in B for all n > 1.

(2.3)

Because of the uniqueness of the solution to (2.2), vn is radially symmetric. Hence, vn(x) = vn(r), 0 < r = Ix -- xol < R and

vn(r) +

N-1 vn(r) = f (vn(r)) r

for all 0 < r < It,

for all 0 < r
vn(r) > 0 vn(R) = n.

( 2.4)

Integrating in (2.4), by continuity arguments it follows that r

vn(r) = r-N+1 j tN-i f(vn(t))dt

for all 0 < r < R.

o

Therefore vn is nondecreasing and V.1 (r) <

f (vV (r))

T

fo

tN-1dt = N f (vn(r))

for all 0 < r
(2.5)

Using (2.5) in (2.4) we obtain

< f (yn(r)) N - vn(r)

for all 0 < r < R.

(2.6)

Blow-up solutions for semilinear elliptic equations

40

We now multiply by vn(r) in (2.6) and then we integrate over [p, r]. We obtain

2 (F(vn(r)) - F(vn(p))) < v'n (r) for all p < r < R. This yields

N

- F(vn(r)) ( F(vn(p))

for all p:5 r < R.

By integrating over [p, R] in the previous inequality we deduce

2(R-p) < 2N

n

dt

,,(p)

F(t) - F(v.(p))

In)OW

F(t) - F(vn(p))

dt

Now Lemma 2.1.3 (ii) implies that the sequence

for all 0 < p < R. 1 is uniformly bounded.

Hence, (vn),,>1 is uniformly bounded in w := B(xo,p). By virtue of (2.3) it follows that the sequence (un),a>1 is uniformly bounded in w. Therefore, there

exists u(x) := u,,(x), for all x E S1. By standard elliptic arguments, u E C2(S2) is a classical solution of (2.1). Because u < un+1 < u in SZ and u = n on O, it follows that u blows up at the boundary. This finishes the proof.

2.2 Blow-up solution for elliptic equations with vanishing potential We are concerned in this section with the existence of positive blow-up solutions of

Du = p(x) f (u)

in S2,

(2.7)

where 1 C RN (N > 1) is either a smooth bounded domain or 0 = RN and f E C1 [0, no) satisfies (f 1) and (f 2). We also assume that p is a nonnegative function such that p E C°"-r(f) if 1 is bounded, and p E C°C (S2), otherwise. The main feature is that the potential may vanish at some points in S2. 2.2.1

Existence results in bounded domains

Lemma 2.2.1 Assume that conditions (f 1) and (f 2) are fulfilled. Then, 11dt

(') o (11)

oo,

7_(t) =

] T (t-)

< oo.

Proof (i) By virtue of (f 1) there exist c, b > 0 such that f (t) < ct for all 0 < t < d. This yields fa dt/ f (t) = no.

Blow-up solution for elliptic equations with vanishing potential

41

(ii) Fix R > 0 and let B be the open ball centered at the origin with a radius R. By Theorem 2.1.2, there exists a positive blow-up solution v of (2.1) with 0 = B. Using Theorem 1.4.1, there exists a unique u,,, E C2(S2) such that in B, in B, on 8B.

I Dun = f (U")

un,>0 un = n

Because f is nondecreasing, by the weak maximum principle it follows that

in B.

un < un+1 < v

This implies that (un)n>1 is uniformly bounded in every compact subdomain of

B. Hence, for all x E B there exists u(x) := limn-,,, un(x). By standard regularity arguments, u is a blow-up solution of (2.1). Notice that by the uniqueness of (2.8), un is radially symmetric in B, which implies that u is also radially symmetric (in this sense we could also apply a well-known result of Gidas, Ni, and Nirenberg j93]). Therefore, u(x) = u(r), 0 < r = jxf < R, and u satisfies in the r variable the equation u"(r) +

N - 1 u'(r) = f(u(r))

for all 0 < r < R.

r

Integrating in the previous equality we obtain r =rl-NJSN-1f(u(s))ds u'(r) for all 0
Similar to the proof in Theorem 2.1.2, we note that u is nondecreasing and

f Mr)) < u"(r)

for all 0 < r < R,

which yields

Thus,

for all0
f r / R we find 00

dt

<

R2

f(t) - 2N U(O)

This ends the proof.

0

42

Blow-up solutions for semilinear elliptic equations

Assume that p satisfies the assumption (p1) for every xo c Il with p(xo) = 0 there exists a domain 11o CC bI such that xo E Sto and p > 0 on 85ta. In this case we have the following result.

Theorem 2.2.2 Assume that S2 is bounded and conditions (f 1), (f2), and (pl) are satisfied. Then equation (2.7) has a positive blow-up solution.

Proof By Theorem 1.4.1, for any n > 1, the boundary value problem Du = p(x) f (u)

u>0

u=n

in Q, in Q, onOQ

has a unique solution un E C2(? ). Furthermore, by the weak maximum principle, the sequence (u,,),,>1 is nondecreasing. We claim that (u, )n>1 is uniformly bounded in every compact subset of Q. To this aim it is enough to show that for all xo E S2 there exists an open set 0 CC S2 containing xo such that (u,,)n>i is uniformly bounded in 0. Let xo E S2 be fixed. In our analysis, two cases may occur. CASE 1: p(xo) > 0. By the continuity of p, there exists a ball B = B(xo, r) CC SZ

such that m := min (p(x) : x E B} > 0. Let v be a positive blow-up solution of Av = m f (v) in B. By the maximum principle it follows that u,, < v in B. Furthermore, v is bounded in B(xo, r/2) so that it suffices to take 0 B(xo, r/2). CASE 2: p(xo) = 0. Assumption (p1) and the boundedness of 11 imply the existence of a domain 0 CC S2 that contains xo such that p > 0 on W. Case 1 shows that for any x E 80, there exist a ball B(x, rx) strictly contained in 52 and a constant Mz > 0 such that un < Mx in B(x, rx/2), for any n > 1. Because 80 is compact, it follows that it can be covered by a finite number of such balls, say B(xi, rx, /2), i = 1, ... , ko. Setting M := max {Mx, , ... , M ,, } we have un < M on 80, for any n > 1. Applying the maximum principle we obtain un < M in 0 and the claim follows. Therefore, for all x E dl there exists u(x) := limn. un(x). Standard elliptic regularity arguments show that u is a solution of problem (2.7). Because u,' < un+1 < u in S2, it follows that u blows up at the boundary. This concludes the

0

proof.

Existence results in the whole space In this subsection we prove the existence of a positive entire blow-up solution of equation (2.7). We require the following condition on p: 2.2.2

(pl)' There exists a sequence of smooth bounded domains (SZn)n>1 such that SZn C dn+1i RN = UO°_1SZn, and (pl) holds in 52,,, for any n > 1;

Blow-up solution for elliptic equations with vanishing potential

43

00

(p2) J rq (r) dr < oo, where q(r) := max {p(x) : [xj = r}. 0

Remark that (pl)' is automatically satisfied if p(x) > 0 for Ix[ sufficiently large. An example of function p that satisfies both (p2) and (pl)', with p vanishing in every neighborhood of infinity, is provided here:

p(r)=O for r= Ix[ E [n-1/3,n+1/3], n> 1; 00

p(r) > 0 in lR+ \ U [n - 1/3, n + 1/31; n=1

p c C1 [0, oo)

and

_

max

2

p(r) n2(2n + 1) Obviously (pl)' is fulfilled for Stn := B(0, n+ 1/2). On the other hand, condition n

1

(p2) is also satisfied because

I

n+l

00

00

rp(r) dr =

rp(r) dr

J

n=111n n+l

00

n=1 n

=E 00

1 n2

2r

n2(2n+1)

dr

<00.

n=1

In this case we have the following existence result.

Theorem 2.2.3 Let 1 = RN and assume that conditions (f1), (f2) and (pl)', (p2) are satisfied. Then equation (2.7) has a positive entire blow-up solution. Proof By Theorem 2.2.2, the boundary value problem f Du = P(--) f (u)

in Stn,

u>0

in S1n,

U=00

on 49p"

(2.10)

has a solution un E C2 (n). Because Stn cc 52n+1, we derive that un >_ U,+1 in SZn, for all n > I. Thus, for all x E R' we can define U(x) := limn_,«, un(x). By standard regularity arguments we find that U E C o, '(R') and AU = p(x) f (U) in RN. Let us now prove that U blows up at infinity. To this aim, it suffices to find a oo. positive function w E C(RN) such that U > w in RN and w(x) --+ oo as This will also imply that U is positive. Set rt

t1-N

I

J0

sN-1 (s) ds dr

for all r > 0.

Obviously, 4) E C2[0, oo). Furthermore, we have the following lemma.

Blow-up solutions for semilinear elliptic equations

44

Lemma 2.2.4 iim 4P(r) is finite if and only if ff rq (r)dr is finite. r-.oo

Proof For any r > 0 we have

N 1 2 (10" t-0(t) dt - rlv-Z f'tN-10(t)dt)

fi(r)

(2.11)

< N1

Jr t(t) dt.

2

f

On the other hand, J

fr

r

N-2

Jff

-

(rN_2 tN-2) to(t)dt r_2 tN_10(t)dt = rrN-2 - (r/2)N-2

fr/2

rN-2

>

The last inequality combined with (2.11) yields

1V 1 2

f'tO(t)dt > I(r) >

1-

(11/2)ZN-2 fr/2

to(t)dt.

The conclusion follows now by letting r -> oo in the previous estimates. This finishes the proof. By virtue of hypothesis (p2) and Lemma 2.2.4 it follows that L := 1imr- o (D(r) is finite. Then, z(r) := L - 0 is positive and verifies

Az=0(r)

z(x)0

r:=lxj >0,

(2.12)

asIxI-4oo.

According to Lemma 2.2.1, the mapping ( 0 , 0 0)

3t!

->

100 f(s)

E (0, oo)

is decreasing and bijective. We implicitely define w as for all x E RN.

(2.13)

w (z)

Note that by (2.12) we have w E C2(RN) and w we have

-Az(x) = Hence,

Aw(x) T(W)

oo as IxI -+ oo. Furthermore;

(w } IOw12 fZw

in I[ '

Blow-up solutions for logistic equations

Ow > p(x)f (w)

45

in RN.

We claim that w <_ un in Stn for all n >_ 1. Obviously this inequality is true on BSln. Using the same arguments as in the proof of inequality (1.35) in Theorem 1.4.1 (with Q replaced with Stn), we obtain that for any e > 0 and n > 1 we have

w(x) < un(x) +,_(I + Ix12)-112

in Stn.

Passing to the limit with e --p 0 we derive w < un in Q. Consequently, U > w in RN and, by (2.13), w(x) -' oo as IxI - oo. This completes the proof. 2.3 Blow-up solutions for logistic equations In this section we discuss the existence of blow-up solutions to the following semilinear elliptic equation: Au + Au = p(x) f (u)

in Q.

(2.14)

We assume that St C RN (N > 3) is a smooth bounded domain, A E R, and f E C' [0, oo) is a nonnegative function that satisfies (f 2) and

(f3) the mapping (0, oo) 3 t -->

ff (t) is increasing.

We also assume that p E C°'7(St) (0 < y < 1) is nonnegative and p $ 0 in Q. Notice that we allow p to vanish in St (or even on (9Sl).

Remark 2.3.1 If f satisfies (f2) and (f3), by Lemma 2.1.3 and l'Hospital's rule we deduce that limt.< f(t)/t = limt.. f'(t) = cc. Remark 2.3.2 Typical examples of nonlinearities satisfying (f 2) and (f 3) are

(i) f (t) = et - 1. (ii) f (t) = tp, p > 1. (iii) f (t) = t[In(t + 1)]P, p > 2. The case of positive potentials We discuss here the existence of blow-up solutions to (2.14) under the additional hypothesis p > 0 in U. More generally, we consider the problem 2.3.1

Au + a(x)u + b(x) = p(x) f (u) in St

(2.15)

where a, b E C°'7(St), b > 0. We first need the following result.

Proposition 2.3.3 Assume that p > 0 in SI and f satisfies (f2), (f3). Let 0 E C°''r(8St) be a nonnegative function such that 0 # 0. Then the boundary value problem

Au + a(x)u + b(x) = p(x) f (u)

u>0 u=4 has a unique solution u E C2 (U).

in St, in St,

on 8Q

(2.16)

46

Blow-up solutions for semilinear elliptic equations

Proof The uniqueness follows directly from Theorem 1.3.17. We will focus on the existence part. Applying Theorem 1.4.1, the boundary value problem in 52,

Au = ((a((.u + ((p((oof (u) U>0

in 52,

u=O

(2.17)

on On

has a unique solution u E C2(52). Clearly u is a subsolution of (2.17), To provide a supersolution of (2.17), let us remark that for all n > 1, the mapping

n(x, t) := a(x)t + b(x) - p(x) f (t) +

-,

(x, t) E 52 x (0, oo),

satisfies the hypotheses of Theorems 1.2.5 and 1.3.17. Hence, there exists a unique solution u E C2(SZ) of the problem

Au + a(x)u + b(x) = p(x) f (u) - n

in 52,

U>0

in 52,

u=0

on W.

By Theorem 1.3.17 we deduce 0 < u < un+1 < un in 52. If u(x) := limn,, un(x), x E 52, by standard elliptic regularity arguments we derive that u E C2(52) is a 0 solution of (2.16). This finishes the proof. Under the assumptions of Proposition 2.3.3, we obtain the following theorem. Theorem 2.3.4 There exists a positive blow-up solution of equation (2.15).

Proof Let 4D(t) := pof (t) - ((a((,,t - b, t > 0, where po = info p > 0 and b := supo b + 1 > 0. Let also r be the unique positive solution of the equation 4?(t) = 0. By Remark 2.3.1 we deduce that limt_,,, 4)(t)/ f (t) = po > 0. By (A2), we further derive that the mapping (0, oo) t ----> 4i(t + r) satisfies the assumptions of Theorem 2.1.2. Hence, there exists a positive blow-up solution of Av = 4b(v + r) in Q. Thus, U(x) := v(x) + r, x E 52, satisfies

AU+{(a((.U+b=pof(U)

in 52,

and blows up at the boundary of 52.

Let un be the unique solution of (2.16) with 0 - n. By Theorem 1.3.17, un < un+l < U in fl. It follows that u(x) := limn. un(x), x E 52, exists and defines a positive blow-up solution of (2.15). Moreover, every positive blow-up solution v of (2.15) satisfies un < v in 52, which yields u < v in fl. This means that u is a minimal blow-up solution of (2.15). This concludes the proof. The case of vanishing potentials We assume in this section that p vanishes in 52. Set 2.3.2

520 := int {x E 52 : p(x) = 01

and suppose that 520 C 52 and p > 0 in 52 \ 520.

(2.18)

Blow-up solutions for logistic equations

47

Let L be the unique self-adjoint operator associated with the quadratic form 0(u) = fo jouj2 dx on HD(S2o) := {u E Ho(S2) : u(x) = 0 for almost everywhere x E 9\ Sto}.

If aSto satisfies the exterior cone condition then, according to Alama and Tarantello [4), Hb(Sto) coincides with Ho (SZo) and L is the classical Laplace operator with the Dirichlet condition on %2o. Let A,,,1 be the first Dirichlet eigenvalue of G in S2o. Set f := limt\o f (t)/t. and denote by Ao,1 the first eigenvalue of the operator -A + fp(x) in Hp (S2). Consider the following nonlinear Dirichlet problem: Au + Au = p(x) f (u)

in 52,

u>0 u=0

in 0,

(2.19)

on as2.

Alama and Tarantello [41 established the following necessary and sufficient condition for the existence of solutions to problem (2.19).

Theorem 2.3.5 Assume f satisfies assumption (f 3). Then problem (2.19) has a solution if and only if A0,1 < A < A,,,,,1. Moreover, in this case the solution is unique.

A related interesting problem is to establish a necessary and sufficient condition for the existence of blow-up boundary solutions of equation (2.14). The following result asserts that such a solution exists for small values of the real parameter A, up to a certain range. However, if the variable potential is positive, then (2.14) has a blow-up boundary solution for any value of A. We observe that our assumptions on the potential p allow that it may vanish on 852. This is the most interesting case, because it corresponds to the "competition" between a vanishing potential and a blow-up nonlinearity.

Theorem 2.3.6 Assume that f satisfies conditions (f 2) and (f 3). Then problem (2.1.4) has a positive blow-up solution if and only if A E (-oo, A"C',1). Proof NECESSARY CONDITION. Let u be a blow-up solution of equation (2.14). As we have already argued, u is positive in Q. Suppose A,,,),1 is finite and let us assume by contradiction that A > A,,.,1. Fix A0,1 < ti < and denote by v a pos-

itive solution of problem (2.19) with A = µ. If M := max {maxi v/ mina u; 1}, then we have A(Mu) + A,,.,1(Mu) < p(x) f (Mu) in S2, in S2, Mu > v Mu = 00 on 852. Hence (v, Mu) is an ordered pair of sub- and supersolution of problem (2.19) with A = A,,,1. Thus, problem (2.19) with A = A.... 1 has at least one positive solution (between v and Mu), which is a contradiction. So, necessarily, A E (-oo, A.,1).

Blow-up solutions for semilinear elliptic equations

48

SUFFICIENT CONDITION. We first need a similar result to that in Proposition 2.3.3.

Proposition 2.3.7 Assume that p > 0 on 9Q and let ¢ E C°'7(& I) be a nonnegative function such that 0 0 0. Then the boundary value problem

IAu + Au = p(x) f (u)

in 9,

u > 0

in S2,

u=q

onl1l

(2.20)

has a solution if and only if) E (-oo, a(,,,,). Moreover, in this case, the solution is unique.

Proof The first part follows exactly in the same way as in the proof of the necessary condition in Theorem 2.3.6. For the sufficient condition, fix A < AQ,,1. By Theorem 1.4.1 there exists a unique classical solution u of the problem

f Du = jA u + IIpII.f (u) I

in 52,

u>0

in 0,

u=0

on 8Q.

(2.21)

Obviously u is a subsolution of (2.20). Let A.,, > A > max {A, Ao,1} and u, be the unique solution of (2.19) with A = A.. Let S2i (i = 1, 2) be two subdomains of 52 such that S2°CCS21CCS22CCS2

_

and S2 \ S21 is smooth.

Because p > 0 in SZ \ Stl, by Theorem 2.3.4 there exists U E C2(i \ fll), a positive solution of (2.14) in ci \ 521. We define v E C2(Q) as a positive function in SZ such that v = U in S2 \ 522

and u = u, in 521. Using Remark 2.3.1 and the fact that info,\n, p > 0, it is easy to check that u := Cv satisfies Du + Au < p(x) f (u) U > max q5

an

00

in S2,

in Sl,

(2.22)

on 8i,

provided C > 0 is large enough. It is clear that u is a positive supersolution of (2.20) and u < maxan 0 < u in Q. Therefore, by the sub- and supersolution method, problem (2.20) has at least a solution u E C2 M) such that u < u < u in Q.

An equivalent criterion to the Keller-Osserman condition

49

Now let -oo < A < A,,.1 and ul, u2 be two solutions of (2.20). Because p may vanish in Q we cannot apply Theorem 1.3.17 directly. However, we can adapt the arguments in the proof of Theorem 1.5.2 to obtain

f(oi)l

2

2

\f (u2) This yields ul = u2 on the set {x E 51 : p(x) > 0}-that is, p(x)f(uj) = u1

U2

p(x) f (u2) in Q. Furthermore, by the strong maximum principle we obtain ul =

u2 in ft In the same manner we obtain that problem (2.19) has at most one solution. The proof of Proposition 2.3.7 is now complete.

Proof of Theorem 2.3.6 completed. Fix A E (-oo,

In our further

analysis, two cases may occur: CASE 1: p > 0 on 851. Denote by uf, the unique solution of (2.20) with = n. For 0 = 1, let u and u be the solutions of problems (2.21) and (2.22), respectively. The sub- and supersolution method combined with the uniqueness of the solution

to (2.20) shows that u < un < U,, 1 < u in 11. Hence U(x) := lima-,, un(x), x E 11, exists and is a positive blow-up solution of (2.14). CASE 2: p vanishes on 851. According to Proposition 2.3.3, for all n > 1 there exists a unique vn E C2(1l) such that Avn + AYln =

((x)+)f(v)

in 51,

vn>0

in 51,

vn = n

on 051.

By Theorem 1.3.17, the sequence (vn)n>1 is nondecreasing. Moreover, (vn)n>1 is uniformly bounded on every compact subdomain of Q. Indeed, let K cc 1) be an arbitrary compact set and d := dist (K, 812) > 0. Choose S E (0, d) small enough such that 'L C Ca, where Ca = {x E SZ : dist (x, 812) > b}. Because p > 0 on 8C5, by the results in Case 1 there exists a positive blow-up solution V of (2.14) for 11 = Q. Using Theorem 1.3.17 in C5, we deduce vn < V in C5, for all n >_ 1. Hence, (vn)n>1 is uniformly bounded in K. By the monotonicity of (v,,),,>, we

conclude that up to a subsequence vn - v in C(K). Finally, standard elliptic regularity arguments show that v is a positive blow-up solution of (2.14). This concludes the proof.

2.4 An equivalent criterion to the Keller-Osserman condition Our aim in this section is to supply an equivalent criterion to the KellerOsserman condition (f 2). We assume that f satisfies the following assumption:

(f4) f fulfills (f 3) and there exists limi_,a, (.E)' (t) := a. Consider the set 9 of all functions g : (0, oo) -+ R satisfying the hypotheses

Blow-up solutions for semilinear elliptic equations

50

(i) there exists 6 = d(g) > 0 such that g E C2(0, S) and g" > 0 on (0, 5); (ii) limt\o g(t) = co; (iii) there exists limt\,o 9(t) g" M Remark first that g is nonempty. Indeed, let 9 E C2(0, oo) be a convex function such that limtNO 9(t) = oo. Because 9' is nondecreasing, it follows that

limt,,o 9'(t) = -oo. Thus, (9'(t))2 + 9"(t)

[9 (t)I

0

as t

o,

which proves that es E t

Remark 2.4.1 ego

g,({t)

=

Jiimo

g"(t}

0 for any function g E

Indeed, if g E G is chosen arbitrarily, then limt\,o g(t) = -oo. Hence, the mapping V(t) = In Ig'(t)) is decreasing in the neighborhood of the origin and by (ii) there exists limt,,o 0'(t). Because limt\p r/0 (t) = oo, it follows that lim ifi'(t) _ -oo. Hence, lim g"(t) = lim Ifi'(t) _ -00, t-"o t'-'O g'(t)

and then by ('Hospital's rule we find g(t) = lim ge(t) = 0. t\,o g'(t) t1-o g"(t) lim

Lemma 2.4.2 Assume that f fulfills (f4). Then, the following hold:

(i) a > 0; (ii) a < 1/2 provided that (f2) is fulfilled.

Proof (i) Suppose that a < 0. Then, there exists ti > 0 such that .f

r {t) < 2 < 0

for any t > ti.

Integrating this inequality over (ti, oo) we obtain a contradiction. It follows that

a>0.

(ii) Using the definition of a we find lim too

F(t) f'(t) = 1 f 2(t)

a.

By Remark 2.3.1 and l'Hospital's rule we obtain tl-'.00

F(t) _ 1 _0 l 2f'(t) f2(t) t-oo

An equivalent criterion to the Keller-Osserman condition and

o

F(t)/ f (t) lim _< t-. ° oo ft, ds/ F(s)

F(t) P(t)

- 2 + tl 1

f 2{t)

-

1

2

51

- a.

( 2 . 23)

This concludes our proof.

Lemma 2.4.3 Assume that f fulfills (f 4). Then the Keller-Osserman condition (f2) holds if and only if (Ag)

lim tf (g(t)) = oo

t\,o

for some function g E 9 .

g (t)

Proof NECESSARY CONDITION. Because (f2) holds, we can define the positive function g as follows:

J

°O

ds

t)

F(s)

= t" for all t > 0,

(2.24)

where t9 > 3/2 is arbitrary. Obviously, g E C2 (0, oo) and limt\,o g(t) = oo. We claim that g E 9 and condition (Ag) is fulfilled. For this purpose, we divide our argument into three steps.

/

Ie(t)

Step 1: i\,o

79

t"

I a - 2)'

(t)) = We derive twice relation (2.24) and we obtain

g'(t) = -t9t"-1 F(g(t)) g"(t) =

t9 _

1

(2.25)

,

g'(t) + t2_2f(g(t))

t 2t2,9-2 f(g(t))

(226) .

2(t9 {

92

1) t25-,(t) f(g(t))

+ 1)

.

By (2.23) and (2.25) we find

g'(t)

t\0

t21-1 f (g(t))

-t9ti9-I

= lim

t,,o

= lim -19

to

F(g(t))

t21- If (g(t))

F(g(t))/ f (g(t))

f (t) ds/ F(s) = 71-00 lira -t9 F(u)/f (u) = t9 (a - 1 f, ds/ F(s) 2 Step 2: g" > 0 in (0, 5) for b small enough. Because a > 0, by the result obtained in Step 1 we find 2(t9 - 1)

t\o

192

g'(t) _ 2(t9 - 1) / - 1\ a 2J t21-If (g(t)) t9

In view of (2.26), the assertion of this step follows.

1 > -1.

(2.27)

Blow-up solutions for semilinear elliptic equations

52

(g(t)) _ 00 lim t f g"(t) Taking into account (2.26) and (2.27) we find Step 3: t\o lim gg(t) (t) = 0

lim

and

t1,o

2t

g'(t)

0 g (t} -

g'(t)

_

1

z92 t2,1 -If (g(t)) 2(

1) t2 v

+1 g'(t) f(9(t))

°,

and, for any t E (0, 8) where 5 > 0 is given by Step 2, we have tf (9 M) =

g"(t)

tf (g(t))

tf (g(t)) lg'(t) + 22t2s-2 f(g(t)) >

2

792t2'0-3'

Sending t to 0, the claim of Step 3 follows. SUFFICIENT CONDITION. Let g E 4 be chosen so that (A9) is fulfilled. By 1'Hospital's rule we find (g'(t))2 =

"o F(g(t))

2

tNp f (g(t))

0.

We choose 5 > 0 small enough such that g'(t) < 0 and g"(t) > 0 for all t E (0, 5). It follows that dt f9(0150)

F(t)

lira

t\0

f ){s) s(t)

ds

(S

ca

-g'(s) ds F(g(s)) -g'(t} < b sup F(g(t)) tE(0,5)

= lim t\O Jt

< 00.

Hence, the growth condition (A2) holds. This ends the proof.

2.5 Singular solutions of the logistic equation on domains with holes Denote by V and R. the boundary operators

Du := u

and

Ru := 8 +'3(3;)u,

where n is the unit outward normal to 852, and,3 E Cl>_'(8St) is a nonnegative function. Hence, V is the Dirichlet boundary operator and 7Z is either the Neumann boundary operator, if 3 = 0, or the Robin boundary operator, if Q 0. Throughout, B can define any of these boundary operators. We are concerned in this section with the following boundary blow-up problem:

Du + Au = p(x) f (u)

in Il,

on BQ, Bu = 0 (2.28) on BSlo, u = 00 where Sto is the interior of the zero set of p as defined in (2.18). We assume that p > 0 on all, 00 CC 52 is nonempty, connected, and has a smooth boundary.

Singular solutions of the logistic equation on domains with holes

53

In (2.28) condition u = 00 on fflo means that u(x) -> oo as x E i \ Sto and d(x) := dist (x, Sto) - 0. Because of the mixed boundary condition imposed on asi, a suitable comparison principle is needed. To this aim, we first provide the following result. Lemma 2.5.1 Assume that p > 0 in 1Z \ SZo, f satisfies (f3), and let u1.. u2 E C2 (P \ S2o) be such that

(i) Dul + Au1 - p(x)f (ul) < 0 < Lu2 + AU2 - p(x)f(u2) in 9 \ Po; (ii) u1, u2 > 0 in 1 \ SIo and Bul > 0 > 13u2 on aQ; (iii) lim supdist (x,8R,,) .o (u2 - u1) (x) < 0.

\Qo. Proof If 13 = D, the conclusion follows by Theorem 1.3.17. We next assume that 13 = R. Let 01, 02 be two nonnegative C2 functions in Il \ Q0 vanishing

Thenul >U2 in

near o Do. Multiplying the first inequality in (i) by 01, the second one by 02, and applying integration by parts together with (ii), we deduce that

-

j(Vu2 ' ©02 - v2L1 - V 1) dx -

i

Jasp 13(x)(u2g2 - ul01) du(x)

(229)

p(x)(f(u2)q52-f(ul)01)dx+A(u141-u202)dx, Ln

where 1 := S2 \ Sto. Let £1 > E2 > 0 and denote fl+(£1, E2) := {x E

: u2(x) + £2 > u1(x) + £1}. vi := (ui + £i)-l ((u2 + £2)2 - (ul + _1)2)+ , i = 1, 2.

Because vi can be closely approximated in the H1 fl L°° topology on Sl \ S2o by nonnegative C2 functions vanishing near a52o, it follows that (2.29) holds for vi taking the place of qi. Because vi vanishes outside the set SZ+(£1, 162), relation (2.29) becomes

-J S2+(El, Ea}

J

(vu2 vv2 - Vu1 Vvj) dx - f O(x)(u2V2 - u1v1) d r(x) re f p(x)(f(u2)v2 - f(ul)vl)dx + a J (u1v1 - u2v2)dx. (2.30)

A straightforward computation shows that the first integral in the left-hand side of (2.30) equals

)n+(1.2)

vu2 -

U2 + £2 u1 + £1

2

Vul

2) +61 vu1 - u1 + vu2 dx<0. U2

£2

Passing to the limit as 0 < £2 < £1 -> 0, the second term on the left-hand side of (2.30) converges to 0. Also, the first term in the right-hand side of (2.30) converges to

54

Blow-up solutions for semilinear elliptic equations

f(u2) U2

f(oi)l (u2-ui1) dx ul )

whereas the other term converges to 0. Hence, we avoid a contradiction only in

the case that SZ+(0, 0) has measure 0, which means that ul > u2 in Q. This concludes the proof.

We start the study of (2.28) with the following auxiliary result.

Proposition 2.5.2 Assume that (f2), (f3) hold and p > 0 on %1. Then, for any positive function 0 E C2'7(8920) and A E ]R, the problem

in 0,

Au + Au = p(x) f (u) Bu = 0

on 892,

u = q5

on 8920

(2.31)

has a unique positive solution.

Proof By Lemma 2.5.1 we find that problem (2.31) has at most one positive solution. To prove the existence of a positive solution to (2.31) we use the suband supersolution method. Let w cc 920 be such that the first Dirichlet eigenvalue of (-0) in the smooth domain 920 \ w is greater than A. Let q E C°°7(92) be such that q(x) = p(x) q(x) = 0 q(x) > 0

in Sl \ 920,

in NO \ w,

in w.

By virtue of Proposition 2.3.7, the problem

Av +.v = q(x)f(v)

v=1

in 11,

on O1

has a unique positive solution v. Let us choose 921 and S22 as two subdomains of 52 such that 920 cc 921 C C 922 C C 52.

Define w E C2(Sl \ 92o) so that

w=1inS2\922i w=vinSi1\920, and let m := min,-ono w > 0. We claim that Mw is a supersolution of problem (2.31), provided M > I is large enough. We first remark that for all M > 1 we have

-A(Mw) = )Mv - Mq(x) f (v) > A(Mw) - p(x) f (Mw)

(2.32) in S21 \ 920.

Singular solutions of the logistic equation on domains with holes

55

Let a := supn\n, (Aw + tlw). By Remark 2.3.1, there exists c > 1 such that

f (mM) >

Ma

min?j,nl p

for all M > c.

Therefore, for all x E I2 \ S2i and M > c we have

p(x) f (Mw) > (min p) f (mM) > M(Aw + ow), Q\nl

which can be rewritten as

-A(Mw) > A(Mw) - p(x) f (Mw) for x E I l \ aj and M > c.

(2.33)

By (2.32) and (2.33) it follows that

-O(Mw) > A(Mw) - p(x) f (Mw) in 0 \ Sto, for any M > c. On the other hand,

B(Mw) > Mmin {1, min 3(x)} > 0 on 5 1, for every M > 0. xEaQ

Now the claim follows by taking M > max {maxan O/m; c}. In view of Theorem 1.4.1, the boundary value problem 1©u = IA(u + IIpIIoof (u)

u>0 u=0

in 52,

on 090,

(2.34)

onOco

has a unique nonnegative solution u such that u > 0 in I2 \ S2o. Because u = 0

on OR we find that Ru = 8u/8n < 0 on 852. It is easy to see that a is a subsolution of (2.31) and it < Mw in S2\S2v for M large enough. The conclusion

of Proposition 2.5.2 follows now by the sub- and supersolution method. This ends the proof.

Corollary 2.5.3 For m > 1 sufficiently large, set

Q,:= {x E Q: d(x) < 1/m}.

(2.35)

If io is replaced by Ilm, then the statement of Lemma 2.5.2 holds.

Proof The proof is very easy in this case. The construction of a subsolution is made as before. As a supersolution we can choose any number M > 1 large enough.

We are now ready to prove the main result of this section.

Theorem 2.5.4 Let (f 2) and (f4) hold. Then, for any A E R, problem (2.28) has a minimal (respectively a maximal) positive solution UA (respectively Ua).

Blow-up solutions for semilinear elliptic equations

56

Proof We first prove the existence of the minimal positive solution for problem (2.28). For any n > 1, let un be the unique positive solution of Au + Au = p(x) f (u) Bu = 0

u=n

in 52,

on 852, on r3S2o.

By Lemma 2.5.1, un(x) increases with n for all x E 52\S2o. Moreover, the following

result holds true. Lemma 2.5.5 The sequence (un(x))n>1 is bounded from above by some function V (x), which is uniformly bounded on all compact subsets of fl \ S2o.

Proof Let q be a C2 function on 0 \ S2o such that

0
forallxES2\52o.

For x bounded away from BSto, it is not a problem to find such a function q. If x is close to the boundary of go we chose 8 > 0 small enough such that x F--> d(x) is a C2 function in the set Qs = {x E S2 : 0 < d(x) < S}. Then, for x e S2b we can define t

jd(x)

q(x) = J 0

min b(z) ds dt.

Jo

d(z)>s

Let g E 9 be a function such that (As) holds. The existence of g is guaranteed

by Lemma 2.4.3. Because q(x) --p 0 as d(x) \ 0, by virtue of hypothesis (f 1) and Remark 2.4.1, there exists b > 0 such that for all x E SZ with 0 < d(x) < b and M > l we have g'(q(x)) inf (Aq(x))+A g(q(x)) (2.36) q(x)f(Mg(q(x))) > sup 1Vq(x)I2+ Mg"(q(x)) 9"(q(x)) g"(q(x)) \00 SE\c2o

Here, 8 > 0 is taken sufficiently small so that g'(q(x)) < 0 and g"(q(x)) > 0 for all x with 0 < d(x) < J. For no > 1 fixed, define V* as follows: (i) V* (x) = uno (x) + 1 for x E Si and near 852;

(ii) V*(x) = g(q(x)) for x satisfying 0 < d(x) < b; (iii) V* E C2 (?j \ ao) is positive in 52 \ Sta.

We show that for M > 1 large enough the upper bound of the sequence (un(x))n>1 can be taken as V := MV*. Because on851 and

lim (un(x) - V(x)) = -oo < 0,

d(x)\0

Singular solutions of the logistic equation on domains with holes

57

to conclude that n, < V in 52 \ lio it is sufficient to show that

-AV -AV +p(x)f(V) > 0

in S2\Sto.

(2.37)

For x E Q satisfying 0 < d(x) < 5 and M > 1 we have

-AV -,1V +p(x)f(V) =M( -Og(q(x)) - Xg(q(x)) + p(x)f (g(q(x))) )

> Mg"(q(x))Oq(x) (-::(x) ) - JVx)12 g(q(x))

-

grr(q(x))

f(Mg(q(x))) q(x)

Mg"(q(x))

By virtue of (2.36) we thus derive

-AV- AV + p(x) f (V) > 0

for all x E 5l \ Qo with 0 < d(x) < b.

For x E n satisfying d(x) > b we also have

-AV - AV + p(x)f (V) = M

(_v-

AV* + p(x)

\

f (V *) I > o,

M

//

for M > 0 sufficiently large. It follows that (2.37) is fulfilled provided M is large 0 enough. This finishes the proof of the lemma. We now come back to the proof of Theorem 2.5.4. By Lemma 2.5.5, UA(x) :_ u, (x) exists, for any x E 12 \ S1o. Moreover, Ua is a positive solution of problem (2.28). Using Lemma 2.5.1 once more, we find that any positive solution

u of (2.28) satisfies u > u on Sl \ 1o, for all n > 1. Hence U,, is the minimal positive solution of problem (2.28). To achieve the existence of the maximal positive solution to problem (2.28), we use essentially the same argument that we use earlier.

Lemma 2.5.6 If S'1o is replaced by defined in (2.35), then problem (2.28) has a minimal positive solution provided that (f2) and (f3) are fulfilled.

Proof The argument used here, which is easier because p > 0 on S2 \ Q,,,, is similar to that in the proof of Lemma 2.5.5. The only difference that appears here (except the replacement of Qo with St,,,) is related to the construction of V*(x) for x near BST,,,. For this purpose, instead of Lemma 2.4.3 we use Theorem 2.3.6,

which says that for any A E R there exists a positive blow-up solution u, of (2.14) in the domain S2 \ 52,,,. We next define V * (x) = u, (x) for x E ) \?!,,, and near 49Q,,,. For M > I and x E S2 \ 52,,, near on,, we have

-AV - AV + p(x) f (V) = -MOV* -,MV* + p(x)f (MV*) = p(x) (f (MV*) This completes the proof of our lemma.

- Mf (V*)) > o. El

Blow-up solutions for semilinear elliptic equations

58

Let v,,,, be the minimal positive solution of the problem considered in the statement of Lemma 2.5.6. It is easy to see that vm > vm+1 > u in 1l\S1,n, where u is any positive solution of (2.28). Hence U),(x) := lim,n_,, vm(x) > u(x), for all x E S2\S2o. A regularity and compactness argument show that U,\ is a positive blow-up solution of (2.28). Consequently, Ua is the maximal positive solution. 0 This concludes the proof of Theorem 2.5.4.

2.6

Uniqueness of blow-up solution

We establish in this section a general uniqueness result for problem (2.28) under

suitable conditions on p and f. Assume that there exists a positive increasing function k E C' (0, So) for some bo > 0 such that p satisfies the following hypotheses: (p3)

p(x) = c for some constant c > 0. lim d(x)\G k(d(x))

(p4) K(t) _

jo

k(s} ds

Also assume that there exist ( > 0 and to > 1 such that

(f 5) f (gt) < '+5 f (t), for all (f6) The mapping (0,1]

E (0,1) and all t > to /e-

l; H A(C) = limt,,,.

f {£t}

is continuous and posi-

tive-

Remark that limt-,,,, f (t)/tl+S exists and is positive because the mapping t r-a f (t)/t1+S is nondecreasing in a neighborhood of infinity. Thus, the hypotheses (f 1) and (f5) imply (f 2). By (f6) we derive that A can be continuously extended in the whole (0, oo) by setting A(1/i;) = for all l; E (0, 1). Moreover, we have the following lemma.

Lemma 2.6.1 The function A : (0, oo) -* (0, oo) is bijective, provided that (f5) and (f6) hold.

Proof Because A is continuous and 1/A(C) for all t; > 0, the surjectivity of A follows at once if we show that limt\o A(C) = 0. To this aim, let l; E (0, 1) be fixed. Using (f 5) we find

fW) <

for all t,>

t

which yields A(C) < 6C. Because C E (0, 1) is arbitrary, it follows that 0.

A(C) _

Uniqueness of blow-up solution

59

We next show that the function f ,) A(l) is increasing on (0, oo), which concludes the proof of our lemma. Let 0 < tt < C2 < oo be chosen arbitrarily. Using assumption (f5) once more, we obtain

Mit} = f 1

2t < (

f (62t)

2}

for all t > to S1

It follows that

AGO Slf(t)

<(Ll)' cftS2f(t)

(e2t)

for all t > t 2

6

S2

Passing to the limit as t --> oo we find S

(1612) A(62) < A(62),

which finishes the proof.

Lemma 2.6.2 Assume that (f 1), (f2), (f 3) with a # 0, (p3), and (p4) are fulfilled. Then, the following hold: (i) K'(0)(1 - 2a) + 2a r= (0, 11. (ii) h E G, where h is the function defined by (2.4O).

Proof (i) Because a

0, by Lemma 2.4.2 we find 0 < a < 1/2. Therefore, the claim of (i) follows if we prove that K'(0) E [0, 1]. To this aim, we first remark that K(0) = 0. Indeed, supposing the contrary, we obtain lim

-In

(

t

k(s) dsl

(t) =

K(0)

E (0, oo),

which contradicts the fact that

In (fo k(s) ds) = -oo. Hence, K(O) 0, which yields K`(0) > 0. Because K E Cl [0, &o), we have fa

_

k(s) ds1 '

so that lim

t-"o

k'(t) fo

k(s) ds = 2 1 - lim

k3/2(t)

t\ O

(

f0t V/ k(s) ds

k(t)

)}

(2.38)

= 2(1 - K'(0)). Furthermore, K'(0) < 1. Indeed, assuming the contrary, relation (2.38) yields k'(t) < 0 in a positive small neighborhood of origin. But this is impossible, because limtNo k(t) = 0 and k > 0 on (0, bo).

Blow-up solutions for semilinear elliptic equations

60

(ii) Using the definition of h, we deduce that h c C2(0, bo) and limt\o h(t) = oo. Then, by twice deriving relation (2.40), we find

h'(t) = - k(t)

2F(h(t))

1

F(h(t))

for all t E (0, s(,)

and

h"(t) = k(t) f(h(t)) -

k'(t)

k(t) k'(t) fo k(s)ds

= k(t)f(h(t}) 1 -

k3/2(t)

F(h(t))/f(h(t)) m(t) ds/

F(s)

Using (2.23) and (2.38), we obtain lim

h'(t)

t\o h"(t)

_

-2

F(h(t))/f (h(t))

urn

K'(0)(1 - 2a) + 2. t\p f(t) ds/ F(s)

lim

f,,t

t\ O

k(s) ds k(t)

2a-1 =0 -2a)+2aK(0)

_

K'(0)(1 and

ego k(t) f ( (t))

K'(0)(1 - 2a) + 2a > 0,

(2.39)

which shows that h" is positive in (0, 61) for some bl > 0. This concludes our

0 Theorem 2.6.3 Assume that conditions (f 1) through (f6), a # 0, (p3), and proof.

(p4) hold. Then for any A E R, problem (2.28) has a unique positive blow-up solution Ua. Moreover, UA(x)

d(x)N0 h(d(x)) _ o> where h is defined by ds Fh(t)

= ft

2F(s)

k(s) ds

for all 0 < t <

o

and o is the unique positive solution of

A() = K'(0)(1 - 2a) + 2a c

Proof By Lemma 2.6.2 we have h E 9. For all C > 0 set

lim p(x) d(x)\O Using (p3) and (2.39) we find

f (h/(d

(x ))C)

l ( ))S

bo,

(2.40)

Uniqueness of blow-up solution

61

p(x) k(d(x))f(h(d(x))) f(h(d(x))e) d(x)\o k(d(x)) h"(d(x)) t; f (h(d(x))) k(t) f (h(t)) lim f (fit) = c lim t\O h"(t) t-.o f (t)

II(l;) = lim

K'(0)(1 C2a) + 2a (0, oo) is bijective. Let Co Thus, by Lemma 2.6.1 we deduce that 11 : (0, oo) be the unique positive solution of the equation f1(1;) = 1-that is,

A(£o) - K'(0)(1 - 2a) + 2a c

Let !;1 := II-1(1 - 4c), respectively e2 := H-1(1 + 4e), for arbitrary 0 < e < 1/4. Using Remark 2.4.1, the hypothesis (p3), and the regularity of 852o, we can choose b > 0 small enough such that (i) dist (x, 8520) is a C2 function on the set {x E 52 : dirt (x, 81o) < 26}; >

(ii)

;, Ad(x) + a h,({t} I < s and h"(t) > 0 for all 0 < t < 26 and x

I

satisfying 0 < d(x) < 26;

(iii) (U(2) - c) h"(d(x)) 6

p(x) < (11(1) + e)

h"(d(x)) 1

for every x

with 0 < d(x) < 2b; (iv) p(y) < (1 + e)p(x),for every x,y with 0 < d(y) < d(x) < 2b.

Let a E (0, S) be arbitrary and define ve(x) := h(d(x) + v)1;1, for any x with d(x) +a < 23, respectively J (x) := h(d(x) - a)6 for any x with a < d(x) < 26. Using (ii), (iv), the first inequality in (iii), and the fact that JVd(x)I = 1, for or < d(x) < 26 we obtain

-©u,(x) - avo(x) + p(x) f (u, (X)) o)x x

v) h'(d(x) - a) Ad(x) - A h(d(x) - 1 + p(x)f (h(d(x) - C)2) h"(d(x) - o) e2 h"(d(x) - v) h"(d(x) - v)

> r;2 h"(d(x) - a) x n(e2) - e

h(d(x) - a)

h'(d(x)

x

- a) Ad(x) - A

h"(d(x) - Q)

h"(d(x) - o)

- 1 +

1+e

>0. Similarly, using (ii), (iv), and the second inequality in (iii), when d(x)+v < 26 we find

Blow-up solutions for semilinear elliptic equations

62

-Av.(x) -.\v-,(x) +P(x)f(,(x)) _ i h"(d(x) + o-) x

(- h'(d(x)

a) Ad(x) - A + < i h"(d(x) + a) x x

x <0.

h'dx) +

h(d(x)

+ a)

Ad(x) - a

{{d(x) +

- 1 + P{ h"(d(x)

E)}

+ {1 +

a)}

+) I}

Define 0s :_ {x E St : d(x) < b} and let w CC 52o be such that the first Dirichlet eigenvalue of (-A) in the smooth domain Sto \w is strictly greater than A. Also, let q E C°'''(S26) be such that

0 < q(x)

P(x)

in Stb \ Kio,

q(x)=0

inSto\w,

q(x) > 0

in w.

Then, by Theorem 2.3.6 there exists a positive blow-up solution of

-Aw = aw - &)f (w)

in Sta.

Suppose that u is an arbitrary solution of (2.28) and let v

u + w. Then v

satisfies

-Av > Av - p(x) f (v) in Sts \ Sto. Because v = oo > v on 0 o and v = oo > vQ on 8S26, we find u + w > vo in Sts \ SZo.

(2.41)

vQ+w>uon QS \Sio.

(2.42)

Similarly,

Letting a -> 0 in (2.41) and (2.42), we deduce h(d(x)) e2 + 2w > u + w > h(d(x)) !sj

in Sts \ S2o.

Because w is uniformly bounded on BSto, it follows that

liminf

h(d(x))

-

h(d()))

2.

(2.43)

Letting E - 0 in (2.43) and looking at the definition of fi respectively e2, we find

u(x)

lim d(x)--,o h(d(x))

(2.44)

Uniqueness of blow-up solution

63

This behavior of the solution will be speculated to achieve the uniqueness of solution to (2.28). Indeed, let u1i u2 be two positive solutions of (2.28). For any e > 0, denote wi := (1 + E) ui, i = 1, 2. By virtue of (2.44) we obtain lim

d(x)NO

ut(x) - u2 (x) h(d(x))

lim U2 (X) - ul (x) _ -Et o < 0, h(d(x))

d(x)\O

which implies

lira (ul(x) - u2(x)) = lira (U2(X) - u1(x)) = -00. d(x)%O

On the other hand, because the mapping t ---> f (t)/t is increasing for t > 0, we have

.

-Aii = -(1 + E) tut = (1 + e) (A ut - p(x) f (ui)) > A u= - p(x) f (ut) in 9 \ ao, taut = Bui = 0 on 852.

By Lemma 2.5.1,

u1(x) < u2(x),

u2(x) < ul(x)

inn \ 52o.

0 we obtain ul = u2. Hence, problem (2.28) has a unique solution. Letting e This finishes the proof of Theorem 2.6.3.

Remark 2.6.4 Some classes of nonlinearities f satisfying the assumptions of Theorem 2.6.3 are

(i) f (t) = t' , p > 1. (ii) f (t) = tp ln(t + 1), p > 1. (iii) f (t) = tp arctan t, p > 1. (iv) f (t) = t lnv(t + 1), p > 2.

Remark 2.6.5 Assume that f satisfies (f3) and (f5). Then equation (2.14) with A = 0 and p - 1 has a unique blow-up solution ii. Moreover, u has the following asymptotic behavior:

u(x)

lim

dist (x,812)-.0 r(dist (x, ml))

= 1,

where F is the function defined as ds J

Ji(t)

2F(s))

=t

for all t > 0.

Let 521 CC S2 be a connected subdomain with a smooth boundary and such that SZo C 01. A direct consequence of Theorem 2.6.3 is seen in the following corollary.

64

Blow-up solutions for semilinear elliptic equations

Corollary 2.6.6 Let (f 6) be added to the assumptions of Remark 2.6.5. Then, for any A E R, problem (2.28) with p - 1 on 8521 and no replaced with S21 has a unique positive solution Ua. Moreover, U,, behaves on 8521 exactly in the same manner as is on 812-that is, lim Uj (x) =1 dirt (x,aSZ,)-'o F(dist (x, 8121))

Proof We use the argument of Lemma 2.5.6 to deduce the existence of a positive solution for the problem considered here. Concerning the uniqueness, let us

remark that conditions (p3) and (p4) are fulfilled by taking c = 1 and k - 1 on (0, oo). It follows that h defined by (2.40) coincides with F. Notice that

F'(t) _ - 2F(I'(t)) and F"(t) = f (F(t)) for any t > 0. Thus, we obtain r E G (without calling Lemma 2.6.2) and II(C) = A(C), for all C > 0. So, by Lemma 2.6.1, II : (0, oo) -* (0, oo) is bijective. From now on, we proceed as in the proof of Theorem 2.6.3, which only requires us to replace h with r and go El with 521. This concludes the proof.

2.7 A Karamata theory approach for uniqueness of blow-up solution In this section we provide a different approach for the uniqueness of a positive blow-up solution of the logistic equation. We are concerned with the following boundary blow-up problem: Du + Au = p(x) f (u)

in 52,

u>0

inc,

2u=oo

(2.45)

on 852,

where A E R, p E C°'7(12) (0 < y < 1), such that clo := int {x E SZ : p(x) = 0}

is smooth (possibly empty), S2° C 52, and p > 0 in Sl \ 52o. The uniqueness of the solution to (2.45) will be achieved using the Karamata regular variation theory, originally introduced by Karamata [1131. We start this section with the following definition.

Definition 2.7.1 A positive measurable function f : [m, co) -+ (0, oo), m > 0, is called regularly varying (at infinity) with index p E R if lim f (it) t-.c f (t)

= l;p

for all Z; > 0.

When the index of regular variation p is zero, we say that the function f is slowly varying.

A Karamata theory approach for uniqueness of blow-up solution

65

We denote by IIBp the class of regular varying functions with index p E R. The canonical p-varying function is f (t) = tp, t > 0. The functions

ln(1 + t), In ln(e + t), exp{(lnt)a} (0 < a < 1), vary slowly, as well as any measurable function on [m, oo), m > 0, with a positive limit at infinity. Furthermore, any function f E IR, can be written in terms of a slowly varying function. Indeed, set f (t) = tpg(t). From the previous definition we conclude that g varies slowly. The basic properties of regular and slowly varying functions are summarized next. For further details, we refer the reader to Seneta [180].

Proposition 2.7.2 We have the following: (i) For any slowly varying function g : [m, oo) -' (0, oo), m > 0, and any q > 0 we have t9g(t) --+ oo and t-4g(t) --* 0 as t --+ oo. (ii) Any positive function g E C'([m, oo)) satisfying tg'(t)/g(t) -> 0 as t -9 00 is slowly varying. Moreover, if the previous limit is p E R, then g E R. (iii) Assume f : [m, oo) - (0, oo), m > 0, is measurable and locally integrable. Then f varies regularly if and only if there exists q E R and Q := lim

t--.oo

t4+1 f(t)

fm sa f (s)ds

> 0.

In this case, f E lRp with p = t:- q - 1; (iv) If f E Rp is Lebesgue integrable on each finite subinterval of [m, oo), then

for all q > -p - 1 we have e+1 f(t)

lim

= q + p + 1. f,t sa f(s)ds

We also have the following equivalence for Cl functions.

Lemma 2.7.3 Let f E C' 10, oo) be a nonnegative function that fulfills (f 3).

t

Then the following conditions are equivalent:

(i) f' E Rp for some p E R. {iz)

There exists ?9

iim

(iii) There exists a := limt-,,,

tf'(t} < 00. f (t)

Ff )j

(t) > 0.

Moreover, p > 0 and a = 1/(p + 2) = 1/(r9 + 1).

Remark that if f' E R. with p > 0, then limt, f (t)/tP = oo, for all 1 < p < 1 + p. This implies that f fulfills (f2). The converse implication is not necessarily true; we can take, for instance, f (t) = u ln4(t + 1), t > 0. Also notice that (f 2)

may fail when p = 0. This is illustrated by f (t) = t or f (t) = t ln(t + 1), t > 0.

Blow-up solutions for sernilinear elliptic equations

66

Proof

Let f E RP. Then f'(t) = tPg(t) where g varies slowly. If p < 0 then, by Proposition 2.7.2 (i) and 1'Hospital's rule we find limt_,"' f (t) /t = limt-,,,, f'(t) = 0, which contradicts (f 3). Fence p > 0 and by Proposition 2.7.2 (iv) (with q = 0) we obtain limt_,,,. t f'(t)/ f (t) = p + 1. (ii)=(i). Using the assumption (f 3) we have t f'(t) > f (t) for all t > 0, which yields 19 > 1. By Proposition 2.7.2 (iii) we now obtain f E Iti9_1. By l'Hospital rule we deduce 1imr ,o F(t)/(t f (t)) = 1/(1+19) and thm00(.f),=1-tlm°°Ff2(/}t)

F(t) tf'(t) = 1 - lim t- o tf(t)

f(t)

1

1+r9 Hence or = 1/(1 -{-19).

(iii)=(ii). From limt_,, (F/ f )' (t) = a > 0 we have a < 1 and tli}m 00

F

f2{t) t)

= 1 - a.

(2.46)

Let us choose to > 0 such that (F/ f )' (t) > a/2, for all t > to. Then,

F(t) > (t - to)a + F(to) f (t)

2

for all t > to.

f (to)

Passing to the limit with t -> oo, we find limt-. F(t)/f (t) = oo. By l'Hospital's rule we deduce tl moo

tf(t) _

1

F(t)

a

(2.47)

Multiplying (2.46) and (2.47) we obtain limt , t f'(t)l f (t) = (1-a)/a. According to Proposition 2.7.2 (iii), it follows that f' E R p, where p = (1 - 2a)/a. This completes the proof. 0 Inspired by the definition of a, we denote by K the Karamata class, consisting of all positive, increasing C' functions k defined on (0, v), for some v > 0, which satisfy f t k(s)ds l i m ( ° k(t) 2Z , i = 0,1. We observe that Po = 0 and Li E [0, 11, for every k E K. Our next result gives examples of functions k E K with limt\o k(t) = 0, for every f, E [0, 1].

Lemma 2.7.4 Let f Cr C' [m, oo), m > 0, be such that f' E RP, for some p > -1. Then we have the following:

(i) If k(t) = exp{- f (1/t)}, 0 < t < 1/m, then k E K with f, = 0.

A Karamata theory approach for uniqueness of blow-up solution

67

(ii) If k(t) = 1/f(1/t), t < 1/m, then k E K with tj = 1/(p+2) E (0,1). (iii) If k(t) = 1/ In f (1/t), 0 < t < 1/rn, then k E K with el = 1. Proof Because p > -1, from Proposition 2.7.2 (i) and (iv) we have

lim tf'(t) = 00

t-.oo and

tf,(t) = p+ 1 > 0. f(t)

lim

(2.48)

Therefore, in any of the cases (i), (ii), or (iii) we have limt\,o k(t) = 0 and k is a Cl increasing function on (0, v), for v > 0 sufficiently small. (i) Using (2.48) we have

\

_

tk'(t)

1/tf'(1/t) = _(P+1).

to k(t) In k(t) to \,

t f (1/t)

Thus, by ('Hospital's rule we obtain

tk(t)

tk(t)/ In k(t) fo k(s)ds (tk'(t) + k(t)) In k(t) - tk'(t)

= lira urn tNO In k(t) fo k(s)ds tNo t\O

_

k(t) In2 k(t)

( tk'(t) 1 tNo t k(t) In k(t) + In k(t)

tk'(t)

- k(t) In2 k(t)

_ -(p+ 1). Hence,

lim

t\a

(fo k(s) ds) k'(t)

-iim In k(t) f,) k(s)ds tk'(t) t\o

k2(t)

tk(t)

k(t)

1,

which implies

el = 1 - t\o

(f o k(s)ds) k(t) k2(t)

= 0.

(ii) It is easy to see that tk(t)) two

=

By 1'Hospital's rule we obtain

_ 21-1

1/f(1/t)t) t\o

t\o

=p+

1.

fo k(s)ds/(tk(t)) = 1/(p + 2). Thus,

ff k(s)ds tk'(t) kt() tk(t)

P+2

Blow-up solutions for semilinear elliptic equations

68

(iii) In a similar manner we have limt\o tk'(t)/k2(t) = p + 1. By 1'Hospital's rule, limtNo fo' k(s)ds/(tk(t)) = 1. Thus, £1 = 1 -limo

-

tk'(t)

fo k(s)ds t

k2(t) = 1.

0

This concludes the proof.

As a consequence of Proposition 2.7.2 and Lemma 2.7.4 we obtain the following corollary.

Corollary 2.7.5 Let f E Cl [m, oo), m > 0. Then f E RF with p > -1 if and only if there exist q > 0, C > 0, and B > D such that

f(t)=Ctgexp{f' ytt)dtI

for alit

B,

(2.49)

111

where y E C[B, oo) satisfies limt_,,,,, y(t) = 0. In this case, f' E R,, with p = q- 1. The core result of this section is the following.

Theorem 2.7.6 Let f E C' [0, oo) be such that (f3) holds and f E R p with p > 0. Assume that p - 0 on 892 satisfies p(x) = c k2(d(x)) + o(k2(d(x)))

as d(x) \,0,

for some constant c > 0 and k E 1C. Then, for any A E (-oo,A,,), problem (2..45) has a unique blow-up solution UA. Moreover,

U,\ () d()NO h(d(x))

0 =

Jh(t)

p) I

,

(2.50)

+

and h is defined by

/°°

\ t/p

2

when eo = G (2

ds

2F(s)

I

= f t k(s)ds

for all t E (0, v).

(2.51)

By Corollary 2.7.5, the assumption f E Rp with p > 0 holds if and only if f satisfies (2.49) with q = p + 1 and for some B, C > 0. If B is large enough (for instance, if y > -p on [B, oo)), then f (t)/t is increasing on [B, oo). Thus, to obtain the whole range of functions f for which Theorem 2.7.6 applies, we have only to "paste" a suitable smooth function on [0, B] in accordance with condition

(f3). A simple way to do this is to define f(t) = tp+'exp{ fo z(s)/sds}, for all t > 0, where z E C[0, oo) is nonnegative and such that limtNo z(t)/t E [0, oo) and limtN,,. z(t) = 0. Clearly, f (t) = t", f (t) = t' ln(t+ 1), and f (t) = tQ arctan t (t > 1) fall into this category. Lemma 2.7.4 provides a practical method to find functions k which can be considered in the statement of Theorem 2.7.6. Here are some examples: k(t) =

A Karamata theory approach for uniqueness of blow-up solution

69

exp {-1/ta}, k(t) = exp {- ln(1 + 1/t)/ta}, k(t) = exp {- [arctan (1/t)] /tQ }, k(t) = -1/ In t, k(t) = to'/ ln(l + 1/t), k(t) = ta, for some a > 0. As we have already seen in the previous section, the uniqueness lies upon the crucial observation (2.50), which shows that all explosive solutions have the same boundary behavior.

Proof Fix A E (-oo,.1,1). By Theorem 2.3.6, the problem (2.45) has at least one blow-up solution.

The uniqueness will follow at once if we prove that (2.50) holds for any solution u), of (2.45). Indeed, if ul and u2 are two arbitrary blow-up solutions of (2.45), then (2.50) yields limd(x)\au1(x)/u2(x) = 1. Hence, for any 0 < e < 1, there exists S = S(E) > 0 such that (1 - E)u2(x) < u1 (x) < (1 + E)u2(x),

(2.52)

for all x E S2 with 0 < d(x) < S. Choosing a smaller S > 0 if necessary, we can assume that 1 o C C5, where Ca :_ {x E 12 : d(x) > S}. It is clear that ul is a positive solution of the boundary value problem in Cs, on 8Cs.

J AO + Aq _ p(x)f (0)

4=u1

(2.53)

By (f3) and (2.52), we see that (1 - e)u2 is a subsolution and 0+ (1 + 6)u2 is a supersolution of problem (2.53). By the sub- and supersolution method, (2.53) has a positive solution 01 satisfying 0- < ¢1 < ¢+ in Ca. Because p > 0 in Ca \ 12o, by Lemma 2.3.7 we derive that (2.53) has a unique positive solution, (i.e., ul - 01 in C5). This yields (1 - -)u2 (x) < ul (x) < (1 + '-)U2 (X) in C5, so that (2.52) holds in Sl. Passing to the limit with e -* 0, we conclude

that ul - u2. To establish (2.50) we first state some useful properties about h.

Lemma 2.7.7 We have (i) h E C2(0, v) and limt-,o h(t) = oo; (ii) for all

> 0 there holds limt\,o

k2(t) f (h(t)t;)

p+l 2 + pl ,

so, h" > 0

on (0, 26), for S > 0 small enough;

(iii) limt\,o h(t)/h"(t) = limt\o h'(t)/h"(t) = 0. Proof (i) Follows directly from (2.51). (ii) Because f E Rp+1, it suffices to prove the claim for C = 1. Clearly h'(t) = -k(t) 2F(h(t)) and

h (t) = k 2(t) f (h(t))

(1_2

k'(t) (fo k(s)ds)

k2(t)

F(h(t))

f(h(t))

(t f 00)

F(s)ds

)

'

(2.54)

Blow-up solutions for semilinear elliptic equations

70

for all 0 < t < v. It is easy to be seen that limt-,,,,

F(t)/ f (t) = 0. Thus, from

1'Hospital's rule and Lemma 2.7.3 we infer that

t-lim'I f (t) f' F(s)-1/gds

2

p 2(p + 2)'

-a

(2.55)

Using (2.54) and (2.55) we derive (ii) and also

F(t) h'(t) = -2(2 + p) . lim fo k(s)ds lim k(t) t-->oo f(t) f too t\o t\o h"(t) 2 + tip F(s)ds lim

(2.56)

-peo =0. 2+tlp

From (i) and (ii) we derive that limt\,o h'(t) = -oo. So, l'Hospital's rule and (2.56) yield limtNa h(t)/h'(t) = 0. This and (2.56) lead to limt\,o h(t)/h"(t) = 0, which proves (iii). This completes the proof of the lemma. Let us come back to the proof of Theorem 2.7.6. We fix e E (0, c/2). Because p =- 0 on aQ, we may take 6 > 0 such that

(i) d(x) is a C2 function on the set {x E RN : d(x) < 25}; (ii) k2 is increasing on (0, 28);

(iii) (c-e)k2(d(x)) 0 for all t E (0, 25). Let a E (0, 5) be arbitrary and define + := [(2 + Lip)/(c j 2,-)(2 + p)] 11P and vo (x) := h(d(x) +Q)1;-, for all x with d(x) + a < 25 respectively, va (x) h(d(x) - v)C+, for all x with a < d(x) < 26. Using (i)-(iv) and the fact that IVd(x)I _- 1, when a < d(x) < 25 we obtain +h"(d(x) - Q)

Ova + Ava-p(x)f (va)

h(d(x) -

v)

(hh'"(d((d(xx))

k2(d(x)

- o) i d(x)+ o,)f (h(d(x)

-

h (d(x) -

h"(d(x) - 6)

v < 25 we find AV + AV,- -p(x) f (vo) > A

h(d(x) + L7)

h"(d(x) + v)

h"(d(x) + v)

h(d(x) + c) Ad{x)+ (h"(d(x) + o)

+ 1- (c + e)

k2(d(x) + ci)f(h(d(x) + o,) h"(d(x) + o,)6-

)

Using Lemma 2.7.7 (ii) and (iii) and by taking 5 > 0 sufficiently small, we can assume that

zva(x) + Ava (x) - p(x) f (va(x)) < 0

for all x with o < d(x) < 2S,

Comments and historical notes

.va (x) + Avo (x) - p(x) f (vQ (x)) > 0

71

for allx with d(x) + a <26.

Let 91 and 02 be two smooth bounded domains such that 52 cc SZ1 CC Q2 and the first Dirichlet eigenvalue of (-0) in the domain Q1 \ Sl is greater than A. Let also a E C°"Y(Sl2) be such that

0 < a(x) < p(x)

a=0 a>0

in Sl \ C2s,

inSZ1\Sl,

in 02\521.

By Theorem 2.3.6, there exists a solution w of the problem Ow + Aw = a(x) f (w)

I

w>0

in Sl \ C28, in Sl \ C26,

w=oo

on8(Sl\C2b).

(2.57)

Let now uA be an arbitrary blow-up solution of (2.45) and set v := ua + w. Then v satisfies Av + Av - p(x) f (v) < 0 in 52 \ C28. Because v = oo > vQ on 8Q and v = oo > v- on 49C25i it follows that

.

uA+w>vQ

in5l\C2S,

(2.58)

v++w>uQ

inCQ\C28.

(2.59)

and similarly

Letting a -; 0 in (2.58) and (2.59), we deduce

h(d(x))e+ + 2w > u,\ + w > h(d(x))C-

for all x E 0 \ C28.

Because w is uniformly bounded on 852, we have < lim inf

ua (x)

< lim sup

uA (x)

d(x)\,o h(d(x)) - d(x)\,o h(d(x))

<+

Letting e - 0 in the previous inequality we obtain (2.50). This concludes the proof.

0

2.8 Comments and historical notes Problems related to large solutions have a long history and are studied by many authors and in many contexts. Singular value problems of this type go back to the pioneering work by Bieberbach [26] in 1916 on the equation Au = e' in a smooth bounded domain 52 C R2. He showed that there exists a unique solution such that u(x) - log(d(x) -2) is bounded as d(x) -+ 0. Problems of this type arise in R.iemannian geometry. If a Riemannian metric of the form IdsI2 = exp(2u(x))Idxl2

Blow-up solutions for semilinear elliptic equations

72

has constant Gaussian curvature -c2, then Du = c2 e2' . We also recall that the equation Au = e" was studied by Max von Lauel (1918) who established, using statistical mechanics, that the density in equilibrium of ideal gases at any point is proportional to an exponential function, which can be described explicitly in terms of the electrostatic potential. Motivated by a problem in mathematical physics, Rademacher [173] continued the study of Bieberbach on smooth bounded domains in R3. These kinds of problems were later studied under the general form Au = f (u) in N-dimensional domains. We refer the reader to [13][16], [491, [501, [53], [65], [70], [128], [129], [1331-[136], and (1911.

We point out that the study of blow-up boundary solutions is motivated by many natural phenomena. For instance, Keller [117] was concerned with an electrohydrodynamic model and his conclusions are the following: as the mass of uniformly charged gas contained in a container 52 C R3 increases to infinity, then its density remains bounded at any interior point, but becomes infinite on the surface 852. In a celebrated paper.. Loewner and Nirenberg [133] linked the uniqueness of

the blow-up solution to the growth rate at the boundary. Motivated by certain geometric problems, they established the uniqueness of blow-up solutions for

Au =

41,(N+2)/(N-2),

N>3.

Bandle and Marcus [15) give results on asymptotic behavior and uniqueness of the blow-up solution for more general nonlinearities including f (t) = tP, where

p> 1. The approach in Section 2.2 is the result of the work by Cirstea and Radulescu

[49]. Similar results were obtained by Cheng and Ni [41) or Marcus [135] but under the stronger assumption p > 0 on 852 instead of (p1). In Section 2.5 we were concerned with the existence of blow-up solutions for semilinear elliptic problems with a mixed boundary condition. Note that the Robin condition R = 0 arises in heat flow problems in a body with constant temperature in the surrounding medium. More generally, if a and 0 are smooth functions on 8Sl such that a, /3 > 0, a + 0 > 0, then the boundary condition f3u = atu/8n + /3u = 0 represents the exchange of heat at the surface of the reactant by Newtonian cooling. Moreover, the boundary condition Bu = 0 is called an isothermal (Dirichlet) condition if a - 0, and it becomes an adiabatic (Neumann) condition if /3 - 0. An intuitive meaning of the condition a +,6 > 0 on 852 is that, for the diffusion process described by problem (2.14), either the reflection phenomenon or the absorption phenomenon may occur at each point of the boundary. If f (t) = tP (p > 1), then the semilinear elliptic problem Au + Au = a(x)uP Bu = 0

in 52,

on 852

(2.60)

'The Nobel Prize in physics, 1914, "for his discovery of the diffraction of X-rays by crystals".

Comments and historical notes

73

is a basic population model (see, for example, (1061) and is also related to some prescribed curvature problems in Riemannian geometry (see, for example, [155] and [1151). The existence of positive solutions of (2.60) has been intensively studied (see, for instance, [4], [7], [58], [62], [79] and [155]). If the variable potential a(x) is positive in fl, then

Du + au = a(x)u"

in St,

is known as the logistic problem. This equation has been proposed as a model for population density of a steady-state single species u(x) when Q is fully surrounded by inhospitable areas. The understanding of the asymptotics for positive solutions of the degenerate logistic equation leads to the study of blow-up boundary solutions. The two approaches for the uniqueness of the blow-up solution to the logistic equation in Sections 2.6 and 2.7 are the result of the work by Cirstea and Radulescu [50], [51].

We have seen in this chapter that if f E Cl [0, oo) is an increasing function such that f (0) = 0, then the problem Au = f (u)

in Q,

u > 0

in Q,

U=00

on all

(2.61)

has a solution if and only if the Keller-Osserman condition

f°° J1

dt

F(t} < °°

is fulfilled, where F(t) := fo f (s) ds. A natural question is to establish a related result provided f fails to exhibit monotone behavior. This problem was studied by Dumont, Dupaigne, Goubet, and Radulescu [71], who formulated an answer in terms of two Keller-Osserman-type conditions introduced by Aftalion and Reichel [1]. In a general setting, we say that a function f : [0, oo) -> [0, oo) (not necessarily increasing!) satisfies the strong Keller-Osserman condition whenever there exists a > 0 such that dt

< 00.

Accordingly, we say that f satisfies the weak Keller-Osserman condition whenever

liminf -P (a) = 0.

a-oo

For instance, the function f (t) = t2(1 +cos t) satisfies the weak Keller-Osserman condition, even if lim supa_,,. oo. The result established by Dumont, Dupaigne, Goubet, and Radulescu [71] is the following.

Blow-up solutions for semilinear elliptic equations

74

Theorem 2.8.1 The following statements are equivalent: (i) f satisfies the strong Keller-Osserman condition. (ii) f satisfies the weak Keller-Osserman condition. (iii) There exists a ball Q = BR such that problem (2.61) admits at least a solution.

(iv) Given any smooth domain St, problem (2.61) admits at least a solution. In particular, Theorem 2.8.1 implies existence of blow-up boundary solutions for oscillating functions such as f (u) = u2 (1 + cos u). The results developed in this chapter also apply to problems on Riemannian manifolds if A is replaced by the Laplace-Beltrami operator

Os VIC

8xi

(Vcaij(x) axi

c

det (aid),

with respect to the metric ds2= cij dxidxj, where (Cij) is the inverse of (ai?). In this case the results developed in this chapter apply to concrete problems arising in Riemannian geometry. For instance (cf. Loewner and Nirenberg [1331), if D is

replaced by the standard N dimensional sphere (S', go), 0 is the LaplaceBeltrami operator logo, f (u) = (N - 2)/14(N - 1)] u(N+2)/(N`2), and A = N(N-2)/4, we find the prescribing scalar curvature equation on SN. For further applications and connections with other fields of mathematics such as Brownian motion and potential theory, we refer the reader to Dynkin [73], Iscoe [110], and Le Gall [130].

3

ENTIRE SOLUTIONS BLOWING UP AT INFINITY FOR ELLIPTIC SYSTEMS The object of pure physics is the unfolding of the laws of the intelligible world; the object of pure mathematics that of unfolding the laws of human intelligence.

James J. Sylvester (1814-1897)

In this chapter, we are concerned with entire radially symmetric solutions of logistic-type systems in anisotropic media. In terms of the growth of the variable

potential functions, we establish conditions such that the solutions are either bounded or blow up at infinity. An important role in the results we establish in this chapter is played by the properties of the central value set. 3.1 Introduction Consider the following semilinear elliptic system

I

AU = p(x)f(v) ©v = q(x)g(u) u > O, v > 0

in RN, in RN, in RN,

where N > 3 and p, q E C°o (RN) (0 < -y < 1) are nonnegative and radially symmetric potentials. We also assume that f, g E C,; 10, oo) (0 < y < 1) are positive and nondecreasing on (0, oo). We are concerned in this chapter with the existence of entire radially sym-

metric solutions of (3.1) that blow up at infinity-that is, radially symmetric solutions (u,v) satisfying u(x) -4 oo and v(x) ---> oo as IxI -k oo. Notice that radially symmetric solutions of (3.1) satisfy the ordinary differential system

u"(r) + v"(r) +

N

r Nr

1 u'(r) = p(r)g(v(r)),

= q(r)f (u(r)), u(r) > 0, v(r) > 0,

for allr>0.

1 of (r)

76

Entire solutions blowing up at infinity for elliptic systems

Our aim is to give a characterization of the central value set g of all (a, b) E 1R+ x R+, where R+ = (0, oo) such that there exists an entire radially symmetric solution (u, v) of (3.1) with the property u(0) = a and v(O) = b. Such a pair will be called the central value of the solution (u, v) of system (3.1). Thus, any radially symmetric solution (u,v) of (3.1) with the central value (a, b) E IR+ x R+ satisfies the following system of integral equations:

u(r) = a + f r tl-N fff0

v(r) = b + Ir tl-N 3.2

sN-1p(s)g(v(s))dsdt,

r > 0,

t SN-iq(s) f (u(s))dsdt,

r > 0.

t

J o

J0

Characterization of the central value set

Bounded or unbounded entire solutions We are first concerned with a framework corresponding to an unbounded central value set. In such a case, taking into account both the decay of the variable potentials and the growth rate of the nonlinear terms, we establish sufficient conditions such that the solutions of the nonlinear system (3.1) are either bounded or blow up at infinity. 3.2.1

Theorem 3.2.1 Assume that tli

oo

9W t(0) = 0 for all c > 0.

(3.4)

Then 9 = ]R+ x R+. Moreover, the following properties are valid: (i) If p and q satisfy 00

00

tp(t) dt = oo

I

and 10

tq(t) dt = oo,

then all radially symmetric solutions of (3.1) blow up at infinity. (ii) If p and q satisfy 00

00

tp(t) dt < oo

and

J0

tq(t) dt < oo,

then all radially symmetric solutions of (3.1) are bounded.

For any r > 0 define

I/ tl-N r

A(r)

t

f sN-Ip(s)dsdt, 0

B(r) :=

1o

r tl-N r SN-lq(s)dsdt. t

(3.5)

Characterization of the central value set

77

Remark 3.2.2 Using Lemma 2.2.4 we deduce the following properties:

(i) Condition (3.5) holds if and only ifr lim A(r) = lim B(r) = oo. oo raw (ii) Condition (3.6) holds if and only if both Jim A(r) and lim B(r) are finite. rco r-oo Proof Fix (a, b) E R+ x R+. A radially symmetric solution (u, v) of (3.1) with the central value (a, b) will be obtained by means of a sequence of approximations ((uk, vk)}k>1, as follows. Set vo - b and let (uk)k>1 and (vk)k>1 be two sequences of functions de ned as

I uk(r) = a + Jr fi

r t'-'fft $N-1p(S)9(vk-l (s))dsdt, 0

ff0

vk(r) = b +

I

r

t

t1-N r3N-1q(S)f(Uk(s))dsdt,

a

for all r > 0. Because vl (r) > b, we find u2 (r) > ul (r) for all r > 0. This implies v2(r) > vl(r), which further produces u3(r) > u2(r) for all r > 0. Proceeding in the same manner we conclude that uk(r) < uk+1(r)

and

vk(r) < vk+1(r)

for all r > 0 and k > 1.

We claim that the nondecreasing sequences (uk(r))k>1 and (vk(r))k>1 are bounded from above on bounded sets. Indeed, for all r > 0 we have uk(r) < uk+1(r) < a + g(vk(r))A(r)

(3.8)

and

vk(r) < b + f (uk(r)) B(r). Let R > 0 be arbitrary. By (3.8) and (3.9) we find

uk(R)
(3.9)

for all k> 1,

or, equivalently,

<

a uk(R)

+ g (b + f (uk(R)) B(R)) A(R) uk(R)

for all k > 1.

-

(3.10)

By the monotonicity of (uk(R))k>1, there exists L(R) := limk-,, uk(R). We claim that L(R) is finite. Indeed, assuming the contrary, letting k -+ oo in (3.10) and using (3.4) we obtain a contradiction.

Because uk(r), vk(r) > 0 it follows that the map (0, oo) E) R i--+ L(R) is nondecreasing in (0, oo), and for all k > 1 we have

uk(r) < uk(R) < L(R)

for all 0 < r < R,

(3.11)

(3.12) vk(r) < b + f (L(R)) B(R) for all 0 < r < R. Thus, there exists L := limj _,, L(R) E (0, oo] and the sequences (uk(r))k>1, (vk(r))k>1 are bounded from above on bounded sets. Therefore, for all r > 0 we

Entire solutions blouring up at infinity for elliptic systems

78

can define u(r) := limk.,,uk(r) and v(r) := limk-0 Vk (r). Passing to the limit in (3.7) we obtain that (u, v) is a solution of (3.1) with the central value (a, b). (i) Assume that p and q fulfill the condition (3.6). By Remark 3.2.2 this also

implies that A := limr_,,, A(r) < oo and B := limn. B(r) < oo. Let (u, v) be a radially symmetric solution of (3.1) with the central value (a, b). This yields

u(r) < a + g(v(r))A(r) v(r) < b + f (u(r)) B(r)

for all r > 0, for all r > 0.

u(r) < a + g(b + Bu(r))A

for all r > 0.

(3.13)

Hence,

Therefore,

1<

a, u(r)

+

g(b + Bu(r)) u(r)

A

for all r > 0

(3

.

.

4)

Now, if u is unbounded we deduce that u(r) - oo as r -> oo. Passing to the limit in (3.14) with r -> oo, we obtain a contradiction. Hence, u is bounded and, using (3.13), v is also bounded in (0, oo).

(ii) Assume now that condition (3.5) holds. In this case, limr.,, A(r) _ limr.,,.,, B(r) = oo. Let (u,v) be a radially symmetric solution of (3.1) with central value (a, b). For all r > 0 we have u(r) > a + g(b) A(r)

and v(r) > b + f(a) B(r).

Letting r -4 oo we derive that both u and v blow up at infinity. This concludes El the proof of Theorem 3.2.1. Some examples of nonlinearities f and g satisfying the assumptions of Theorem 3.2.1 are stated here:

(i) Let aj, bk, aj, 13k > 0 and a/3 < 1, where a = maxi<j<1 aj and f3 = maX1
At) =

0

if t<0,

0

if t<0,

ajt"i

if t > 0, and g(t) _

bkt)'

if t > 0-

j=1

(ii) f (t) = (1 + t2)a/2 and g(t) = (1 + t2)a/2 for t E R, with a, 6 > 0 and

a,3<1. (iii) Let a, 3 > 0 such that a/3 < I and define

if t<0,

0

if t < 0,

t"

if 0 < t < 1, and g(t) =

t'0

to

if t > 1,

to

if 0 < t < 1, if t > 1.

0

At) =

Charactenzation of the central value set

79

(iv) Letg(t) =t for tER, f(t)=0fort<0and f (t) = t [ - In ((..) arctan t) ) a

fort > 0,

where a e (0, 1/2). Concerning the bounded radially symmetric solutions of (3.1), we establish the following auxiliary result. Proposition 3.2.3 Assume that condition (3.6) holds and let f and g be locally Lipsschitz functions on (0, oo) that fulfill (3-4). Let (u, v) and (ii, v) be two bounded

radially symmetric solutions of (3.1). Then there exists a positive constant C such that for all r > 0 there holds max {Iu(r) - u(r)I, Iv(r) - v(r)I} < C max{Iu(0) - u(0)I, Iv(0) - v(0)I}.

Proof Set K := max {Iu(0) - u(O)I, Iv(0) - v(0)I}. From the first equation in (3.3) we find

r

rl-N

u'(r) - W(r) =

sN-lp(s)(g(v(s))

J0

- g(v(s))) ds.

Hence,

It

Iu(r) - ii(r)I < K + J r tl-N It sN-lp(s)Ig(v(s)) - g(v(s))1,dsdt. o

(3.15)

Because (u, v) and (ii, v) are bounded, for any r > 0 we have Ig(v(r)) - 9(v(r))I

mlv(r) - v(r) 1,

If (u(r)) - f (u(r))I

mIu(r) - u(r)I,

where m denotes a Lipschitz constant for both functions f and g. Therefore, by (3.15) we find

Iu(r) - u(r)1 < K + m

JJr

tl-N

ft sN-lp(s)Iv(s) - v(s) I ds dt,

(3.16)

for all r > 0. In the same manner we obtain pr

Iv(r) - v(r)I < K + m J

ft

tl-N

sN-lq(s) Ju(s) - ii(s)I dsdt,

0

o

for all r > 0. Define next t

X(r) := K + m J r tl-N o

Y(r) := K + m

f

sN_lp(s)Iv(s) - v(s)I dsdt,

JO

tl-N r sN-lq(s) lu(s) - ii(s) I 0

Clearly X and Y are nondecreasing functions and X(0) = Y(0) = K.

(3.17)

Entire solutions blowing up at infinity for elliptic systems

80

A simple computation together with (3.16) and (3.17) leads us to

(rN-1X')'(r) = mrN-lp(r)Iv(r) - v(r)j < mrN-lp(r)Y(r), (rN-lY')'(r) = mrN-lq(r)I u(r) - fl(r) ` C mrN_1q(r)X (r),

(3.18)

for all r > 0. Because Y is nondecreasing, we have

X (r) < K + mY(r)A(r)

Y(r) r tp(t) dt J0 < K + mCpY(r) for all r > 0,

< K + Nm

where

Cp =

(3.19)

2

N1 2 J

00

tp(t) dt. 0

Using (3.19) in the second inequality of (3.18) we find

(rN-lY')'(r) < mrN-1q(r)(K + mCpY(r))

for all r > 0.

Integrating twice from 0 to r in the previous inequality we obtain

Y(r) < K(1 + mCq) +

m2

N-2

r

tq(t)Y(t) dt

Cp

where Cq =

N

for all r > 0,

0

2

/ 0

'

tq(t) dt.

Next we apply the Gronwall inequality in the following form: assume that 0 < T < +oo and let cp, a, 6 : [0, T) -> Ilk be continuous functions, with a nondecreasing and /3 > 0. If $(s)cp(s)ds

for all t E [0,T),

W(t) < a(t) exp (fo J3(s)ds/ t )

for all t E [0, T).

W(t) < a(t) +

J 0t

then

Thus, by Gronwall's inequality, we deduce that for all r r> 0, Y(r) < K(1 + mCq) exp

- 2)-lCp f

r tq(t) dt 1

(3.20)

< K(1 + mCq) exp (m2CpCq) ,

and similarly for X. The conclusion now follows from (3.20), (3.16), and (3.17). This finishes the proof.

Characterization of the central value set 3.2.2

81

Role of the Keller-Osserman condition

We assume in this section that f and g satisfy the stronger regularity f, g E C1 [0, oo). We drop assumption (3.4) and we require in turn that f and g fulfill

(Hl) f (0) = g(O) = 0 and there exists a := lim inft-,,. L (H2) g satisfies the Keller-Osserman condition dt

(°°

J1

/G --(t-)

where G(t) :=

< cc,

rt

J

> 0;

g(s) ds.

Remark that assumptions (Hi) and (H2) imply that f satisfies condition (H2), too. Set'v := max{p(0),q(0)} > 0 and g := min {p,q}. Throughout this section we assume that v > 0, rl is not identically zero at infinity, and condition (3.6) on p and q holds. Before the main result is stated, we provide several auxiliary results concerning the set of central value. Lemma 3.2.4 G ,-E 0. Proof Notice that p + q satisfies the assumptions (pl )' and (p2) in Section 2.2. Thus, by Theorem 2.2.3 and the fact that p and q are radially symmetric, there exists a radially symmetric solution rfi E C0 ,(RN) of the problem

d _ (p + q)(x)(f + 9)(V)

in RN,

0>0

inR

-0 = 00

as IxI - oo.

,

Hence, ib satisfies the integral equation

ON =i/(0)+ (rtl-N f s'-'(p'+ q)(s)(f+g)('+/)(s))dsdt, o

00

for all r > 0. We claim that (0, r/,(0)] x (0, V,(0)] C G. To this aim, let us fix 0 < a, b < g(0). Define the sequences (uk)k>1 and (vk)k>o by vo(r) := b for all

r>0and

t

uk(r) = a+ (

in

tl-N f sN-lp(s)9(vk-I (s)) ds dt,

t jr tl-N J sN-lq(s) vk(r) = b + f ((s)) ds dt, fffo

Jo

(3.21)

f

o

for all r > 0 and k > 1. We first remark that vo < v1, which produces u1 < u2. Consequently, vi < v2i which further yields u2 _< u3. With the same arguments, we obtain that (uk)k>1 and (vk)k>o are nondecreasing sequences. Because

jb"(r) > 0 and b = vo < '(0) < ?P(r) for all r > 0, we find

Entire solutions blowing up at infinity for elliptic systems

82

ul (r) < a+

J0

r

(0) +

ti-N f isN-lp(s)g(b(s)) ds dt

jt1j 0

s(p+q)(s)(f +g)((s))dsdt

=,O(r)Thus, ul < 9fi. It follows that i

vl(r) < b + jo r ti-N

J0

sN-iq(s)f ('(s)) ds dt

«(0) + fof r ti-N f sN-i(p+ q)(s)(f +9)(i(s)) dsdt 0

= O(r).

A further induction argument yields

uk(r) < b(r) and vk(r) < O(r)

for all r > 0 and k > 1.

Thus, {(uk, Vk)}k>i converges and (u(r), v(r)) := limk-0(uk(r), vk(r)), r > 0 is a radially symmetric solution of (3.1) with central value (a, b). This completes the proof. A direct consequence of Lemma 3.2.4 is the following corollary.

Corollary 3.2.5 If (a, b) E 9, then (0, a} x (0, b) C g.

Proof The process used before can be repeated by taking uk(r) = ao + vk(r) = b0 +

J0

r ti-N r ti_ N

J0

t SN-ip(s)g(vk_i(s)) ds dt

for all r > 0,

t $N-iq(s) f (u (S)) ds dt

for all r > 0,

J0 J0 where 0
Now let (U, V) be a radially symmetric solution of (3.1) with central value (a, b). As in Lemma 3.2.4, we obtain

uk(r) <'uk+i(r) < U(r)

for all r > 0 and k > 1,

vk(r) < vk+i(r) < V(r)

for all r > 0 and k > 1.

Thus, for all r > 0 we can define (u(r),v(r)) := limk__,,(uk(r), vk(r)). Then u < U, v < V on 10, oo) and (u, v) is a radially symmetric solution of (3.1) with the central value (ao, bo). This shows that (ao, b0) E 9, so that our assertion is proved.

Lemma 3.2.6 G is bounded.

Characterization of the central value set

83

Proof Set 0 < A < min {a, 1} and let 6 = 6(A) be large enough so that

f(t)>)tg(t)

for allt.>J.

(3.22)

Because r1 is radially symmetric and not identically zero at infinity, we can find an open ball BR centered at the origin and of radius R > 0 such that rj > 0 on BBR. Furthermore, according to Theorem 2.2.2, there exists ( E C2(BR) such that

in BR,

L1( = )17(x)g (2}

>0

in BR,

(--+ oo

a s l x l -* R.

Arguing by contradiction, let us assume that G is not bounded. Then there exists (a, b) E G such that a + b > max {26, ((0) }. Let (u, v) be a radially symmetric solution of (3.1) with the central value (a, b). Because u(x) + v(x) > a + b > 26 for all x E RN, by (3.22) we find

f (u(x)) > f

(U(X)

v(x)!

> Ag { u(x)

2

and

g(v(x)) > g [

u(x)

v(x))

2

v(x))

if u(x) > v(x)

v(x)

if v(x) > u(x).

2 > Ag (tz(x)

2

It follows that d(u + v) = p(x)g(v) + q(x)f(u)

>'1(x)(g(v) + f(u))

> \n(x)g (u 2 v)

in RN.

On the other hand, ((x) -f oo as jxj -> R and u,,v E C2(RR). Thus, by the weak maximum principle, we conclude that u + v < Sin BR. But this is impossible O because u(O) + v(0) = a + b > ((0). This finishes the proof.

Lemma 3.2.7 F(G) C G. Proof Let (a, b) (=- F(G) and no > 1 be such that no min {a, b} > 1. We claim

that

(a - 1/n, b- 1/n) E G

for all n> no.

Indeed, if this is not true, by Corollary 3.2.5 we find

D := (a - 1/n, oo) x [b - 1/n, oo) C (R+ x R+) \ G, for some n > no. Hence, we can find a small ball B centered in (a, b) such that B CC D; in other words, B n G = 0. But this contradicts the choice of (a, b).

84

Entire solutions blowing up at infinity for elliptic systems

Consequently, for all n > no there exists (un, vn), a radially symmetric solution of (3.1) with the central value (a -1/n, b -1/n). Thus, for any n > no and r > 0 we have

+ J ti_N J

un(r) = a -

t

sN-1p(s)9(vn(s)) ds dt,

r0

vn(r) = b -

n

+

Jo

r r tl-N J t sN-1 q(s)f (un(s)) ds dt. 0

We observe that (un)n>n0 and (vn)n>n0 are nondecreasing sequences. Next, we prove that (un)n>n0 and (vn)n>n0 converge in RN. To this aim, let xo E RN be arbitrary. Because t is not identically zero at infinity, we may find R > 0 such that x0 E BR and rl > 0 on MR. Because o, = lim inft.,,,, f (t)/g(t) > 0, we find T E (0,1) such that

f (t) > Tg(t),

-

b f o r all t > a +

1

no

2

Therefore, on the set where un > vn we have 2 f(un)>f un+vn\

Tg(un+yn\ 2

lJ

Similarly, on the set where un < vn we have

>Tgfl\ un+yn 9(vn)>9 un+ynl 2 ) 2 /J Now it is easy to see that for all x E l[PN we have ©(un + vn) > Tr](x)g

Un + vn 1 2

On the other hand, by Theorem 2.2.2 there exists (E C2(BR) such that Ot; = TTl (x)9 G

(>0 (

oo

J

in BR, in BR, as IxI -r R.

The weak maximum principle yields un + vn < ( in BR. So, it makes sense to define (u(xo),v(xo)) := limn-,,(un(xo),vn(xo)). Because xo is arbitrary, the functions u, v exist on RN. Hence, (u, v) is a radially symmetric solution of (3.1) with central value (a, b)-that is, (a, b) E G.

For (c, d) E (R+ x R+) \ 9, denote by R,,d the supremum over r > 0 such that there exists a radially symmetric solution of (3.1) in B(0, r) so that (u(0), v(0)) _ (c, d).

Characterization of the central value set

85

Lemma 3.2.8 For all (c, d) E (R+ x R+) \ 9 we have 0 < Rc,d < 00-

Proof Because v > 0 and p, q E C[0, oo), there exists E > 0 such that (p + q)(r) > 0 for all 0 < r < E. Let 0 < R < E be arbitrary. Hence, there exists a positive radially symmetric solution of

ay'R = (p+q)(x)(1 + g) (OR) in BR, which blows up as jxj -+ R. Furthermore, for any 0 < r < R we have r

t

f s'-'(p + q)(s)(f + 9)(OR(s)) ds dt.

OR (r) = OR(0) + r tl-N 0

Obviously, 0R > 0. Thus, we find

0R (r) = rl-N J r sN_1 ('+ q)(s)(f + 9)(OR(s)) ds < C(f + g)(OR(r)), 0

where C > 0 is a positive constant such that f, ,(p + q)(s) ds < C. Because f + g satisfies the hypotheses (H1) and (H2), by Lemma 2.2.1 (ii) we derive. °°

dt

< oo.

J1

(f + 9)(t)

Therefore,

d

_

ds

dr +Grz(r) (f + 9)(s)

OR jr)

(f + 9)(')R(r))

C

for all 0 < r < R.

Integrating in the last inequality and taking into account the fact that VR(r)

oo as r / R, we obtain ds fo""O(O)

(f + 9)(s)

< CR.

Now, we let R \ 0 in the previous relation and we have ds lint W =0 R%04"'(0) (f + g)(s)

This implies that l'R(0) -+ oo as R \ 0. Thus, there exists 0 < p < e such that

0
fr

vk(r) = d +

0 t

ti_N j sN-lq(s) f (up(s)) dsdt, 0

(3.23)

86

Entire solutions blowing up at infinity for elliptic systems

for all r > 0. As in the proof of Lemma 3.2.4 we deduce (uk)k>1, and (vk)k>o are nondecreasing, and for all k > 1 there holds max{uk(r),vk(r)} < Op (r)

for all 0 < r < p.

Thus, for any 0 < r < p there exists (u(r),v(r)) := limk_,,,,,(uk(r),vk(r)) that, moreover, is a radially symmetric solution of (3.1) in Bp such that (u(0),v(0)) _ (c, d). This shows that R6,d > p > 0. By the definition of R,,d we also derive lim u(r) = oo

lim v(r) = oo.

and

r/&,d

rJ'Rte,d

(3.24)

we conclude that Rc d is finite. This

On the other hand, because (c, d) completes the proof.

U

The main result in this section is the following.

Theorem 3.2.9 Assume that v > 0, rl is not identically zero at infinity and (Hi), (H2), and (3.6) hold. Then any entire radially symmetric solution (u, v) of (3.1) with (u(0), v(0)) E F(CQ) blows up at infinity.

Proof Let (a, b) E F(G). By Lemma 3.2.7, (a, b) E G so that there exists (U, V), a radially symmetric solution of (3.1) with (U(0), V(0)) = (a, b). Obvi-

ously, for any n > 1, (a + 1/n, b + 1/n) E (R+ x R+) \ 9. By Lemma 3.2.8, Rn := Ra+1/n,b+l/n is a positive number. Let (Un, Vn) be the radially symmetric

solution of (3.1) in BR, with the central value (a + 1/n, b + 1/n). Thus, for all 0 < r < R, (Un,V,,,) is a solution of the system of integral equations r

U(r) = a +

t

+ Jl t1I SN-1p(S)g(Vn(s)) dsdt, n °jr

V(r) = b +

+

r(3.25)

t1-N J sN-1q(s)f (Un(s)) ds dt. o

In view of (3.24), we have

lira Un(r) = lim VV(r) = oo

rZRn

TZRn

for all n > 1.

We claim that the sequence (Rn)n>1 is nondecreasing. Indeed, if (uk)k>1 and (vk)k>1 are the sequences of functions defined by (3.23) with c = a + 1/(n + 1)

andd=b+1/(n+1), then for all k > 1 and 0 < r < Rn we have uk(r) < uk+1(r) < U,(r),

vk(r) < vk+1(r) < V,(r).

(3.26)

This implies that (uk(r))k>1 and (vk(r))k>1 converge for any 0 < r < R. vk) is a radially symmetric solution of (3.1) in BR with the central value (a+1/(n+ l ), b+l/(n+l)). By the definition of Rn+1, it follows that Rn{1 > Rn for any n > 1. Moreover, (Un+1, Vn+1) :=

Characterization of the central value set

87

Set R := Rn and let 0 < r < R be arbitrary. Then, there exists nt .n1(r) such that r < Rn for all n > n1. From (3.26) we derive that Un+1 <_ Un V. on 10, Rn), for all n > n1. Thus, for all 0 < r < R we can define and Vn+1

U(r) := lim Un(r) n-.oo

V(r) := Iim Vn(r). n-roo

and

(3.27)

Because UU (r) > 0; from (3.25) we find

Vn(r) < b + _ + f (Un(r)) fo"O t1-N r tsN_lq(s) ds dt. n

0

This yields

for all 0 < r < R,

VV(r) < C1Un(r) + C2 f (Un(r))

(3.28)

where C1 is an upper bound of (V(0) + 1/n)/(U(0) + 1/n) and C2 =

f 0o t1-N [ sN-lq(s) dsdt <

1

t

N_2

f

00

tq(t) dt < oo.

Define h(t) := g(C1t + C2 f (t)), t > 0. It is easy to see that h satisfies the hypotheses of Lemma 2.2.1. Hence, we may define

r(t) = /

forallt>0.

ds)

Notice that Un verifies

AU, = p(x)g(Vn)

in BR.,

which combined with (3.28) implies AU,, < p(x)h(UU)

in BR,,.

A simple computation yields

Ar(Uu) = r`(Uu)AUu + rl"(Un)IvUnl2 h(UU)

AU,

IVUn12

+ (h(Un))2

> h(I p(x)h(Un)

= -p(x)

in BR,,.

Therefore,

[ rN-1 L r(Um)

Ir

>

-rN-ip(r)

for any 0 < r < R,

88

Entire solutions blowing up at infinity for elliptic systems

Fix 0 < r < R. Then r < Rn for all n > n1, provided n1 is large enough. Integrating the previous inequality from 0 to r we deduce

drr(Un) > -r 1-N J0r sN-1p(s)ds

for all 0 < r < Rn.

Integrating the last inequality over [r, Rn - s] we obtain

-e

s)) - r(Un(r)) > -

Jr

ti-N f t sN-lp(s) dsdt

for all n > nl.

0

Recall that UU(Rn - s) -> oo as s --+ 0 and r(t)

0 as t -+ oo. Therefore, passing to the limit with s -> 0 in the last inequality we deduce that rRrt

r(Un(r)} _<

t1-N

Jr

It

sN-lp(s) ds dt

for all n > ni.

0

By letting n - co in the previous relation, we obtain R

r(U(r)) < f t1-N fo s N-lp(s) ds dt

for all 0
r

U(r) > r-1 (J R tl-N f t sN-lp(s) ds dt) r

for all 0 < r < R.

0

Passing to the limit as r / R, and using the fact that limt,,o r-1(f.) = oo, we deduce R

lim U(r) > lim r-1

r/R

r/R

J$Np(s) ds dt = oo.

t1-N

r

But (U, V) is an entire solution of (3.1), so that we conclude R = oo and limr__,,,, U(r) = oo. Using (3.6) and the fact that V'(r) > 0 we find 00

U(r) < a + 9(V (r)) fo t1-N f'SN- ip(s) ds dt

< a + g(V(r)) N

1

2

J0

tp(t) dt,

for all r > 0. This last estimate leads to limr_. V (r) = oo. Consequently, (U, V) 0 is an entire blow-up solution of (3.1). This concludes the proof.

Comments and historical notes

89

3.3 Comments and historical notes We have provided in this chapter a characterization of the central value set of radially symmetric solutions to a nonlinear elliptic system of logistic type. Our study has pointed out the role played by the Keller-Osserman condition in this characterization. First, if the nonlinear terms f and g have almost sublinear growth, then the central value set consists of entire R+ x R+, and the existence of radially symmetric solutions blowing up at infinity depends on the growth of potentials p and q. In turn, if f and g are comparable, in the sense that lim inft, f(t)/g(t) E (0, oo), and satisfy the Keller-Osserman condition at infinity, then the central value set is bounded. Furthermore, any element on the positive boundary of c-that is, on 8q fl (R+ x R+),--is a central value of an entire blow-up solution. The results in this chapter are the work of Cirstea and Radulescu [52]. We also refer to Lair and Shaker [126] for the particular case of pure powers in the nonlinearities.

In Part II of this volume (Chapters 2 and 3) we have been concerned with boundary blow-up solutions of some classes of nonlinear elliptic equations and systems. Our approach was essentially based on the maximum principle, in combination with other ingredients, such as elliptic estimates, regularity arguments, and differential equations techniques. In all the results we have established in these two chapters, we studied only positive solutions, especially because of the physical meaning of the corresponding unknowns. A different approach was developed by Aftalion and Reichel [1], who argued the existence of multiple boundary blow-up solutions of the problem

Au = f (u)

in 1?,

where Q is a bounded and convex domain. Taking into account the growth of f, they distinguished two distinct situations: (i) f is a sign-changing nonlinearity. In such a case there is both a positive and a sign-changing blow-up boundary solution. (ii) inf f > 0. In this case, Aftalion and Reichel [1] considered the bifurcation problem .

Au = A f (u)

in cZ

(3.29)

and showed that there is some critical value A* > 0 such that problem (3.29) has blow-up boundary solutions if and only if 0 < A < A*.

PART III

ELLIPTIC PROBLEMS WITH SINGULAR NONLINEARITIES

4

SUBLINEAR PERTURBATIONS OF SINGULAR ELLIPTIC PROBLEMS In mathematics the art of proposing a question must be held of higher value than solving it. Georg Cantor (1845-1918)

From now on, we are interested in the qualitative analysis of solutions to semilinear elliptic equations or systems involving singular nonlinear terms, such as u_a (with a > 0) or gradient terms like IVula (with 0 < a < 2). We are concerned with existence and uniqueness properties, but also with the asymptotic behavior of solutions. A special feature will be played by the influence of one or several real parameters, which usually are referred as bifurcation parameters. The term bifurcation is one that is also used in topological dynamics and catastrophe theory. In such cases, it is usually considered a family of functions or vector fields dependent on parameters. The associated bifurcation values of parameters are those in which the system is not actually stable, in the sense that small variations of the parameters change the topological nature of the collection of orbits. Our setting in this volume is quite closely related to the study of stability phenomena, by means of the first eigenvalue of the associated linearized operator. Singular problems arise in the study of non-Newtonian fluids, boundary laver phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrically conducting materials. The associated singular stationary or evolution equations describe various physical phenomena. For instance, superdiffusivity equations of this type have been proposed by de Gennes [60] as a model for long-range Van der Waals interactions in thin films spreading on solid surfaces. Singular equations also appear in the study of cellular automata and interacting particle systems with self-organized criticality (see [39]), as well as to describe the flow over an impermeable plate (see 138]). 4.1 Introduction This chapter is devoted to the study of elliptic problems of the following type:

1-Du = 4)(x, u, A)

in Sl,

u>0

in D,

U=0

on %I

(4.1)

94

Sublinear perturbations of singular elliptic problems

in a smooth domain Sl C RN (N > 1), where : Si x (0, oo) x 1W -+ R is a Holder continuous function. The main feature of this chapter, as well as of the following

ones, is that we allow 4i to be unbounded near the origin with respect to its second variable-that is, 4P is a singular nonlinearity. Therefore, the singular character of our problem lies not in the prescribed behavior of the solution at the boundary, as we required in Chapters 2 and 3, but in the nonlinearities that govern problems of type (4.1). To make these arguments more transparent, let us present a simple example. Consider the problem in Q, I Du = up

u>0

in 9,

U=+00

on 992.

If p > 1, by Theorem 2.1.3, the previous problem has a solution u. By the change of variable u = 1/v we derive that v satisfies

-AV = v2-p -

2 1ov12

in 9,

v>0 v=0

in Q,

on00.

This equation contains both singular nonlinearities, like v-1 or v2-p (if p > 2), and a convection (gradient) term, denoted by 1 Vv12. The influence of the gradient term in problems like (4.2) will be emphasized in Chapter 9.

By classical solution of problem (4.1) we mean a function u E C2(S1) n C(SC) that satisfies pointwise (4.1). Because of the presence of the singular term D (x, u, A), solutions to (4.1) are not C2(SZ) functions. However, in some particular cases, which will be discussed herein, the regularity of classical solutions to (4.1) can be improved up to the class u E C1'7(SZ) and Au E L1(Sl). The simplest model that falls within the theory we develop in this chapter is

Au = g(u) u>0 u=0

in Q,

in 1, on BSl,

where 1 C Rl'(N > 1) is a smooth bounded domain and g c- C1(0, oo) is a decreasing function such that limt\o g(t) = no. Existence of a classical solution to (4.3) follows directly from Theorem 1.2.5. In particular, if SZ = (0, 1) C IW then there exists a unique y E C2(0,1) n C[O,1] such that -y"(t) = g(y(t)) 0 < t < 1, y(t) > 0 0 < t < 1, y(0) = y(1) = 0.

Introduction

95

A surprising result is that boundary estimates for the solution it of (4.3) in higher dimensions are expressed in terms of y. More exactly, setting d(x) dist(x, &S2), we have the following result.

Theorem 4.1.1 Problem (4.3) has a unique solution it E C2(11) n C(a) and there exist c, m, M > 0 such that my(cd(x)) < u(x) < My(cd(x))

for all x E Sl,

(4.5)

where y is the unique solution of (4.4). Moreover, if g E L1(0, 1), then the solution of problem (4.3) verifies cid(x) < u(x) < c2d(x)

for all x E 9

(4.6)

for some positive constants ci and c2.

Proof Because y is concave, there exists y'(0+) := limt\o y(t) E (0, oo]. Thus, we can find 0 < a < 1 such that y' > 0 in (0, a). That is, y is strictly increasing on (0, a). From Corollary A.1.3 in Appendix A, there exist two positive constants, Ci and C2, such that Cicpi(x) < d(x) < C2Soi(x)

in Q.

Let us take c > 0 such that cC2coj < a/2 in Q. We first prove that there exists M > I large enough such that u := My(ccpi) is a supersolution of (4.3). Indeed, we have

-Llu = Mc2]DVij2g(y(ccpi)) +)iMcpiy'(ccpi)

in 11.

By virtue of the strong maximum principle, there exist 11o CC S2 and 6 > 0 such that (4.7) IVcpi1>6 inc\SZ0. Thus, we can choose M > 1 such that Mc]Qcpil > 1

in SZ \ 11

(4.8)

.

Therefore,

-Au > g(y(ccpi)) ? g(My(ocpi)) = g(u)

in S2 \ Slo.

(4.9)

Because cpiy'(ccpi) is bounded away from zero in 110, we can take M > 1 such

that Mcaicpiy'(cpi) > g(y(cpi))

in Q o.

(4.10)

Hence, (4.11) -©u > g(y(coi)) ? g(My(ccPi)) = g(u) in 1o. From relations (4.9) and (4.11), it follows that u = My(cpi) is a supersolution of problem (4.3), provided M > 1 satisfies (4.8) and (4.10). In a similar way

Sublinear perturbations of singular elliptic problems

96

we deduce the existence of a constant 0 < m < 1 such that u := my(WI) is a subsolution of (4.3). Thus, u < u < u in S2 and (4.5) follows.

We claim that if g E L'(0,1) then y'(0+) < oo and y'(1-) < oo-that is, y E C1[0,1]. To this aim, we multiply by y'(t) in (4.4) and we integrate over [e, t], where 0 < e < t < 1. We find -y,2(t) + y/2(E) = 2

rt JE

V(t)

g(y(s))y'(s)ds = 2

J y(E)

g(r)dr,

for all 0 < e < t < 1. Because f0 g(r)dr < oo, we can let E --- 0 in the previous equality and we obtain y'(0+) < oo. In the same manner we derive y'(1-) < oo. Hence, there exists cl, c2 > 0 such that

cit
for all 0 < t < 1. Using the previous estimate in (4.5) we easily deduce (4.6). The proof is now complete.

Estimate (4.5) allows us to say more about the regularity of the solution to

(4.3). To illustrate this matter, we consider the case g(s) = s-', a > 1. Thus, problem (4.3) becomes

1-Au=u «

2u>0

inc,

in 9, u=0

(4.12)

on OSl.

Theorem 4.1.2 For all a > 1, problem (4.12) has a unique solution u E C2 (Q) fl C(fl). Moreover, we have

(i) u C1 (St); (ii) uEHH(St) if and only if a<3. Proof (i) Existence follows from Theorem 1.2.5 (i). Note that if a > 1 then y(t) =

(2 + a) 2 2/(1+a) t2/(1+a)

for all 0 < t < 1,

(2(a - 1)

is the unique solution of (4.4). Therefore, by (4.5) we have cjco2/(1+a)

< u < c2(pi/(1+a)

in Ii

(4.13)

Fix x0 E aSZ and let n be the outer unit normal vector on 8Sl at x0. Using (4.13) we have u(xo + tn) - u(xo) au t an(x0) = tfo ` 1(xo + tn) - cpi(x0) (1-a)/(1+a) .

< c1 lim

tfo

-oo. Hence, u Cl(S2).

(Pi

(x°+ tn)

Introduction

97

(ii) Assume first that 1 < a < 3. Using Theorem 1.2.5, the approximated problem

-Au = (u + 1/k) I

in S2,

u>0 u=0

in Q,

on 80

has a unique solution Uk E C2(St) for all k > 1. It is easy to see that uk < uk+1 << u

in Q.

(4.14)

Moreover, Vk := Uk + 1/k verifies

-Avk = vk

in S2,

vk>0

in Q, on OP.

vk = 1/k Hence,

1 1 uk+1 > uk+1+k-I-1 >- u

(4.15)

in Q-

Therefore, by (4.14) and (4.15) we obtain

J IVukI2dx = -

r ukAukdx = f uk(uk + 1/k)`'dx <

u1-adx.

By (4.13) and Corollary A.1.3 in Appendix A we have J

IVuk12dx < c (

V1

2(a-1)/(a+l)dx

< 00.

Hence, (uk)k>1 is bounded in Ho (S2). Then, passing to a subsequence we see that (uk)k>1 is weakly convergent in HH(Sl) to some v E HH(Sl). Because (uk)k>1 converges pointwise to u in Q, we conclude that u = v E H o (Il). Assume now a > 3 and that problem (4.12) has a solution u E Ho (Q). Taking

into account estimate (4.13) and the fact that 2(a -1)/(a+l) > 1, by Corollary A.1.3 (ii), it follows that

fc

ul-adx = oo.

(4.16)

Let (Wk)k>1 C CC (S1) be such that wk -* u in Hol(Sl) as k --> oo. For each k > 1 set wk = max{wk, 0}. Then (wk)k>1 C Hfl (f1) and Vwk = VwkX{,Uk>o} . Passing to a subsequence if necessary, we may assume that (wk )k>1 converges to u almost everywhere in Q. By (4.16) and Fatou's lemma we find

wk u-adx = oo.

lim

k--oo

SE

Sublinear perturbations of singular elliptic problems

98

Then,

f Vu Vwtdx =

- f wudx =

/ wka-adx.

It follows that

L

IVuf 2dx

= lim J Vu Owk dx = oo, k-oo

which is a contradiction. Hence, u V Ho (1) and the proof of Theorem 4.1.2 is now complete.

4.2 An ODE with mixed nonlinearities To see more clearly the nature of the problems we are dealing with, let us consider the following ODE:

I -y"(t) = y(t) + Y, (t) y(t) > 0

- 1 < t < 1, - 1 < t < 1,

(4.17)

Y(-') = y(1) = 0, where 0 < a, p < 1. The existence of a solution to (4.17) follows by virtue of Theorem 1.2.5. To achieve the uniqueness, we effectively construct a solution y of (4.17) such that y" E L1 (-1, 1). Then, by Theorem 1.3.17, we derive that y is the unique solution of (4.17). We assume that y' is odd, which yields y'(0) = 0

and y(-t) = y(t), for all -1 < t < 1. Multiplying by y'(t) in (4.17) and then integrating over [0, t], 0 < t < 1, we obtain y(0)

(y')2(t) = 2 jy(t) f (s) ds

for all 0 < t < 1,

(4.18)

where f (t) = t-a+tP. Because fo f (t) dt < oo, it follows that y'(1-) = limt\l y'(t) is finite. Similarly, we obtain y'(-l+) E ]R so that y E C'[-l, 1]. This yields

cl(1-It])
for all -1
where cl and c2 are positive constants. Combining the last inequality with (4.17) we deduce that y" E Ll(-1, 1) and the uniqueness follows by Theorem 1.3.17. Actually, we shall see in the next section that y E C2(-1, 1) n C1,1-a[-1, III.

Let c = y(O) > 0 and F(t) = ft f (s)ds, 0 < t < c. From (4.18) we have

y'(t) = - 2F(y(t))

for all 0 < t < 1.

It follows that

y'(t) =1 2F(y(t)) An integration in (4.19) yields

for all 0 < t < 1.

(4.19)

A complete description for positive potentials

for all 0 < t < 1,

t

99

(4.20)

where

{t)= Ic

1

ds 0
2F(s) Notice that 91 [0, c] - [0, W(0)] is bijective and (4.20) holds for all 0 < t < 1. In this way, the unique solution y of (4.17) has the form t

:

y(t)

if -1 < t < 0,

- (t)

if0
4.3 A complete description for positive potentials Motivated by the study of (4.17) we now consider the following more general boundary value problem:

-Du = p(x)g(u) + of (x, u) + ltk(x) u>0

u=0

in SZ,

in S2,

(4.21)

on W,

where S2 is a smooth bounded domain in RN (N > 1) and A, p > 0. We assume that p : S2 - (0, oo) and f : SZ x [0, oo) -> 10, oo) are Holder continuous functions

such that f > 0 on a x (0, oo). We also assume that f is nondecreasing with respect to the second variable and is sublinear. That is, (f 1) the mapping (0, oo) E) t --> 1-(' t)

(f2) L

o

is nonincreasing for all x E S2;

f ( , t) = oo and tltrn f ( , t) = 0 uniformly for all x E S2.

t t We suppose that g E C'(0, oo) is a nonnegative and decreasing function satisfying

(gl) lirnt\o g(t) = oo, (g2) there exists 0 < a < 1 such that g(t) = O(t-a) as s \ 0. Also we assume that k : St -4 (0, oo) is a Holder continuous function that satisfies infrE-a k(x) > 0 and

(kl) there exists 0 < 13 < I such that k(x) = O(d(x)-p)

as d(x) \ 0.

Obviously problem (4.17) is a particular case of (4.21), because f (x, t) = tP and g(t) = t-a, 0 < p, a < 1 satisfy (f 1) - (f 2) and (g1) - (g2) respectively. As we have already argued in the beginning of this chapter, we cannot expect to find solutions in C2 (1i). Define

E_ {u E C2(st) n Cl-ry(si) : Au E L'(sl)}, where -y = max{a, /3}. Here, and in the rest of this chapter, we are looking for solutions of (4.21) in the class E.

100

Sublinear perturbations of singular elliptic problems

Remark 4.3.1 Let u e C2(f) n C(U) be a classical solution of (4.21). Then Au E L1(1) if and only if p(x)g(u) E L'(SZ). The main result of this section is stated next. Theorem 4.3.2 Assume that p > 0 in SZ and (f 1) - (f 2), (gl) - (g2), (kl) hold. Then, for all A, it > 0 problem (4.21) has a unique solution uau, E S. Moreover, u,\, is increasing with respect to A and A. For a fixed k > 0, the dependence on A of the unique solution ua = u,\,, is depicted in Figure 4.1.

(0,0)

A

FIGURE 4.1. The dependence on A > 0 in Theorem 4.3.2.

Proof For our convenience, we divide the proof into three steps. Step 1: Existence. Fix A, i > 0. Remark first that the result in Theorem 1.2.5 does not apply here directly, because the mapping T (x, t) = p(x)g(t) + Af (x, t) +,uk(x) is not defined on 8Il x (0, oo). By Theorem 4.1.1, there exists a unique solution v E C2(Q) n C(l1) of the problem

-Ov = g(v)

v>0 v=0

in 1, in D, on On.

Because g E L1(0,1), from (4.6) we also have

mid(x) < v(x) < m2d(x)

in S1,

(4.22)

for some m1, m2 > 0. Clearly, is := v is a subsolution of (4.21). To provide a supersolution, we need the following auxiliary result.

A complete description for positive potentials

101

Lemma 4.3.3 There exists a unique positive solution w E C2(1) fl C(SZ) of the problem

Aw = 2 + k(x)

in SZ,

W=0

on 89.

(4.23)

Moreover, w < cd(x) in 1, for some c > 0.

Proof Obviously w := 0 is a subsolution of (4.23). Consider the problem

1-Az = z-0

in 0,

z>0

in S1,

z=0

on BSZ.

Because 0 < 3 < 1, by Theorem 4.1.1, the previous problem has a unique solution z E C2(ST) fl C(SZ) such that

Cld(x) < z(x) < C2d(x)

in SZ,

(4.24)

for some Cl, C2 > 0. Hence, -Az > C2 3d(x)-13 in SZ. In view of the assumption

(kl) we can find M > 1 large enough such that -r,(Mz) > 2 + k(x) in Q. This means that w := Mz is a supersolution of (4.23). Hence, (4.23) has a solution w E C2 (Q) nC(ST) such that w < w < w in SZ. By the strong maximum principle,

w is the unique solution of (4.23). The fact that w < cd(x) in SZ follows from (4.24). This finishes the proof of Lemma 4.3.3. Now we can proceed to construct a supersolution of (4.21). Because f is sublinear, we have

lf c

0

00

uniformly for x E 0.

Thus, there exists M > I such that

M>.)f(x,Mw)

in Q.

(4.25)

Furthermore, by taking M > 1 large enough, we can assume that M > max f (x, Mv)

and M> max{1, p, IIpIIoo}.

(4.26)

XESt

We claim that ii := M(v + w) is a supersolution of (4.21) provided that M satisfies (4.25) and (4.26). Indeed, we have

-Du = Mg(v) + 2M + Mk(x) > p(x)g(v) + A(f (x, Mv) + f (x, Mw)) + pk(x) > p(x)g(M(v + w)) + Af (x, M(v + w)) + µk(x) = p(x)g(u) + Af (x, u) +,uk(x) in SZ. Hence, a is a supersolution of (4.21) and u < u in Q. By Theorem 1.2.3 it follows that problem (4.21) has at least one solution, uaµ E C2 (0) fl C(ST), such that

Sublinear perturbations of singular elliptic problems

102

u _< uau, < i in St. Using (4.22) and Lemma 4.3.3, there exist c1, c2 > 0 such

that cld(x) < ua,(x) < c2d(x)

in Q.

(4.27)

Step 2: Uniqueness and dependence on A and p. To achieve the uniqueness, let us first remark that by (4.27) and (g2) we have p(x)g(u,\u) < p(x)g(cid(x)) < Cd(x)-'

in St

for some positive constant C > 0. Thus p(x)g(uau) E L'(St), and by Remark 4.3.1 we derive that Au,,, E L1(Sl). The uniqueness of the solution follows now from Theorem 1.3.17. Let us obtain the dependence on A and lc. Fix p > 0 and Al > A2 > 0. Let ua,,, and ua2, be the solutions of (4.21) obtained for the pairs (Al, pc) and ('2, p), respectively. Then, by Theorem 1.3.17 for t) = p(x)g(t) + Alf(x, t) + p.k(x), we deduce uaZu < ua101 in S2. By virtue of the strong maximum principle we find UA,A < ua,A in Q. In a similar way we obtain the dependence on p. Step 3: Regularity of solution. We have already proved that the unique solution of (4.21) verifies u .,,, E C2(St) fl C(SQ) and Aua, E Ll(1). It remains to show that ua,,, E Cl,l_''(1). To this aim, let Q be the Green function associated with the Laplace operator in Q. We shall use the following estimates, which is the work of Widman [195]. Lemma 4.3.4 There exists a positive constant c > 0 such that for all x, y E S2, x # y we have min{lx - y1,d(y)} (i)

(ii) xx(x,y)

]x-y1N c

,

min{]x - yC>d(y)} Ix - y1N+1

Then u;kw(x) = - in C(x, y),P(y)dy z

for all x E 1,

where F(y) = a(y)g(ua,+(y)) + A(y, u(y)) + pk(y). Hence

_-

f

9x(x, y)4 (y)dy

for all x E Q.

(4.28)

A very useful tool in our approach is the following technical result.

Lemma 4.3.5 There exists co = co (Sl) > 0 and 8o = So(Q) > 0 such that for all xl, x2 E Q, 0 < Jxl - x21 < bo there exists a C' path [0,1] -> St with the properties

(i) e(0) = xl and 1;(1) = x2i (ii) W(t)I < ccIxl -x21, for all 0 < t < I.

A complete description for positive potentials

Fix x1i x2 E Q, 0 < Ixl - x21 < bo and let path in Lemma 4.3.5.

:

103

[0, 1] -* Q be the corresponding

By (4.28) we have

lVuar,(x1) - Vuau(x2)I 5 f I9x(xl,y) - 9x(x2,y)I'D(y)dy l

S

Icx(x1,y) - 9x(z2,y)I"D(y)dy

B ,-(xi)

+

(4.29)

f

lgx(x1, y) - 9x(x2, y)I -D(y)dy if

where r = (co + 1)lxi - x21 and co is the constant appearing in Lemma 4.3.5. Before evaluating I and II, let us remark that by (g2), (kl), and (4.27) there cld(y)_Y for all y E Q. Then exists cl > 0 such that fi(y) <

I < Ci

B ,(xi)

cl

l ggx(xl, y) - gx(x2, y) l d-7(y)dy IJCc(xl,y)ld-Y(y)dy + cl

fk(x2)

19.(x2,y)ld Y(y)dy,

where R = r + 1x1 - X21. Let y E B,-(xl). If d(y) > l xl - yl then, by Lemma 4.3.4 (i), we have -yl-N+1d-Y(y)

Igx(x1,y)Id-1(y)
< clxl

-yl-N+1-Y

If d(y) < lxl - y1 then, by Lemma 4.3.4 (i), we obtain IQlx(x1,y)l d-Y(y)

clxi

-

yl-Ndl-Y(y) < Clxl -

y]-N+1-7

Therefore, for all y EBr(xl) we have IJCx(xl,y)Id-Y(y)

< Clxi .._.

yI-N+1-7,

and similarly 19x(x2, y)l d-Y(y) G CIx2 _ yl-N+1

for all y E BR(x2).

Hence,

I
lxi -yl-N+1-Ydy+C2 ,.(xi)

< c3 f D

R

t-Ydt < C41x1

-

x211-7

f

Bu(x2)

1X2

_y1-N+1-7dy

(4.30)

104

Sublinear perturbations of singular elliptic problems

To evaluate II, we first apply the mean value theorem. We have

II < cl

"\B,-(xi)

IGxW0), y) - 9x(6(1), y)Id- r(y)dy

r

< cl J

S\B,-(xi)

((t),y)II'(t)Id(y)dtdy

J

< cslxi - x21 f

t\B,.(xl)

f

y)ld-7(y)dtdy. 1

0

As shown earlier, by Lemma 4.3.4 (ii) we obtain y) Id-7(y) <- CIC(t) - yI -N-'

for allyEBr(x1)and0
Let c6 = 1/(1 +co), where co is the constant from Lemma 4.3.5 that depends only on Q and not on x1, x2. Then for all y E 12 \ Br(x1), K(t) - yl ? Ixi - yl - 10) - x11= Ixl - yl - 10) - Va)I Ixl - Ill - tle'(ct)I ? Ixl - yl - calx1 - X21 ? c61x1- yl Combining the last two estimates we obtain Icx(S(t),y)Id-7(y) < C71x1

-

yl-N-7

Hence, we may write

II < c71x1-x21 / < c71x1 - x'21

i

\B,(xi)

lxl -yl-'-Idy

00

t-l-7dt < c8lx1

-

x2lr-7

(4.31)

< C91x1 - x211-7

The conclusion follows now from (4.29), (4.30), and (4.31). This finishes the proof of Theorem 4.3.2.

4.4 An example Consider the problem

-AU =u-a+f(x,u)

u>0 u=0

in n,

in0,

(4.32)

on a1,

where 0 < a < 1 and f is a Holder continuous function of class'C0'7 (0 < -y < 1), which satisfies (f 1) and (f2) stated in the beginning of Section 4.3.

An example

105

Proposition 4.4.1 For all 0 < a < 1 problem (4.32 has a unique solution ua E C2(Q) fl Cl,'--(?!) with Aua E L' (n). Furthermore we have Ilua - vIIc, (?) - 0

as a \ 0,

(4.33)

where v is the unique solution of the problem

1-dv = 1 + f (x, v)

in Sl,

v>0

in Q,

V=0

on 0,11.

(4.34)

Proof The existence and uniqueness of solution ua E C2(Q) fl C1,1-a(St) with Dua E L'(Sl) follows directly from Theorem 4.3.2. To prove (4.33) we first remark that for all 0 < a < 1 we have ua > in Sl, where E C2(a) is the unique solution of

-at; = AX, 0)

>0 =0

in Sl,

inSl, on 09.

Using the regularity of t; we deduce that for all 0 < a < 1,

cld(x) < fi(x) < ua(x)

in dl ,

(4.35)

where cl is a positive constant that does not depend on a. Let us fix q > p > N/(1 -,y). We prove in the following that there exists c2 > 0 not depending on a such that for all 0 < a < 1/q there holds ua(x) < c2d(x)

in dl.

(4.36)

For this purpose consider the problem

t1w=w-11q+1+f(x,w)

w>0 w=0

in Q, in Q,

(4.37)

on aft

By Theorem 4.3.2, there exists a unique solution w E C2(d) fl C1 (?j) of (4.37)_ Furthermore, there exists c2 > 0 such that w(c) < c2d(x) in St. Let

F(x, t) = t_a + f (x, t), (x, t) E SZ x (0, oo). Then F satisfies the hypotheses of Theorem 1.3.17. Note also that

t-0
for all t>0and 0
Hence,

in il. Because w, ua > 0 in 1, Aua E L1(S2) and w = ua = 0 on ad, by Theorem 1.3.17 we obtain ua < w in Sl for all 0 < a < 1/q, and (4.36) follows. ©w + F(x, w) < 0 < Dua + F(x, ua)

Sublinear perturbations of singular elliptic problems

106

From (4.35) and (4.36) we deduce that the sequence (F(x, u, (x)))o0

as a\0.

Furthermore, by Schauder estimates, (ua)o
up to a subsequence, we have ua --+ v in C2 (W) as a \ 0. Because w CC S2 was arbitrary, passing to the limit with a \ 0 in (4.32), it follows that -Av = 1 + f (x, v) in ft Moreover, from (4.35) and (4.36) we derive cld(x) <'v(x) < c2d(x)

in 0

Thus, we can extend v = 0 on 852, which yields v E C(852). Hence, V E C2(1) fl C(91) is a classical solution of (4.34). In conclusion, any subsequence of (ua)o
0

4.4.1.

For N = 1, S2 = (-1, 1), and f (x, t) = V, convergence (4.33) in Proposition 4.4.1 is depicted in Figure 4.2. The continuous line represents the solution v of problem (4.34) (which corresponds to the case a = 0) whereas the dashed lines represent the solution ua of problem (4.32) plotted for a = 0.1 and a = 0.3. 12

0.8

45o 0.6

3

0.4

0.2

0

4

-05

0

0.5

X

FIGURE 4.2. Convergence (4.33) in a one-dimensional case.

Bifurcation for negative potentials

107

4.5 Bifurcation for negative potentials This section is devoted to the case p < 0 in (4.21). For convenience, let us pass the singular nonlinearity p(x)g(u) from the right-hand side to the left-hand side. Therefore, we shall be concerned with the problem

-Au + p(x)g(u) = Af (x, u) + pk(x) u>0 U=0

in Q, in Q,

(4.38)

on 00,

where f , g, and k satisfy (f I)-(f 2), (g1)-(g2), and (ki), respectively, and p S2 -; (0, oo) is a Holder continuous potential.

Theorem 4.5.1 Assume that p > 0 in !l and conditions (f 1)-(f2), (gl)-(g2), and (kl) are fulfilled. Then, there exists A. > 0 and u* > 0 such that problem (4.38) has at least one solution in .6 if A > A. or p, > p*; problem (¢.38) has no solution in £ if 0 < A < A,, and 0 < p < it.. Moreover, if A > A or p. > p,R, then (4.38) has a maximal solution in £, which is increasing with respect to A and p. The diagram of dependence on A and a in Theorem 4.5.1 is depicted in Figure 4.3.

At least one solution

p.

No solution

A

Ficuitu 4.3. The dependence on A and p in Theorem 4.5.1.

Proof We split the proof into several steps. Step 1: Existence of a solution for large A. Let ti > 0 be fixed. Repeating the arguments in the proof of Theorem 4.3.2 we find that for all A > 0 there exists a unique solution vau, e £ of the problem

Sublinear perturbations of singular elliptic problems

108

in 1, in 9,

-Ov = A f (x, v) + µk(x) v>0

v=0

(4.39)

on M2.

Obviously, ua,,, := vas, is a supersolution of (4.38). The main point is to find a subsolution uau of (4.38) such that uar, vau, in 0. For this purpose, let 4i : [0, 00) -+ [0, 00),

4) (t) t

ds.

1

2 fo g(r)dr

0

Remark first that 4? is well defined, because g E L1(0, 1). Indeed, there exists m > 0 such that g(s) > m, for all 0 < s < 1. This yields 1/2

/ r8

1

_<( ms)-1

g('r)d-r1

for all 0<s<1,

which implies /

o

g(s)ds j

+ J

\o

(4.40)

dt < oo.

We claim that 4? is bijective./Indeed, 4) is increasing and, if M := g(1), then

f s g(T)d-r < J 1 g(r)dr + M(s - 1) Jo

for all s > 1.

J0

Thus, there exists c > 0 such that 3

g(r)d-r<Ms +c

for all s>1.

It follows that

fi(t)

>

Jl

2(Ms + c)

ds >

(

2(Mt + c) - cl)

for all t > 1.

This yields limt_.-,0,, 4?(t) = oo, and the claim follows. Let h : [0, oo) -+ [0, oo) be the inverse of 4?. Then h satisfies

in (0,00),

h>0

fo' (t)

g(s)ds h'(t) = il2 h"(t) = g(h(t))

in (0, oo), in (0, 00),

h(0) = h'(0) = 0. Hence, h E C2 (0,00) fl C' [0, oo).

The key result for this part of the proof is the following lemma.

(4.41)

Bifurcation for negative potentials

109

Lemma 4.5.2 There exists a positive constant M > 0 (which depends on A) such that uau := Mh(W1) is a subsolution of (4.38) provided A > 0 is large enough.

Proof In view of the strong maximum principle there exist b > 0 and lb cc S2 such that IVWiI > S

in 0 \ Sbo.

(4.42)

We choose M > 1 such that MIQWjj2>1+IIpIIoo

inlb\S1o.

(4.43)

Moreover, because -g(h(Wi))+MAiWi h'(o1) - -oo as d(x) \ 0, we can assume that -g(h(Wi)) + MA1W1h'(W1) < 0

in 0 \ lo.

(4.44)

Now we are in position to show that ua,, := Mh(Wj) is a subsolution of (4.38) for large A > 0. First we have in Q.

-Aua, = -Mg(h(coi))I ©coi I2 + MAicih'(Wi) Because g is decreasing and M > 1, it follows that

-Au,,, +p(x)g(ux,,) <(IIplk«> - MIVW1J2)g(h(Wi)) + MAiWih'(Wi) in Q.

(4.45)

From relations (4.43) through (4.45) we have

-1\u aµ + p(x)g(ua,,) < -g(h(Wi)) + MAicplh'(Wi) < 0

in S2 \ Qo.

Hence,

+ p(x)g(uxw) < Af (x, u,\u) + pk(x)

in Q \ S1o.

(4.46)

Furthermore, from (4.45) we have -4uau + p(x)g(uxi,) < IEplloog(h(Wi))IQWi l2 + MAiWih'(wwi)

in S1.

(4.47)

Because Wi > 0 in Sto, we can choose A > 0 such that

A min f(x, h(Wi)) > max{JIpjI.g(h(Wi)) + MAiWih'(Wi)}. xESEo

(4.48)

xE?Yo

Combining (4.47) and (4.48) we derive -Duau, + p(x)g(uau,) < ,\ f (x, u,N,) + p,k(x)

in S1o.

(4.49)

From (4.46) and (4.49) we note that uau is a subsolution of (4.38), and the proof of our lemma is now complete.

Sublinear perturbations of singular elliptic problems

110

To end the proof in this step, it remains to show that uAu _< u,\,, in P. This follows immediately by using Theorem 1.3.17 for the mapping

t) _

Af (x, t) + p.k(x), (x, t) E Sl x (0, oo). Thus, problem (4.38) has at least one solution, ua., such that Mh(cpi) < u,\, < va, in ft. The regularity of u,\,, and the fact that ua,, e E follows in the same manner as in the proof of Theorem 4.3.2. Note that from the previous inequality we derive the same estimates as in (4.27), which are necessary to achieve the regularity ua,, E C1,1-7(St)

Step 2: Existence of a solution for large µ. The proof is the same as that shown earlier. We have only to replace (4.48) with

p min k(x) > max{IIpll.g(h(w1)) + xEi o

zE12n

Step 3: Nonexistence for small A and p.. Let m = inft>o{tA1 + p*g(t)}, where P. = minxE) p(x) > 0. Because g is positive and decreasing on (0, oo), we have m > 0. Denote by ( E S the unique solution of (4.39) for A = p = 1. Because infxE-a k(x) > 0, there exists c > 0 such that f (x, () < ck(x) in 0. We claim that (4.38) has no solutions for A and p in the following range:

0
and

(Ac+u)lIkI1111w111o.. <mII

iIIi.

(4.50)

Suppose by contradiction that problem (4.38) has a solution u,\ , E E for A, A that satisfies (4.50). In this case, ( is a supersolution of (4.38) and, by Theorem in Q. We multiply by W in (4.38) and then we 1.3.17, we deduce that u,\, integrate over Q. Therefore, (A1u + p*9(u))cp1dx <

f(Af(x)+k(x))idx.

This yields m11o1111 <_ (Ac+p)J k(x)cpldx < which contradicts (4.50). Hence, problem (4.38) has no solutions, provided A and p fulfill (4.50).

Step 4: Existence of a maximal solution for problem (4.38). We show that if problem (4.38) has a solution ua, E E, then it has a maximal solution that belongs to E. Let A, p > 0 be such that (4.38) has a solution ua,, E E. If vat, is the corresponding solution of (4.39), in view of Theorem 1.3.17 we deduce ua, < vas, in

0. For n > I large enough, define

9n:={xEfl:d(x)> n1}. Let uo = vas, and un be the solution of

Bifurcation for negative potentials

Aw +

111

in SZn,

p(x)g(un_i) = Af (x, un_i) + 1ih(x)

tU = un_1

in SZ \ SZn.

Using the fact that the mapping (x, t) = A f (x, t) + tck(x), (x, t) E SZ x (0, oo) is nondecreasing with respect to the second variable, we obtain u,\1,<
in Q.

Define

U,\,(x) := n_oo lim un(x)

for all x E U.

By standard elliptic arguments it follows that U,\ , is a solution of (4.38) and, obviously, U,;,, is the maximal solution of (4.38). Moreover, with the same reasoning as in the proof of Theorem 4.3.2, we have Uai, E E.

Step 5: Dependence on A and p.. We first prove the dependence on A of the maximal solution Uar, E £ of (4.38). For this purpose, define

A:= {a > 0: for all µ > 0, problem (4.38) has at least one solution in £}. Let A := inf A. From the previous steps we have A # 0 and A > 0. Let A, E A, p > 0, and U.,,,, be the maximal solution of (4.38) for A = A1. We prove that (A1, oo) c A. Indeed, if A2 > At, then Ub,,, is a subsolution of (4.38) with A = A2.

On the other hand, if v,\,,, is the solution of (4.39) with A = A2, then Ova2u + 4'a2/,(x,vA,2M) < 0 < AUa,,, + vA2 ,, Ua11 > 0 VA2w ? U,,,

U,\,,,)

in Q,

in 5i,

on

,

©U,x,,,, E L1(0)

By Theorem 1.3.17 we deduce Ua,,, < vat,, in U. Therefore, by the sub- and supersolution method, problem (4.38) has a solutionnua2u E £ such that u')2j1 <

in U.

Hence, A2 E A and so (A., oo) C A. If U,,2u E £ is the maxima] solution of (4.38) with A = A2, the previous relation implies U,\,,, < U1\211 in SZ and, by virtue of the strong maximum principle, it follows that UA,, < U,,21 in Q. Thus, U,\,, is increasing with respect to A. To achieve dependence on i we define

B := J ;z > 0 : for all A > 0, problem (4.38) has at least one solution in £}.

Let u := inf B. The conclusion follows in the same manner as just demonstrated. El The proof of Theorem 4.5.1 is now complete.

Sublinear perturbations of singular elliptic problems

112

4.6

Existence for large values of parameters in the sign-changing case

In this section we study problem (4.21) assuming that the potential p changes sign in 0. We shall obtain the existence of solutions in the class £ for large values of parameters A and it. Let p* := ma_xp(x) XEn

and p* := minp(x). XJ

Theorem 4.6.1 Assume that p* > 0 > p*. Then there exists A*, p* > 0 such that (4.21) has at least one solution uAK E E if A > , or it > p.. Moreover, for A > A. or p > it., ua,, is increasing with respect to A and p.

Proof Because p* < 0, Theorem 4.5.1 implies that there exist A*, p* > 0 such that the problem

-Au = p*g(u) + of (x, u) + pk(x)

u>0 u=0

in St,

in P,

one

has a maximal solution U,\,. E £, provided A > , or p > p.. Then U,\,, is a subsolution of (4.21). Moreover, Ua,, is increasing with respect to both A and p. Consider the problem

-Av = p*g(v) + Af (x, v) + pk(x) v>0

v=0

in S2,

in n,

(4.51)

onl3l.

Because p* > 0, by Theorem 4.3.2, problem (4.51) has a unique solution vau, E E. Obviously, va , is a supersolution of (4.21). Note also that 4)A,,, (x, t) = p*g(t) + A f (x, t) + pk (x), (x, t) E 1 x (0, oo), fulfills the hypotheses of Theorem 1.3.17. According to this result, Ua,, < va, in Q. Hence, by Theorem 1.2.2, there exists

a minimal solution uau, of (4.21) with respect to the ordered pair (U,\, ,v,\,,). Furthermore, using Theorem 1.3.17 it is easy to see that uau is minimal in the class of solutions u E E of (4.21) that fulfill Uaµ < u in St. Now let us prove the dependence on A. Fix Al > A2 > a*, p > 0 and let u,\,,,, ua2u E £ be the solutions of (4.21) for A = Al and \ = A2, respectively, that we obtained earlier. Note that ua,, is a supersolution of (4.21) with A = A2. Moreover, because Uaµ is increasing with respect to A > a*, we obtain uA14 > Ua1M > Ua2µ

in Q.

Using the minimality of ua2,, from the previous relation we deduce UA,u >

uA2u

in 9 and, by the strong maximum principle, we deduce that uA12 > ua2, in Q. The dependence on p follows in the same way. This completes the proof of Theorem 4.6.1.

Singular elliptic problems in the whole space

113

Singular elliptic problems in the whole space

4.7

In this section we extend the study of singular elliptic problems to the case where the domain St is the whole space RN. We shall be concerned with problems of

the type

1-Du = 4i(x, u) u>0

u(x) - 0

in RN, in RN,

(4.52)

as jxj -> oo,

where (: RN x (0, oo) - [0, oo), N > 3, is a Holder continuous function having a singular behavior at the origin with respect to the second argument. Existence of entire solutions We are concerned here with the existence of classical solutions to (4.52)-that 4.7.1

is, functions u E CZ(RN) (or even u E Cj , (RN) for some 0 < ry < 1) that verify (4.52) pointwise in RN. A solution of (4.52) is often called a ground-state solution. Roughly speaking, such a solution achieves a minimal level of energy as a result of the prescribed condition at infinity.

A natural way to construct solutions to (4.52) is to provide a monotone sequence of positive functions that are solutions of similar problems to (4.52), but in bounded domains. The main point is to supply a supersolution w of (4.52) that must satisfy w(x) - + 0 as lxi oo. Then, standard elliptic arguments will imply that the pointwise limit of the previously mentioned sequence is, in fact, a genuine solution of (4.52). This procedure clearly fails in the absence of a comparison principle for problems like (4.52) in bounded domains. Bearing these points in mind, we start out the study of existence by considering first the case where 4* in (4.52) is monotone with respect to its second variable. This particular situation will help us to construct in an easier way the monotone sequence with a limit that is the solution of (4.52). Let us consider the problem

-du = p(x)g(u) U>0

u(x)-'0

in R v, in RN,

(4.53)

asjxi -'oo,

where g E C1(0, oo) satisfies g > 0, g' < 0 in (0, oo) and limt\o g(t) = oo. We shall also assume that p : RN -> (0, oo) is a Holder continuous function. Let

O(r) := maxp(x) and O(r) := min p(x), r > 0. IzI=r

Izl=r

Theorem 4.7.1 The following properties are valid: (i) If f O° oo, then problem (1.55) has no classical solutions. (ii) If f000ro(r)dr < oo, then problem (4.53) has a unique classical solution.

Sublinear perturbations of singular elliptic problems

114

Proof (i) Assume that (4.53) has a solution u E CZ(RN) and let U be the spherical average of u defined as

U(r) =

N_1

wNr

f

u(x)du(x)

IxI=r

for all r > - 0,

where wN is the surface area of the unit sphere in RN. Because u is a positive ground-state solution of (4.52), it follows that U is positive and U(r) -- 0 as r -* oo. Let m := minxERN g(u(x)). Because u(x) -4 oo as lxi - oo, it follows that m > 0 is finite. With the change of variable x = ry, we have

f

U(r) = -1

WN

u(ry)dc(y)

for all r > 0

u1=1

and

U'(r) =WN1 f' 1=1 Vu(ry) ydo(y)

for all r > 0.

Hence,

U,(r) =

1

WN

8u (ry)du(y)

( 1711=1

Oar

f

_' (x)do, (x ) 1 = WNrN-1 I.,=r Or

.

That is,' 1

U1(r) = WNTN-1

1

This yields

_(rN_1U'(r))' = J

1

d(

wN d r

`

Irk
1

wN ICI=r

-Au(x)dx)

-Au(x)dcr(x)

for all r > 0,

which finally leads us to

-(rN-1U'(r))' > mrN-1 O(r)

for all r > 0.

(4.54)

Integrating in (4.54) we find

U(0) - U(r) > m

jr t1-N

ft sN-10(s)dt

for all r

0.

(4.55)

a

Because fo rib(r)dr = oo, according to Lemma 2.2.4 we deduce that the righthand side in (4.55) tends to infinity as r -> oo. Thus, passing to the limit with r - oo in (4.55) we obtain a contradiction. Hence, problem (4.53) has no solution in this case.

Singular elliptic problems in the whole space

115

(ii) We first apply Theorem 1.2.5 for Bn := {x E RN : Ix I < n}. Thus, for all n > 1 there exists un E C2'Y(B) fl C(B,ti) (0 < -y < 1) such that

-dun = p(x)g(un)

in Bn,

un>0

in B.,

un = 0

on f3Bn.

We extend un by zero outside of B. Because g is decreasing, we deduce

U1 < u2 < ... < Un < un+1 < ...

in RN.

We next focus on finding a supersolution w of (4.53). To this aim, define

((r) := rl-iv jtN_1(t)df

for all r >_ 0.

(4.56)

By Lemma 2.2.4, and that fact that f oo rq5(r)dr < oo, we find f ow rc (r)dr < oo. Consider now fi(x) ((t)dt for all x E RN. llfxl

Then

satisfies

{

-d6 = O(Ixl)

in RN,

t; > 0 fi(x) -> 0

in RN, as lxl -> oo.

Because the mapping [0, oo) D t H fo ds/g(s) E [0, oo) is bijective, we can implicitly define w : RN --> (0, oo) by a'(x)

dt = fi(x) g(t)

for all x E RN .

It is easy to see that w E C2(RN), w > 0 in RN, and w(x) -> 0 as Ixl -> oo. Furthermore, we have

-off = g'(w) I©w12

g(w) = c(IxI)

in RN.

Because g' < 0, it follows that -Llw > O(Ixl)g(w) > p(x)g(w)

in RN.

(4.57)

Therefore, w is a supersolution of (4.53). Now, it is easy to deduce that un < w in

RN. By standard elliptic arguments we find that u(x) := limn" un(x), x E RN, satisfies u E Coy (RN) for some 0 < y < 1, and -Du = p(x)g(u) in RN. Because u,, < w in RN, we obtain that u < w in RN, which yields u(x) -> 0 as jxl - 00. Hence, u is a solution of (4.53). Finally, if ul and u2 are two solutions, by the maximum principle and the fact that ul - u2 tends to zero at infinity we derive ul =- u2-that is, problem O (4.53) has a unique solution. This concludes the proof.

116

Sublinear perturbations of singular elliptic problems

Next we study the existence of classical solutions for the problem

-Au = p(x) (g(u) + f(u)) U>0

u(x) -s 0

in 1[ N,

in RN,

(4.58)

as 1xI --> 00.,

where g and p are as considered earlier and f : [0, oo) - [0, oo) satisfies the sublinear conditions (f 1) and (f2) introduced in the beginning of this chapter. As a result of the lack of monotonicity, the uniqueness is not so obvious as in the study of problem (4.53). This matter will be discussed in the last part of this section for radially symmetric solutions.

Theorem 4.7.2 Assume that foo r¢(r)dr < oo. Then problem (4.58) has at least one classical solution.

Proof We use the same approach as in Theorem 4.7.1 (ii). By Theorem 1.2.5, for all n > 1 there exists un E C2°"r(Bn) n C(B,) such that Aun = p(x)T (un)

un>0

in Bn, in B.,,,

onBBn, where W(t) = g(t) + f (t), t > 0. We have Aun+1 + p(x)%P(un+i) < 0 < iu,i +p(x)qf(un)

in Bn,

un = 0 < un+1 on OBn. Notice that Au,,+1 E L'(B,,), because un+i is bounded away from zero in B,,. Thus, by Theorem 1.3.17 it follows that un < un+1 in B. We extend u, by zero < un < un+1 < . . . in RN. It remains to outside of B. Then 0 < ul < find an upper bound for (un)n>1. By Theorem 4.7.1 with g replaced with g + 1, there exists v E C2(RN) such that

-Av = p(x)(g(v) + 1)

v>0 V(X) -} 0

in RN,

in R"',

(4.59)

as jxI - oo.

Because f is sublinear and v is bounded in RN, we can find M > 0 such that M > f (Mv) in RN. Hence, w := My satisfies -iw = p(x) (g(w) + 1 (w))

in RN ,

W>0 W(X) -. 0

in lE$N

As seen earlier, we find un < w in RN and u(x) is the solution of (4.58). This finishes the proof.

as jxI -* oo.

un(x), for all x E RN,

Singular elliptic problems in the whole space

117

Uniqueness of radially symmetric solutions We discuss in the sequel the uniqueness of a radially symmetric solution to (4.53). More precisely, we shall be concerned with the uniqueness of classical solutions 4.7.2

to the problem

-u"(r) - Nr 1u'(r) =p(r)*(u(r))

in [0>00),

u>0 u'(0) = 0,

in [0, oo),

u(r)-f0

asr -goo,

where p : (0, oo) -> [0, oo) and IF functions such that

:

(4.60)

(0, oo) -> (0, oo) are Holder continuous

is decreasing; (Al) the mapping (0, oo) D t (A2) there exists to > 0 such that %P is decreasing on (0, to) and increasing on [to, oo);

(A3) there exists

to r (t) E (0, oo) for some a > 0. i o Notice that these hypotheses are quite natural (see Figure 4.4). Typical examples

of nonlinearities encountered so far, such as '11(t) = t--a+t', a > 0, 0 < q < 1, or W(t) = t-a+ln(l+t), a > 0, satisfy (Al) through (A3). Obviously, any solution u of (4.60) provides a radially symmetric solution u(x) = u([xI), x E 1[l;^`, of problem (4.52) with t (x, t) = p(I xI)'1(t).

(0,0)

to

t

FIGURE 4.4. The shape of function T.

Theorem 4.7.3 Assume that (Al) through (A3) hold. Then, problem (4.60) has at most one solution.

Sublinear perturbations of singular elliptic problems

118

Proof Assume that there exist u, v E C2[0, oo) two distinct solutions of (4.60). By virtue of Theorem 1.3.17 we easily deduce the following useful result.

Lemma 4.7.4 If there exists R > 0 such that u(R) = v(R), then u - v in [0, R).

By virtue of Lemma 4.7.4, one of the following three situations may occur: 1. u > v in [0, oo). 2. u(0) = v(0) and u > v in (0,co).

3. There exists R > 0 such that u =- v in [0, R) and u > v in (R, oo). We shall discuss these three situations separately. CASE 1: u > v in [0, oo). Let us notice that u and v verify

-(rN-lu'(r))' = rN-1p(r)'y(u(r)),

(4.61)

-(rN-lv'(r))' = rN-1p(r)`y(v(r)),

(4.62)

for all r > 0. We multiply the first equality by v', the second one by u', and then subtract. We obtain

(rN-1(u'v - UV'))' = rN-1p(r)uv(%P(u) u

- %F(v)I v

for all r > 0.

"' (v)}dt

for all r > 0.

An integration over [0, r], r > 0 yields

frtN-1P(t)UV(T(u) rN-1(u'v -7GV') = l

U

v

-

Therefore, u'v - uv' > 0 in [0, oo), which implies

v is increasing and

v > v'

in (0, oo).

(4.63)

Hence, there exists t:= limr_,oo u(r)/v(r). Because u(0) > v(0), it follows that

1 0, we have

u'(r) = -rl-N

r

(4.64)

0

r

W(r) =

-r1-N

tN-1p(t),>i(v)dt.

(4.65)

0

Assume first that fO° tN-1p(t)i(u)dt = oo. Because u > v in [0, oo), and both u and v tend to zero as r --+ oo, by (A2) we find 41(u(r)) for

Singular elliptic problems in the whole space

119

r large enough. This yields fo tN-lp(t)J1(v)dt = oo. Then, by l'Hospital's rule we find

= Jim u'(r) = Jim f0 1<e= lim u(r) r-'oo W(r) r-.oo f tN - 1p(t) 41 (v)dt v(r)

lim CUM) r--T(v(r))

By (A3) we have lim

f(u(r)) = lim (u(r))_a i(v(r))

vaW(v) -

which is a contradiction. Let us assume now that fOO tN-lp(t) (u)dt < oo. If f O oo, as before we have

(u)dt u(r) u'(r) =0 = lim 1 < f = lim = lim f° tN-lp(t} r-'oo v(r) r-oo v1(r) r__ f° tN-lp(t)T(v)dt which is again a contradiction. It remains to analyze the case when both integrals fo tN-lp(t)%F(u)dt and f OO tN-lp(t)'(v)dt are finite.

Lemma 4.7.5 There exists a unique a > 0 such that

1'(u(r)) > T(v(r)) for all 0 < r < a, '1(u(a)) = ql(v(a)), I'(u(r)) <'I'(v(r)) for all r > a. Proof Let us first remark that 41(u(0)) >''I(v(0)). If we would have the converse inequality, then, by (A2) and u(0) > v(0), we obtain v(0) < to. It follows that for all r > 0, W(v(r)) > f(u(r)), which yields 00

00

tN-1p(t)W(u)dt < J 0

tN-lp(t),Q(v)dt.

0

Again by 1'Hospital's rule we find

1 < t = lim

u(r)

r-+oo v(r)

Jim

=

u'(r) _ fa tN-lp(t)I1(u)dt < 1, oo r-+oo v'(r) (v)dt ftN-lp(t)°

which is a contradiction. Hence, 1F(u(0)) > W(v(0)). Furthermore, because u > v

and u(t) --> 0, v(r) -> 0 as r - oo, for r > 0 large enough we obtain f(u(r)) < W(v(r)). Thus, by continuity arguments, we can find-a > 0 such that xk(u(a)) W(v(a)).

Sublinear perturbations of singular elliptic problems

120

Assume that there exists two real numbers al > a2 > 0 such that 41(u(ai)) _

W(v(ai)), i = 1, 2. Then we must have u(ai) > to > v(ai), i = 1, 2. Because al > a2, we deduce u(al) < u(a2), and by (A2) we further obtain IP(v(al)) = W(u(al)) < T(u(a2)) = W(v(a2)).

Because v(ai) < to, i = 1, 2, the last equality implies v(al) > v(a2), which in turn implies a1 < a2, but this is clearly a contradiction. Now, the rest of the proof follows from the uniqueness of the number a > 0 with W(u(a)) ='I'(v(a)). By ]'Hospital's rule we have found °° tN-lp(t)W (u)dt P

- f °O tN-lp(t)I(v)dt

Thus, fort 0 < E < 1 there exists ro > a large enough such that J

r tN-lp(t)41(u)dt > (B - r) JOrtN_1P(t)(V)dt

for all r

ro.

0

By Lemma 4.7.5 we have '(u(t)) < T(v(t)), for all t > a, which yields

a tN-lp(t)w(u)dt + J0

Ja

rtN-lp(t)'F(v)dt > (t - E) J r 0

for all r > ro. This leads us to ftN_1p(t)(W(u) - W(v))dt > (e - 1 - E) f r tN-1p(t)T(v)dt

for all r > TO.

Passing to the limit with r -> oo, and because 0 < E < 1 was arbitrarily chosen, we obtain

fa

tN-1p(t)(T(u) - IF(v))dt > (e - 1)

J0

tN-lp(t)T (v)dt.

That is, aN-1 (v' (a) -

u'(a)) > (t - 1) J r tN-1p(t)WY(v)dt. 0

On the other hand, from (4.63) we have u'/v' < u/v < t. This yields v'(a) - u'(a) = v'(a) I 1 - -I(a) J <

Hence,

(4.66)

Comments and historical notes

aN-1(v'(a)

121

f)aN-1v'(a) - u'(a)) < (1 -

f

d

1)

tN-1p(t)T (v)dt

0

< V - 1)

f

(4.67)

oo

tN-1p(t)T(v)dt.

0

Ftom (4.66) and (4.67) we obtain the desired contradiction. This concludes the proof in Case 1.

CASE 2: u(0) = v(0) and u > v in (0, oo). Let w = u - v. Then w verifies

-(rN-1wr)r = rN-1p(r)(W(u)

- `y(v))

w>0 w(0) = w'(0) = 0, w(r)-=+0

in (0, oo), in ]0, oo), (4.68)

asr -+oo.

Integrating in (4.68) we have

w(r) = - f t1-N

f sN-1p(s)((u) - W(v))dsdt

for all r > 0.

(4.69)

0

We claim that u(0) = v(0) < to. If u(O) = v(0) > to, then u(t) > v(t) > to for t > 0 small enough, and by the hypothesis (A2) on %F we deduce *(u) > %k(v) in a small neighborhood of the origin. But using (4.69) this yields w(t) < 0 for some t > 0, which is a contradiction. Hence, u(0) = v(0) < t0. Then

v(r) < u(r) < u(0) < to

for all r > 0,

which implies T(u) - W(v) < 0 in (0, oo). Now, to obtain a contradiction we look at (4.69). The left-hand side tends to zero as r oo whereas the right-hand side tends to a positive quantity as r -+ oo. This is clearly a contradiction.

CASE 3: There exists R > 0 such that u = v in [0, R) and u > v in (R, 00). We proceed exactly in the same manner as we did in Case 2 for ii(r) = u(r + R) and ii(r) = v(r + R), r > 0. This finishes the proof of Theorem 4.7.3.

4.8

Comments and historical notes

Singular elliptic problems arise in various branches of mathematics (see [38], [56], [134], and [160] for more details). To our best knowledge, the first study in this direction is from Fulks and Maybee [81] and have been intensively studied after the pioneering work by Crandall, Rabinowitz and Tartar [57].

In their work [57], a similar problem to (4.3) has been studied for more general differential operators. The existence of a classical solution to (4.3) has been obtained by considering the perturbed problem

122

Sublinear perturbations of singular elliptic problems

-Du = g(u +,-), 0 < e < 1

in S2,

u>0

in S2,

U=0

on 852.

(4.70)

Then (4.70) has a unique solution uE E C2 (?I), which is decreasing with respect to e. If u(x) := limb\o u,(x), x E S2, then u E C2 (Q) n C(S2) is the unique solution of problem (4.3). In the case of pure powers in nonlinearities and p = 0, problem (4.21) was studied by Shi and Yao [182] and was then generalized by Ghergu and Radulescu in [83]. For more results concerning singular elliptic equations in bounded domains we refer to [40], [42], [48], [55], [61], [67], [84], [85), [86], [96], [101], [127], [174], [201], and [202] as well as to recent surveys (102], [175]. If a < 0, problem (4.12) is called the Lane-Emden-Fowler equation and arises in the boundary-layer theory of viscous fluids (see Fowler [78] and Wong [197]). As we have already mentioned, as a result of the meaning of the unknowns (concentrations, populations), the positive solutions are relevant in most cases. But nonnegative solutions may also arise in some situations. In fact, the presence of the singular term g(u), which is not locally Lipschitz near the origin, may give rise to dead core solutions-that is, solutions identically zero on subdomains of S2 with positive measure. For more results on dead core solutions involving singular nonlinearities, we refer the reader to Davila and Montenegro [59], where the following problem was considered: Au = X{u>o} u=0

(-u-a

+ AuP)

in S2, (4.71)

on 852,

with 0 < a!, p < 1. It is proved in [59] that for all A > 0, problem (4.71) has a maximal solution ua. Moreover, there exists A* such that for all A > A* the maximal solution ua is positive and stable. Another direction regarding the study of singular elliptic problems refers to sign-changing solutions for such problems and was considered by McKenna and Reichel [137]. In this respect, the precise asymptotic behavior of a solution at the boundary where it changes sign plays an essential role in the study of existence. The approach in obtaining the C" (11) regularity of the classical solution to (4.21) follows the method in Gui and Lin [99] in the case A = u = 0, g(t) = t-a,

and p(x) behaves like d(x)°, 0 < a < 1 near 852. The method in [99] can be extended to more general nonlinearities than those we dealt with in this chapter. As a conclusion, Cl"'Y(S2) (0 < 7 < 1) is the best regularity of solutions we can

expect for these kinds of problems. We have seen in Theorem 4.1.2 that if g decays fast in the neighborhood of the origin, the solutions may not be in Cl (S2) or, even more unexpected, they do not belong to the usual Sobolev space Ho (52). In other words, it may happen that a classical solution is not a weak solution. We will come back to this issue in Chapter 7.

Comments and historical notes

123

The growth condition (g2) on g is a natural assumption. This allowed us to show that if problem (4.38) has a classical solution u E C2(f) f1 C(Sl), then u E E. We shall see in Chapter 9 that condition g E L1(Q) is necessary to have the existence of a classical solution. Condition (4.40) is often called the Keller-Osserman condition around the origin. As proved by Benilan, Brezis, and Crandall [23], condition (4.40) is equiv-

alent to the property of compact support. That is, for every k E L'(RN) with compact support, there exists a unique u E W"' (RN) with compact support such that Au E Ll(IRN) and

-Du + g(u) = k(x)

almost everywhere in RN.

Theorem 4.5.1 leaves open the question of multiplicity. This is clearly a delicate issue even in simpler cases. In this sense we refer the reader to Ouyang, Shi, and Yao [156], who studied the existence of radially symmetric solutions of the problem -Au = A(ur - u-°) in B, u>0 in B, (4.72)

u=0

on 49B,

where B is the unit ball in RN, (N > 1), 0 < a, p < 1 and A > 0. Using a bifurcation'theorem of Crandall and Rabinowitz, it has been shown in [156] that there exists Al > AO > 0 such that problem (4.72) has no solutions for A < A0, one solution for A = \o or A > A1i and two solutions for al > A > .10. Finally, the results obtained for singular elliptic problems in bounded domains allowed us to extend the study to the case when the domain is the whole space. The first results in this sense concern problem (4.53) and have been obtained by Kusano and Swanson [122] and Edelson [74]. Condition f rq5(r)dr < oo in the statement of Theorem 4.7.1 was first supplied by Lair and Shaker [124] (see also [125] also [200]), where the existence of a ground-state solution of (4.53) is obtained under a weaker assumption on the potential p, namely, p satisfies the assumption (pl) in Section 2.2. That is, p is nonnegative and any zero of

p is surrounded by a region where p > 0. In the case '(x, t) = p(x)(t,-° + tp), 0 < p, a < 1, the uniqueness of a radially symmetric solution to problem (4.60) has been obtained in [199].

5

BIFURCATION AND ASYMPTOTIC ANALYSIS: THE MONOTONE CASE A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. Godfrey H. Hardy (1877-1947)

In the current chapter and in the following one, we are concerned with bifurcation problems of the type Au = Af (u)

in Il,

u>0

in Q,

u=0

on 852,

where 0 is a smooth bounded domain in R N (N _> 1), A is a positive parameter, and f : [0, oo) -> (0, oo) is a continuous function. We study the existence and the qualitative properties of solutions u E C2(Sl) n C(Sl) (provided such solutions exist), as well as their asymptotic behavior with respect to A. To give an elementary example, let us consider the class of functions f with

linear or superlinear growth-that is, there exists a > 0 such that f (t) > at, for all t > 0.

(5.2)

We observe that if f satisfies condition (5.2), then there exists 0 < A0 < Ao < 00 such that (i) problem (5.1) has at least one solution if A E (0, Ao); (ii) problem (5.1) has no solution if A > Ao. Of course, natural questions are the following: Is Ao = Ao?

If not, what happens if A E (Ao, Ao)? Assertion (i) follows by the implicit function theorem (Theorem B.1 in Appendix 13). To argue (ii), let
126

Bifurcation and asymptotic analysis: The monotone case

to the first eigenvalue ai of (-A) in Hp (Il). Thus, by multiplication with cpl in (5.1) and integration, we find Al in ucpidx = A fo f(u) Aa sz

f ucojdx.

in

This relation shows that problem (5.1) has no solution if A > \i/a. An important feature in our analysis concerns the notion of stability, which is viewed in terms of a related linearized problem. More precisely, to any solution (u, A) of the nonlinear problem (5.1), we associate the linear eigenvalue problem

Aw - Af'(u)w = uw

in Q,

W=0

on 852.

<1u/ 4n ,--- -4 oo, Problem (5.3) has a sequence of real eigenvalues pi < u2 < ... < and ui is a simple eigenvalue with a corresponding eigenfunction Y'i such that z/ii > 0 in Q. If it, > 0, then solution u of problem (5.1) is said to be stable; if µl < 0, then u is unstable; and if ui = 0, then the solution u is neutrally stable. Usually, a neutrally stable solution is referred as an unstable solution. If ui = 0, then solution u is degenerate; otherwise, it is nondegenerate. If ui < 0, then the number of negative eigenvalues uk (counting their multiplicity) of problem (5.3) is the Morse index of solution u.

5.1 Introduction Consider the bifurcation problem

Au = \f (u)

u=0

in S2,

onof2,

where Q C RN (N > 1) is a smooth bounded domain and A > 0. We assume that f : [0, oo) --> (0, oo) is a C1 convex function such that f'(0) > 0 and f is asymptotically linear. That is, lim f-(t) = a E (0, 00).

1-400

t

A first existence and bifurcation result concerning problem (5.4) is given by Amann's bifurcation theorem in Section 5.2 in a more general framework. Notice that because f is convex, by (5.5) we have limt_,00 f'(t) = a, and the mapping g : [0, oo) --* [0,-00), g(t) = f (t) - at is nonincreasing. Hence, there exists E := tliim g(t) E 1-00, 00).

The sign oft plays a crucial role in the analysis of (5.4). The current chapter is concerned with the case t > 0. Thus, we study here the monotone case on f

A general bifurcation result

127

f(t}t

t

FIGURE 5.1. The monotone case on f.

because the mapping t --> f (t)/t is nonincreasing, so that a = inft>o f (t)/t (see Figure 5.1).

In turn, if f < 0 then m := inft>o f (t)/t > a, and both numbers a and m will play an important role in the study of (5.4). This fact will be emphasized in Chapter 6. For a E L°° (1l) we denote by Al (a) and co, (a) the first eigenvalue and the first eigenfunction, respectively, of the operator -A - a in Ho (S2). By means of the variational characterization of the first eigenvalue of a linear operator, AI (a) is defined as

At(a) =

inf

J (IDu12 - a(x)U2)dx.

uEHo(A),IiulI2=1 n

(5.6)

Definition 5.1.1 A solution u of (5.4) is called stable if Ai(Af'(u)) > 0, and, is unstable otherwise.

5.2 A general bifurcation result The main result in this section is the following.

Theorem 5.2.1 (Amanri's bifurcation theorem) Assume that f : R -* R is a C2 function that is convex, positive, and such that f'(0) > 0. Then there exists A* E (0, oo) such that the following holds: (i) If A > A*, then problem (5.4) has no classical solutions. (ii) If 0 < A < A*, then problem (5.4) has a minimal solution u;k E C2(S2) such that (iil) ua > 0 in &1; (112) the mapping (0, A*) E) A ua(x) is increasing, for every x E S2; (ii3) ua is the unique stable solution of the problem (5.4) for all 0 < A < A* .

Proof We divide the proof of Theorem 5.2.1 into several steps.

Bifurcation and asymptotic analysis: The monotone case

128

Step 1: Definition of At. To apply the implicit function theorem (see Theorem B.1) to our problem (5.4), define F : Co'a(S2) x R -> C°,a(Sl) by

F(u, A) = -Au - o f (u),

(u, A) E C2,' (Sl) x 1EF,

where 0 < a < 1. It is clear that F verifies all the assumptions of the implicit function theorem. Hence, there exists a maximal neighborhood of the origin I and a unique map u = ua that is the solution of problem (5.4) for all A E I and such that the linearized operator -A - )if'(ua) is bijective. In other words, for every A E I, problem (5.4) admits a stable solution, which is given by the implicit function theorem. Let A* := supl < oo.

Step 2: The mapping (0, A*) E) A --v u,\ (x) is increasing and ua > 0 in 0. For x E i, denote by v,\(x) the derivative with respect to A of the mapping (0, A*) D A 1

) u (x). We differentiate in (5.4) with respect to A and we obtain

(-A - Af'(ua))va = Af (u,\) va = 0

in 92,

on acl.

Note also that the operator -A-A f'(ua) is coercive. Therefore, by Stampacchia's maximum principle (see Theorem 1.3.10), either VA = 0 in Sl or va > 0 in 9. The first variant is not convenient, because it would imply that f (uA) = 0, which is impossible, by our initial hypotheses. Hence, va > 0 in i, which then yields

ua>0in0.

Step 3: ua is stable. Set O(A) := Al(Af'(uA)). Thus, Vi(0) = Al > 0. Furthermore, by the implicit function theorem the linearized operator -A-A f'(ua) is bijective, which yields i1(A) # 0 for all A < A*. Now, by the continuity of the mapping A i--- A f'(ua), it follows that 0 is continuous, which further implies that 0 > 0 on [0,A*).

Step 4: A* < oo. Because f is convex, it follows that f'(ua) _> f'(0) > 0, which leads to A1(Af'(0)) > Aj(Af'(ua)) > 0 for every A E I. Notice that A1(Af'(0)) = Al - Af'(0). This yields Al - Af'(0) > 0 for any A < A*-that is, A* < Al/f'(0) < oo. Step 5: Problem (5.4) has no classical solutions provided A > A*. Assume that there exists IL > At and a corresponding solution v to problem (5.4). We first prove that for every A < A* we have ua < v in Q. Indeed, by the convexity of f it follows that

-A(v - u>,) = lf(v) - Af(ua) > A(f(v) - f(ua)) > A f'(ua) (v - u,\)

in n.

Hence,

(-A - A f'(ua))(v - ua) > 0

in 11.

(5.7)

Existence and bifurcation results

129

Because the operator -A - \f'(uA) is coercive, by Stampacchia's maximum principle, we deduce v > u,\ in 0 for all A < A*. Therefore, ua is bounded in LOO(S2) by v and there exists u* = lima/. ua in n and u* = 0 on 852. We claim that \,(A*f'(u*)) = 0. We already know that Aj(A*f'(u*)) > 0. Assume by contradiction that Ai(A* f'(u*)) > 0. In other words, the operator -A-A* f'(u*) is coercive. By applying the implicit function theorem to F(u, A) = -Au-,\f (u) at the point (u*, A*), there exists a curve of solutions of the problem (5.4) passing through (u*, A*), which contradicts the maximality of A*. Hence, al(A*f'(u*)) = 0, which implies the existence of 0 in ci such that cpl = 0

on 8i and -Awe

-

0

(5.8)

in 92.

Passing to the limit with A / A* in (5.7), we find

(-A - A* f'(u*))(v - u*) > 0

in Q.

Multiplying this inequality by cpl and integrating over S2, we obtain

- J (v - u*)Av,dx - A* J f'(u*)(v - u*)cpldx >_ 0. According to (5.8), the previous relation is in fact an equality, which implies that -A(v - u*) = A* f'(u*)(v - u*) in Q. It follows that pf(v) = A* f (u*) in Q. But p > A* and f (v) > f (u*) so that f (v) = 0, which is impossible. Hence. problem (5.4) has no solutions for A > A*.

Step 6: ua is the minimal and the unique stable solution of (5.4). Fix an arbitrary A < \* and let v be another solution of problem (5.4). We have

-A(v - ua) = of (v) - Af(ua) ? Af'(ua)(v - u,\)

in Q.

Again, by Stampacchia's maximum principle applied to the coercive operator -A - Af'(ua), we find that v > ua in 92. Now let w be another stable solution of (5.4) for some A < A*. With the same reasoning as applied earlier to the coercive operator -A - A f'(w), we derive that 0 ua > w in S2 and, finally, uA = w. This concludes the proof.

5.3 Existence and bifurcation results By Amann's bifurcation theorem we have the following information about problem (5.4): (i) There exists A* E (0, oo) such that (5.4) has solution when A E (0,,\*), and no solution exists if A E (A*, cc). (ii) For .1 E (0, A*), among the solutions of (5.4) there exists a minimal one, say ua. (iii) The mapping A'--p ua is a C' convex and increasing function.

Bifurcation and asymptotic analysts: The monotone case

130

(iv) u,\ can be characterized as the only solution u of (5.4) such that the operator -0 - A f `(u) is coercive; that is, ua is the only stable solution of (5.4).

Assuming now (5.5), we aim to discuss in an unified way some natural questions raised by (5.4), namely what can be said when A = A*? what is the behavior of ua when A approaches A*? are these results still valid if f is unbounded around the origin? is there other solution of (5.4) excepting ua? if so, what is their behavior? In the rest of this chapter we try to answer these questions. The approach we present in the following is the result of the work by Mironescu and Radulescu [142], {143]. We are first concerned with the monotone case, corresponding to

2 := tlirn (f (t) - at) > 0, where a:= limt.(,,, f (t)/t E (0, oo).

Theorem 5.3.1 Assume f > 0. Then we have (1) A* = Ai/a; (ii) The problem (5.4) has no solution for A = A*; (iii) ua is the unique solution of (5.4) for all 0 < A < A*; (iv) lima/A' ua = oo uniformly on compact subsets of Q. The bifurcation diagram in the monotone case is depicted in Figure 5.2.

A=At/a

it

FIGURE 5.2. The bifurcation diagram in Theorem 5.3.1.

Existence and bifurcation results

131

Proof (i)-(ii) If 0 < A < Al/a, then Af satisfies the hypotheses of Theorem 1.2.5. Hence, problem (5.4) has at least one solution for all 0 < A < Al/a, so A* > Ai/a. We claim that problem (5.4) has no classical solutions for A > Ai/a. Assume that there exists A > Al/a such that (5.4) has a solution u. Because the mapping s H f(t)/t is nonincreasing, it follows that A f (u) > Al u in Q. Then, multiplying by Cpl in (5.4) and integrating by parts, we have

Al I ulpldx = A f f(u)Vldx > Ai

r

J

This yields A = Ai/a and f (u) = au; that is, f (t) = at for all 0 < t < maxxC?j u(x). But this is a contradiction because f (0) > 0. Therefore A* = At/a and (5.4) has no solutions for A = A*. (iii) As we have argued in the previous section, the fact that f is convex and fulfills (5.5) implies that g : [0, oo) -> [0, oo), g(t) = f (t) - at is nonincreasing. Then ua satisfies -Aua - Aaua = Ag(ua) in Q. Let vA be another solution of (5.4). According to the minimality of UA, we have ua < va in Q. Thus, w := VA - UA > 0 satisfies -Ow - Aaw < 0 in S1. Because

as < Al, the strong maximum principle yields w = 0 in Q and the uniqueness follows.

(iv) Because of the special character of our problem, we will be able to prove that, in certain cases, L2 boundedness implies Ho boundedness! Assume by contradiction that (ua)o
u*(x) = lim' u,\ (x)

ala

for all x e 1.

According to our assumption, u* t= no. Because ua is increasing with respect to A, it follows that u* is lower semicontinuous. Let xo E 0 be such that u*(xo) no and denote by ica the spherical average of ua with respect to x0-that is,

ua(r) =

1 wN

ua(xo + ry)dv(y)

/

0 < r < dist(xo, 8Sl).

IyI=1

Because ua is a superharmonic function we have

ua(r) =

1

wN JIuI=1

Vua(xo+ry).yda(y) aua (xo

1

WN JIy1=1 an

=

Dug (xo + ry) d6 (y)

1

WN

<0.

+ ry) do-(y)

Il=1 y

Bifurcation and asymptotic analysis: The monotone case

132

Hence, ua is decreasing in (0, R), which implies

ua(xo) = ua(xo) > 1 f, ,1=1 ua(xo + ry)da(y)

for all 0 < r < dist(xo, 811).

WN

Integrating in the previous relation we obtain ua(x)dx < RWNua(xo) < Cu*(xo) < oo.

By Fatou's lemma we derive u* E L1(BR(0)). Let X := {x E S1 : u* is integrable in a neighborhood of x}.

From the previous arguments we deduce that X is open and nonempty. Furthermore, because u* is lower semicontinuous, it follows that X is also closedBecause St is connected we derive that X = S2-that is, u* E L oc (S2). This implies that (uA)o oo as A / A* and w,\ E L2(9),

11W)1112 =1.

(5.9)

Notice that f (t) < at + b for all t > 0, where b = f (0). This implies Af(ua) k(A)

-0

in L oC(9) as A / V.

That is, (5.10)

0

By Green's first identity, for all 0 E CO '(Q) we have Vwa -Vodx =

-fn c6Awadx = -J

i0wadx.

(5.11)

Supp

J0

Using (5.10) we obtain I0IIowaj dx

0 £w,\ dx Supp

f

(5.12) f AwA I dx -4 0

as A / A*.

Supp 46

Combining (5.11) and (5.12) we derive that for all 0 E Ca (S2) we have

i

n

asA/A*.

(5.13)

Existence and bifurcation results

133

By definition, the sequence (wA)o
10

=

k(A) f WAf(UA)dX

wA (aua + b)dx 2dX

*a /

A*b

+ k(A) 10 wadx

< \*a+cil01i/2, where cl > 0 does not depend on A. From the previous estimates it is easy to see that (wa)o
w. -> w

weakly in Ho (l), strongly in L2(fl).

(5.14)

On the one hand, by (5.9) and (5.14), we derive that 11w112 = 1. Furthermore, using (5.13) and (5.14), we infer that

Vw-©odx=0 forallOECa (Sl).

II

Because w E Ho (0), using the previous relation and the definition of Ho (1), we find w = 0. This contradiction shows that (ua)o
ua

u*

weakly in Ho (fl),

(5.15) strongly in L2 (Sl), ua -+ u* almost everywhere in Q. Now we obtain the desired contradiction in the same manner as in the proof

ua --> u*

of (i)-(ii). We first multiply by Aa sz

si

f

uMoidx

for all 0 < A < A*. (5.16)

st

Passing to the limit in (5.16) with A J' A*, by virtue of Lebesgue's theorem on dominated convergence we find

ai

fz

u*(Pi =

A1

a

*

f ('a )cpidx..

Hence, f (u*) = au*, which contradicts the fact that f (0) > 0. This shows that lima/a. uA, = oo uniformly on compact subsets of Q, and the proof is now complete.

0

134

Bifurcation and asymptotic analysis: The monotone case

Remark 5.3.2 The results in Theorem 5.3.1 hold in a little more general context -namely, the mapping (0, oo) 9 t 0 1 f (t)/t is nonincreasing, f satisfies (5.5), and f (0) > 0. By Theorem 1.2.5 and the strong maximum principle we may show that (i)-(iv) in Theorem 5.3.1 hold. However, the convexity assumption on f provides a complete description of the unique solution ua to problem (5.4). We will see in Chapter 6 that the hypothesis f convex is strongly needed when the issue of multiplicity arises.

Example 5.1 Let us consider the one-dimensional problem

-u" = a

u2 + u + 1

in-(0,7r),

u(0) = u(7r) = 0.

(5.17)

Clearly, f (t) = 02 -+t + 1, t > 0, obeys the monotone case described in this chapter. In this case AI = 1, and by Theorem 5.3.1 we have that (5.17) has a solution if and only if 0 < A < 1. Moreover, for all 0 < A < 1, problem (5.17) has a unique solution u),, which is stable, and lim. fi u), = oo uniformly on compact subsets of (0, 7r). Using the collocation method we have computed the solution ua of (5.17) when A approaches A1. The solution is plotted in Figure 5.3.

FIGURE 5.3. The unique solution of problem (5.17).

5.4 Asymptotic behavior of the solution with respect to parameters In this section we discuss the asymptotic behavior of the unique solution ua of (5.4) with respect to A. We have seen that limA fx. ua = oo uniformly on compact subsets of 9.

Asymptotic behavior of the solution with respect to parameters

135

We consider the normalized sequence (w,\)0<,\<.\- of (U'\)0<'\<'\- in L'(Q). That is, ua = k(A)w,\

with IIWA 12 = 1.

Because lima/a. ua = oo uniformly on each compact subset of Sl, we deduce that k(A) -> oo as A J' A*. From the proof of Theorem 5.3.1 we have a uniform bound for (w))o
Theorem 5.4.1 We have wa -+ cp1 in Cl(12) as A / A*. Proof Because (wa)o<, <X. is bounded in Ho (Sl) as A / A*, there exists w E Ha (SZ) such that, up to a subsequence, we have as A / A*: wa ' W

weakly in H, '(Q),

waw strongly in L2(c ), WA - W

(5.18)

almost everywhere in Q.

Then -Aw(A) -> -Ow in D'(SZ). On the other hand, we have

-Owa = A k((A))

i n Q.

(5 . 19)

We claim that f (uA)/k(A) -> aw almost everywhere in fl as A / A*. If x E Q is such that w(x) > 0, then, by (5.5), we obtain lira

'\/A

f (k(A)wa(x)) k(A)

= lira f (k(A)w,\ (x))wa(x) = aw(x). ala k(A)w,\(x)

If w(x) = 0, let e > 0 and 0 < Ao < A* be such that w,\(x) < a for all Ao < A < A*. Because f (t) < at + b for all f (k(A)WA(x)) k(A)

< ae +

b

k(A)

0 (with b = f (0)) we deduce for all Ao < A < A*.

This yields f(wa(x))/k(A) -* 0 as A / A*, which proves the claim. Passing to the limit with A / A* in (5.19) we obtain -OW = A1w, W E HH(fI), w > 0, IIwI12 =1.

But this means exactly that w = cp1. Moreover, (wa)o
Bifurcation and asymptotic analysis: The monotone case

136

Theorem 5.4.2 The sequence ((pi/wa)o<,,
Proof The strong maximum principle implies that 8w,\/an < 0 on 812. Therefore,
for all x E 852.

8wa (x) /,9n

Using the regularity of 4952 we can find b > 0 such that if

A5 :_ {x E RN : d(x) < b}, then (i) for each x E A6 there is a unique xo E aft such that d(x) = 1x - xoI; (ii) if 7r : A6 -> 85t is defined by ir(x) = xo, then Tr E C'(A6); (iii) if Ix-ir(x)I = e then x = lr(x)-en(7r(x)) or x = 7r(x)+en(ir(x)), according to the case xEQor xIV 5l; (iv) if x E SZ then [x, ir(x)) C 1. Let 526 = 52 It Ad. Because wa -' cpi uniformly in S2, it is easy to see that cal/wA is uniformly bounded in 12 \ 526. If x E 526, let xo = 7r(x). Then cai(x) - cpi(xo) _ acai(xo +T(x - xo))/an(xo) awa(xo +T(x - xo))/an(xo) wa(x) - wa(xo)

col(x)

w,\(x)

for some T E (0, 1). Taking a smaller 5 if necessary, we may assume that awa (x) <0 8v(ir(x) )

on 526.

Thus, the quotient WI/w,\ is uniformly bounded in 126 for A near A*. This con0 cludes the proof.

Theorem 5.4.3 We have

Proof Let to be a limit point of (A* - A)Ilua112 as A / A*. Multiplying by cpi in (5.4) and then integrating by parts we obtain Al fz uAcpldx = A

f

n

f(ua)cpldx,

which may be rewritten in the form

fwi(Ai - aA)uadx = Af (f(uA) - aua)cpidx. i

st

Examples

137

Hence,

coi(A1- aa)IIu.\ II2w,dx = J \vl(f(ua) - aua)dx.

(5.20)

sl

Note that the right-hand side integrand in (5.20) converges dominated to A*2cp1 A*. Also the left-hand side integrand tends to eocp1. Now, passing to the as a limit in (5.20) we find to = (A12/a) IIWi II j . This finishes the proof.

5.5 Examples Theorem 5.4.3 describes the growth rate of (a* - A)IluaII2 as A / a*. If 2 = 0 the convergence depends heavily on f, and from Theorem 5.4.3 we deduce only that 1Iua112 grows slower than (A* - A)-'. In the curent section we illustrate this aspect by the following two examples. 5.5.1

First example

Proposition 5.5.1 Let f (t) = t + 1/(t + 2), t > 0. Then, the unique solution ua of (5.4) satisfies A/AI lim

Al - AIIua112 =

aljnj-

Proof With the usual decomposition ua = k(A)w,\ and dividing (5.20) by A1-Awehave al - Ak(A)wx
Ac01

A-1 --A (k (A) wA + 2)

dx.

(5.21)

We first infer that lim infA/,\l Al - Ak(A) > 0. Otherwise, passing to a subsequence we may suppose that al - Ak(A) -40. Then al - Ak(A)wxcpl --+ 0

uniformly in SZ

and

A1 - A(k(A)wa + 2) --> 0

uniformly in S2,

which contradicts (5.21). We also claim that limsup,\/Al Al - Ak(A) < oo. Supposing the contrary, and passing to a subsequence, we have al - Ak(A) ---> oo as A / A1. Then the left-hand side in (5.21) tends to oo whereas the right-hand side remains bounded because by Theorem 5.4.2, the quotient cpl/wa is uniformly bounded in a.

Let c E (0, oo) be a limit point of A, - Ak(A) as A / A1. Then the lefthand side of (5.21) tends to c, whereas the right-hand side integrand converges dominated to Ai/c. Hence, c = A, 101/c, which finishes the proof. Remark 5.5.2 A similar computation can be made for f (t) = t2 + 1, t > 0.

Bifurcation and asymptotic analysis: The monotone case

138

Second example

5.5.2

If f (t) - at decays to oo faster than 1 It, then. the behavior becomes more complicated. This is illustrated by the following example.

Proposition 5.5.3 Let f (t) = t + (t + 1)-2. Then Ilua II2 tends to oo as A / Al like no power of (A -- A). More precisely, we have

(i) im (A - A)"IIuA II2 = oo if a < 1/3; (ii)

lim (Al - A)'Ilua112 = 0 if a > 1/3.

alal

Proof We first need the following auxiliary result.

Lemma 5.5.4 There exists eo > 0 and two positive constants c1, c2 > 0 such that 1

c1I lnel <

dx < c2l In El

for all 0 < E < eo.

(5.22)

in Q.

(5.23)

{w1>e} P1

Proof Consider c > 0 such that cd(x) < cp1 < 1 d(x) c

Let b > 0 and Ab be as in the proof of Theorem 5.4.2. Define 4; : Ab -+ t9

x (-b, 5)

and

W : 8St x (-S, b)

Ab

by

4i(x) = (7r(x), (x - ir(x), n(7r(x))))

Then 4, T are smooth and

and

$(xo,e) = xo +En(xo).

= I -1. Replacing b if necessary, by a smaller

number, we may assume that there exist C1, c2 > 0 such that 0 < c1 < IJ(WY)I < c2

in 8!2 x

(5.24)

where J(W) is the Jacobian of W. Let Eo

min{ inf cp1, c5}.

Now, for 0 < E < Eo we decompose

f

dx = w1>E} Y1

1 dx + J{fGwi <eo} W11 dx.

J{t't>eo} 1

Note that by (5.23) we have {E/c < d(x) < ceo} C {e < cp1 < Eo} C Ice < d(x) < eo/c}.

Hence,

(5.25)

Examples

139

I

1 dx<{e«i<eo} (pl dx

cJ

{e/c
(5.26)

<1f C

1

/ fce
dx.

With the change of coordinates x = T (xo,77) and by virtue of (5.24) we obtain 1

dx =

{e/c
IJ()I dv xo)dr}

I

77

(5.27)

c2e0

> C1IM I In

Similarly

f

c
e

I dx = J

} d(x)

IJOY) Ido,(xo)dn anx(ce,

)

(5.28)

71

< c2lasiI h. '0

C26

Now the claim follows from (5.25) through (5.28).

U

Let us come back to the proof of Proposition 5.5.3. (i) It is enough to show that

lim (A1- \)1,3IIuAII2 = 00.

A/Al

Supposing to the contrary, there exists c E [0, oo) such that up to a subsequence we have (a1 - A)1/3k(1) - c as A / A*, where k(A) = IIua112. Dividing by (a1 - A)2/3 in (5.20) we obtain

L (A1 -

A)1/3'lk(A)wAdx

=

A

f(

1)2dx.

1 -,

A)2/3(k(A)wa +

(5.29)

If c = 0 then the left-hand side in (5.29) converges to 0, whereas the righthand side tends to oo. Hence, c E (0, oo). Let e > 0 be fixed. Because (A1 A)2/3(k(A)wa + 1)2 -->

as A / A1i we can find 0 < Ao < Al such that

(Al - A)2/3(k(A)wa + 1)2 < c2(cp2 + e)

for all AO < A < A1.

Then (5.29) yields

L Because the limit of the left-hand side is c, we deduce

c >C2-

'r1

J11'P2+S

dx

for all F > 0.

Letting s \ 0, by Lemma 5.5.4 we obtain c = oo, the desired contradiction.

140

Bifurcation and asymptotic analysis: The monotone case

(ii) We argue again by contradiction. Thus, there exists a > 1/3 such that up to asubsequencewehave (A1 - A)1-a in (5.20) we obtain

(.11 - A)alplk(A)dx = A in (.\1

-

A)2c,-j3(k(A)w,\ +

1)2dx.

(5.30)

The limit of the left-hand side is c E (0, oo]. To estimate the right-hand side, we have

r

Cpl dx 0<J{w(A1 - A)1-a(k(A)wa + 1)2

ft(A1-'\1 A)1dx

A)aII I -40

as A / A*.

By Lemma 5.5.4 we also obtain

0<J

X01

C(A1

<

(i11 - .1)1-a(k(A)wa + 1)2 )3a-1

dx

1dx--0 asA/A*,

J{cP1>ai-a} 'p1

where C = sup0
5.6

O

The case of singular nonlinearities

Our aim in this section is to supply a similar result to that in Section 3.3 in case f is singular at the origin. Assume that f satisfies the following: (f 1) The mapping (0, oo) 3 t --- f( t) is decreasing and there exists lim ff (t) =a E (0, oo)

t-+0o

t

(f 2) lims\o f (s) = oo and there exists 0 < a < 1 such that

f (t) = O(t-a)

as t \, 0.

Thus, we are dealing with the monotone case, but in its nonsmooth variant. Note that, because of the assumptions (f 1) and (f2), the nonlinearity f is no longer monotone. Also notice that condition (f 1) allows us to relax the convexity

requirement on f . Furthermore, in view of (f 1) we have f (t) - at > 0 for all t > 0 and we allow f := limt_ ) (f (t) - at) to be oo. As may be easily verified b y the reader, the mapping f (t) = 1 / ' + f + t, t > 0 satisfies (f 1) and (f2), is not convex, and f = oo (see Figure 5.4). The convexity assumption of f is needed in the framework of Amann's theorem and provides the stability of the minimal solution. In our context we prove the uniqueness of a classical solution whereas the stability of the solution will be achieved in a more general setting in Chapter 8.

The case of singular nonlinearities

141

f(t)

FIGURE 5.4. The singular monotone case on f with t < oo respectively f = 00.

Theorem 5.6.1 Assume that condition (f 1) and (f2) are fulfilled. (i) If A > At/a, then problem (5.4) has no classical solutions. (ii) If 0 < A < Ai/a. then problem (5.4) has a unique solution ua such that (iii) ua E C2(S2) n C1,1-- (P); (ii2) lima/A. ua = oo uniformly on compact subsets of Q. Proof Existence and nonexistence of a classical solution as well as the uniqueness follows in the same manner as in the proof of Theorem 5.3.1. The regularity of the solution in (iii) is similar to that obtained in Theorem 4.3.2. For the proof of (ii2) we claim that the arguments used in Theorem 5.3.1 (iv) still work here. Indeed, we have only to show that the sequence wa = UA/Ilux112 (0 < A < Ai/a) is bounded in Hol (S2). Using the assumptions (f 1) and (f2), there exists b, c > 0

such that f (t) < at + bt_a + c for all t > 0. Then, by Holder's inequality we may write

jIvwAI2 = - fwiw < _ A*a

(A)

k(a)

J waf (U,\)

J wa(aua + bua- + c)

fn

w2 + A*b

A*c r A*b W1_0, wa + a Jo k(A) J k(a) Iq(1+a)/2 + A*c I911/2.

< A*a + k(A)1+a

k(A)

From now on, we follow the proof of Theorem 5.3.1 line to line. This completes 0 the proof.

Finally, let us remark that the same result holds for bifurcation problems of the type

Bifurcation and asymptotic analysis: The monotone case

142

U

10

(0,0)

A

1 = A,/ a

FicuRE 5.5. The bifurcation diagram for the singular problem (5.31).

-Du = g(u) +.\f (u)

u>0 u=0

in Sl,

in Q, on B1,

(5.31)

where g is a singular nonlinearity that satisfies (f 2) and f is a C' function having a sublinear growth. The bifurcation diagram is depicted in Figure 5.5.

5.7 Comments and historical notes The constant a` in the statement of Theorem 5.2.1 is also called extremal value (or the frank-Kamenetskii constant in the combustion literature). The class of functions satisfying assumption (5.2) includes the family of convex mappings f : 10, oo) --+ (0, oo), which are increasing in a neighborhood of 00.

The important example corresponding to f (t) = et is related to the celebrated Liouville-Gelfand problem

Du=Ae"

in St,

u=0

on 0"0.

(5.32)

This problem was initially studied by Liouville [132) if N = 1 and, subsequently,

by Gelfand [82]. The solutions of (5.32) arise as steady-state solutions of the nonlinear evolution problem (called the solid fuel ignition model)

vt = Av + A (I -

v=0

in d, on 811,

within the approximation e << 1, where A is known as the Frank-Kamenetskii parameter (see [80]), v is a dimensionless temperature, and s-1 is the activation energy.

s

BIFURCATION AND ASYMPTOTIC ANALYSIS: THE NONMONOTONE CASE We are servants rather than masters in mathematics. Charles Hermite (1822-1901)

6.1 Introduction The monotone case in problem (5.4) corresponding to an increasing, positive,

and convex nonlinear term f satisfying f (t) > at for all t > 0, where a :_ limt . f (t)/t E (0, oo), was studied in Theorem 5.3.1. The proof of this theorem relies essentially on the maximum principle. The complementary nonmonotone case is more difficult and more rich in information. This different setting is analyzed in this chapter by means of combined arguments, including variational methods like the mountain pass theorem of Ambrosetti and Rabinowitz.

(0,0)

t

FIGURE 6.1. The nonmonotone case on f. More precisely, we study the bifurcation problem (5.4) if the nonlinear term f obeys the nonmonotone case (see Figure 6.1)-that is, f : (0, oo) - (0, oo) is a Cl convex function such that f`(0) > 0, there exists a := limt- 3 f (t)/t. E (0, oo), and

144

Bifurcation and asymptotic analysis: The nonmonotone case

t:= tl (f (t) - at) E [-oo, 0). The basic hypothesis 2 < 0 will change-radically the study of (5.4). At the same time, new and different methods are needed to deal with the multiplicity of solutions to (5.4).

Auxiliary results Lemma 6.2.1 Let a E L°O(Q) and w c- HH(St) \ {0}, w > 0, be such that 6.2

A1(a) < 0 and -Ow > aw in 9. Then the following properties are valid: (i) A, (CO = 0.

(ii) -Ow = aw in Q. (iii) w > 0 in St.

Proof Let us multiply the inequality -Ow > aw by col(a) and then integrate by parts. We obtain

acpl(a)wdx+A1(a) fz

f

cpi(a)wdx >

r

Jn

a(pl(a)wdx.

This implies that A, (a) = 0 and -Ow = aw in Q. Because w > 0 and w O_ 0, we deduce w = cV1(a) for some c > 0, which concludes the proof. 0

It is .obvious that A f satisfies the hypotheses of Theorem 1.2.5 for all 0 < A < Al/d. This yields A* > Al/a. Lemma 6.2.2 The following properties hold true: (i) If problem (5.4) has a solution for A = A*, then this solution is necessarily unstable.

(ii) Problem (5.4) has at most one solution when A = A*. (iii) u,, is the only solution u of problem (5.4) such that A,(Af'(u)) > 0. Proof (i) Suppose that (5.4) with A= A* has a solution u* with A1 (A* f'(u*)) > 0. Thus, by the implicit function theorem applied to the mapping G : CQ'1/2(?) x l1 , C°°1/2(52),

G(u, A) = -Du - Af(u),

it follows that problem (5.4) has a solution for A in a neighborhood of A*, contradicting by this the definition of A*. (ii) Let u be a solution of (5.4) for A = A*. Then u is a supersolution of (5.4) for all 0 < A < A*. Using the minimality of ua, it follows that u > ua in 92 for all 0 < A < A*. This shows that ua (which increases with A) converges to a certain u* in L'(Q). Let us multiply by ua in (5.4) pand integrate by parts. We obtain

f

IVuaj2dx = A J f (ua)u. dx < A* o

Hence, (ua)o<j
f uf(u)dx. s1

Auxiliary results

145

We claim that ua - u* in Ho (ft) as A / A*. Indeed, if v is a limit point of (ua)o v almost everywhere in Q. But ua -> u* a.e in ft. Hence, u* E H0(9). The proof will be concluded when we show that u = u*. Let w = u - u* > 0. Then

-Aw = A* (f (u) - f (u*)) > A* f'(u*)w

in Q.

(6.2)

Because A, (A* f'(u*)) < 0, by Lemma 6.2.1 it follows that either w - 0 or

w > 0,

Al (A* f'(u*)) = 0, and - ©w = A* f'(u*)w in 9.

Assuming w $ 0, by (6.2) we deduce that f is linear in all the intervals of the

form [u*(x),u(x)], x E Q. It is easy to see that this forces f to be linear in [0, max.-u]. Let a, f3 > 0 be such that f (u) = au + ,Q and f (u*) = au* + /3. We have

0 = A1(A*f'(u*)) = A1(A*a) = Al - A*a,

that is, A* = Al/a. Hence, u satisfies

-Du = A zu + A1-

in

ft ,

in ft, onoSl,

U>0

u>0

which is a contradiction. Thus u* = u. (iii) Suppose that problem (5.4) has a solution u # ua with A,(Af'(u)) > 0_ By the strong maximum principle we derive u > ua in Si. Let w = u - ua > 0. Then -Aw = A(f (u) - f (ua)) < A f'(u)w in ft. (6.3)

Next, we multiply by cp := ccl(Af'(u)) in (6.3) and we integrate by parts. We find

AJ f'(u)cpwdx + A,(Af'(u)) n

J

cpwdx < A f

s?

f'(u)cpwdx. t

Thus, A1(Af'(u)) = 0 and in (6.3) we have equality that is, f is linear in [0, maxi u]. Let a, /3 > 0 be such that f (u) = au +,3, f (ua) = aua + /3. Then 0 = A1(Af'(u)) = A1(Af'(ua)),

which is a contradiction. This finishes the proof.

Lemma 6.2.3 The following conditions are equivalent: (i) A* > Al/a. (ii) Problem (5..¢) has exactly one solution for A = A*. (iii) The sequence (ua)5
Bifurcation and asymptotic analysis: The_nonmonotone case

146

Proof (i)=(ii) Assume that (5.4) has no solution for A = A*. We claim that ua -> oo as A / A* uniformly on compact subsets of Q. Supposing the contrary, with the same arguments as in the proof of Theorem 5.3.1 (iv) it follows that

(ua)o oo as A / A* uniformly on compact subsets of Q. Let

ua=k(A)w.,with IIUA112=1, forall0
w

weakly in Ho (O),

wa

w

strongly in L2 (1l), almost everywhere in fl.

w), - to

Then -aw), - -Aw in D'(SZ). On the other hand, we have -Ow), = A

in ft.

(6.5)

As in the proof of Theorem 5.3.1, we further deduce 1

k(A)

f(ua)-*aw inL2 (S1)

asA/°A*.

Passing to the limit in (6.5) we obtain -Aw = A*w, W E Ho(fl), w > 0, 11wII2 = 1.

This yields w = cpl and A* = Al /a, which is a contradiction. Hence (5.4) has at least a solution, say u*, when A = A*. By Lemma 6.2.2 (ii), it follows that u* is the unique solution of (5.4) for A = A*. (ii)=(iii) We have seen in Lemma 6.2.2 (ii) that if (5.4) has a solution,u* for A = A*, then u,, -+ u* almost everywhere in fl as A / A*.

It suffices to prove that ua has a limit in C(fl) as A / A*. Even less, it is enough to prove that (ua)o
Existence and bifurcation results in the nonmonotone case

147

The following result asserts that the limit of a sequence of unstable solutions corresponding to problem (5.4) is also unstable.

Lemma 6.2.4 Let un be a solution of (5.4) for some A = An > 0 such that A1(A.n f'(un)) < 0 and as n -> oo we have An - A, un - u in HH(0). Then u is an unstable solution of problem (5.4) -that is, A1(Af'(u)) < 0.

Proof The fact that u is a solution of (5.4) follows by standard elliptic arguments. To prove that u is unstable, let us first notice that by the minmax characterization (5.6) of the first eigenvalue, condition A1(a) < 0 for some a E L°°(52) is equivalent to the existence of E Ho(Q) such that IIVII2 = 1 and

< fs

;2fIvI2dx

Therefore, for all n >/ 1 there exists /f n E Ho (f l), II'n II 2 = 1 such that

J IVVnl2dx <

An f'(un)V) dx.

(6.6)

t

sa

Because f' < a, from (6.6) we derive that (zfin)n>1 is bounded in HO(SE). Let V) E Ho (SE) be such that, up to a subsequence, 7l' - ' in Ho (S2) and Yin --> ip in L2(1). Furthermore, up to a subsequence, (V)n)n>1 is dominated in L2(l). This yields IIb1I2 = 1 and

j

J1n f'(un)Vindx -* Af'(u)zb2dx

as n -> oo.

(6.7)

By (6.6) and Fatou's lemma we obtain

L We have thus obtained the existence of 0 E HH(SE) such that 110112 = 1 and fsi I©/I2dx < f0A f'(u)02dx-that is, u is unstable. This finishes the proof of 0 our lemma. 6.3 Existence and bifurcation results in the nonmonotone case In the framework of the curent chapter, existence and bifurcation results are completely different to those presented in Theorem 5.3.1 in the monotone caseFurthermore, the uniqueness no longer holds in a neighborhood of the bifurcation point A*. This will be proved by means of the mountain pass theorem of Ambrosetti and Rabinowitz (see Theorem D.1.3). Let m := mint>o f (t)/t. As a result of (6.1) we have m > a. Theorem 6.3.1 Assume that f verifies (5.5) and (6.1). Then the following hold: (i) A* E (A1/a, Al/m).

Bifurcation and asymptotic analysis: The nonmonotone case

148

(ii) Problem (5.4) has exactly one solution u* for A = A*;.

(iii) lima/, . ua = u* uniformly in N. (iv) ua is the unique solution of (5.4) for all 0 < A < Ai/a. (v) For all Al/a < A < A*, problem (5.4) has at least an unstable solution va. For any solution v,, 0- ua we have (vi) lima\al/a va = oo uniformly on compact subsets of Il; (vii) limA fa. va = u* uniformly in U.

The bifurcation diagram in the nonmonotone case is depicted in Figure 6.2.

FIGURE 6.2. The bifurcation diagram in Theorem 6.3.1.

Proof We first prove that A* _< Al /m. To this aim, it suffices to show that problem (5.4) has no solution for A = Al/m. Suppose the contrary. Let u be a solution of (5.4) for A = Al/m. Then, multiplying in (5.4) by cpl and integrating by parts we find Al

J

cprudx = A f cot f (u)dx.

(6.8)

Furthermore,

Al J cc1ndx =

in

1m

J ,1f(u) ? Al f o1udx, fn n

which forces f (u) = mu. This clearly contradicts the fact that f (0) > 0. The remaining parts of (i), (ii), and (iii) are equivalent in view of Lemma 6.2.3. In this sense, it is enough to prove that A* > Al/a. Assume by contradiction that A* < Al/a. By Lemma 6.2.3 we derive that A* = Al/a. Then, problem (5.4) has no solution for A = A*. Indeed, if u* would be such a solution, then by Lemma

Existence and bifurcation results in the nonmonotone case

149

6.2.2, u* is necessarily unstable. On the other hand, because f'(u*) < a we obtain

0 < a1(A* f(u*)) < a1(A*a) = 0. Hence, A (A* f'(u*)) = 0-that is, f'(u*) = a. We now proceed as in the proof of Lemma 6.2.2 (ii) to obtain a contradiction. Hence, (5.4) has no solution for A = )11/a. With the same arguments as in the proof of Lemma 6.2.3, we derive limA/A. ua = oo uniformly on compact subsets of Q.

From (6.8) we have

0=

Js

Jn

o1 [A1uA - Af (ua)] dx

cpl [(A1 - aA)uA - A(f (uA) - auA))]dx

-J

0

(6.9)

W i If (ua) - au,\] dx.

Because lima/a. ua (x) = oo for all x E 9, it follows that f (ua (x)) - aua (x) e as A / A*, for all x E Q. Passing to the limit in (6.9) we obtain the contradictory inequality 0 > -DA* fn cpi > 0. We have proved that Al/a < A* < Aj/m, and by Lemma 6.2.3, we have that problem (5.4) has a solution when A = A*. This shows that )1* < Ai/m. (iv) Taking into account Lemma 6.2.2 (iii), it is enough to prove that for all 0 < A < Ai/a, any solution u of (5.4) verifies a1(Af'(u)) > 0. Because f'(u) < a we obtain A1(Af'(u)) > A1(Aa) > 0 for all 0 < A < aI/a. (v) We want to find solutions of (5.4) different from ua, which are critical points other than u\, of the energy functional J : Ho (12) -4 R,

J(u) = 2

J

[Dul2dx

-

fF(u)dxw

F(t) = A f f f (s)ds. In the following, for each Al/a < A < A* we will take ua as uo in the mountain pass theorem of Ambrosetti-Rabinowitz. Clearly, J E C1(Ho (S2), R) and uo is a local minimum for J. To apply the Ambrosetti-Rabinowitz theorem, we transform uo into a local strict minimum by modifying J. Let E E (0, 1) and define

JE:Ho(se)-R, JE(u)=J(u)+2f IV(u-uo)I2dx. Then JJ E C1(Hfl (S2), R) and for

u, v E Hp (I) we have

JE(u)v = j Vu Vvdx - A J f (u)vdx + E in O(u - uo) Ovdx. t

insz

s2

Moreover, uo is a local strict minimum for JE provided e > 0 is small enough.

Lemma 6.3.2 Let Eo = (Aa - )11)/(2x1). Then there exists vo E Ho (1) such that JJ(vo) < JE(uo)

for all 0 < E < Eo.

Bifurcation and asymptotic analysis: The nonmonotone case

150

Proof Because JE is nonincreasing with respect to e, it suffices to prove that

-oo.

lim JeO (tcpl)

1C'O

Notice that JE(tcci) = 21 t2 + 2 ait2 - exalt in (pluodx

+ 2 j IVuoI2dx a

f F(t5o1)dx.

in

Let a = (3a.\ + )1)/(4A). Because a < a, there exists a E R such that f (t) > at +,0 for all t > 0, which implies that F(t) > as/2t2 + J3At when t > 0. Then (6.10) shows that

i--

t2 A (tto1) <-

lim sup

1

al + eo,\l - as < 0, 2

0

because of the choice of a. Hence, vo := tcpi satisfies the conclusion of lemma for

larget>0.

0

The following result states that the Palais-Smale condition (see Appendix D) on Je is satisfied uniformly in E.

Lemma 6.3.3 Let (ur)n>1 C HO '(Q) and 0 < en < eo be such that and

is bounded in R

J(un) -3 0 in H-1(52).

(6.11)

Then (ur)n>i is relatively compact in Ho(c2).

Proof It suffices to prove that (ur)n>1 contains a bounded subsequence in Ho(c2). Indeed, suppose we have already proved this claim. Then, there exist 0- < e < eo and u E Ho (52) such that, up to a subsequence and as n - oo we have en -* a and

un - u

weakly in HQ (52),

un -* u

strongly in L2(52), almost everywhere in ft

un

u

Because JE, (un) -> 0 in H-1(92) we deduce

-Dun - A f (un) - En0(un - uo) - 0

in D'(52) as n

oo.

(6.12)

Note that f' < a implies F

if (un) - d (u)) < aIun - ul

In 0,

which yields f (u,,) --+ f (u) in L2(Q) as n --+ oo. Thus, from (6.12) we have

-Du-Af(u)-e0(u-uo) = 0.

(6.13)

Existence and bifurcation results in the nonmonotone case

151

Multiplying (6.13) by u and integrating over Il we obtain

(1+E)i IVul2dx-AJ of(u)dx.-eAJ of(uo)dx=0. in n in

(6.14)

From (6.11) and the boundedness of (un)n>1 we also deduce JE(un)un

0

as n -+ oo.

Therefore, as of -+ oo we have

(1 + En) in I Vun12dx - A in unf (un)dx - EnA in unf (uo)dx - 0.

(6.15)

Combining now (6.14) with (6.15) we obtain un -f u in HO'(Q) as n -+ oo. We now establish that (un)n> 1 is, up to a subsequence, bounded in HI(Q)To this aim, it suffices to show that passing to a subsequence (un)n>1 is bounded in L2(1). Then, by virtue of (6.11) we deduce that (un)n>1 is bounded in Ha(1). Let un = knwn with IIwn112 = 1, kn > 0 and assume by contradiction that up to a subsequence we have kn -* oo as n -p oo. We also may assume that En --+ e. By (6.11) we derive JE,. (un) k2n

-40

asn ->oo.

That is,

2f

J Vwn 12dx -

kn

f

2

F(un)dx + 2

dx -+ 0

st

(6.16)

as n --+ oo. Notice that n IV

2J

(Wn

- !!-0) kn

2

dx = 2 J IVwnl2dx sa f En 2 uo dx + 2n kJ

Af - En kn J

w.f(uo)dx,

so that (6.16) produces l m.

1+2En

('

J JVwn12dx - k2 12

f F(un)dx = 0.

(6.17)

2

Because F(un) = F(knwn) < Aaknw2/2+Abknwn (where b = f(0)), from (6.17) we deduce that (wn)n>1 is bounded in Ha(ft). Let w E HH(S1) be such that up to a subsequence and as n - oo we have

w,, - w wn

w

wn -+ w

weakly in Ho (1), strongly in L2(St), almost everywhere in ft

Bifurcation and asymptotic analysis: The nonmonotone case

152

We claim that -(1 + E)Aw = Aaw.

(6.18)

Indeed, dividing (6.11) by k,,, we have

hm {( 1 + -,,. )

j

V wn - V v dx -

A / .f(u") v dx }=0 , kn

lS2

ffff n

ll

(6 . 19)

JJJ

for each v e Ho (1). This implies (1

+ E) in Vw . V v dx =

A h 1 f (unn) vd x 0

for all v E HQ (0) .

(6 . 20 )

With the same arguments as in the proof of Theorem 5.4.1, we derive that (f (u,,)/kn)n>1 converges almost everywhere to aw in Q. Moreover, because f is asymptotically linear and up to a subsequence (wn)n>1 is dominated in L2(St), we obtain f (un)/kn - aw in L2(0) as n - oo. Now, from (6.20) we obtain

(6.18). Furthermore, from (6.18) we deduce that w = '1 and Aa/(1 + E) = al, which contradicts the choice of Eo. This completes the proof.

0

Let us come back to the proof of (v) in Theorem 6.3.1. By the AmbrosettiRabinowitz theorem, for all 0 < E < Eo there exists ve E Ho '(!Q) such that Ef (vv) +

1

+Ef (uo)

in Q.

(6.21)

Set Ce = Jelve)

Because JE increases with E, we have co < cE < cep-that is, is bounded. Furthermore, by Lemma 6.3.3 there exists v E Ha (f) such that, up to a subsequence, vE - v in H01(9) as e -> 0. Now (6.21) implies -AV = A f (v) in Sl and, by standard regularity arguments, v is a classical solution of (5.4). To conclude the proof of (v) it remains only to show that v # uo = uA. This will be achieved by Theorem 5.2.1 and the fact that v is an unstable solution of (5.4). It can be readily seen that uo is the minimal solution of (6.21). Therefore, by Theorem 5.2.1 applied to the mapping

RDt,--ti

A

'\S

+

Ef(t)+l+Ef(uo),

it follows that vE is unstable in the sense that Al

(-_.j'V) < 0.

By virtue of Lemma 6.2.4 we derive that A, (A f ` (v)) < 0 and thus uo completes the proof of (v).

v. This

Existence and bifurcation results in the nonmonotone case

153

(vi) Suppose the contrary. With the same arguments as in the proof of Theorem 5.3.1 there exist (An)n>1 C (0, oo) such that An \ Al/a and vn E C2(SI) fl C(Q) a solution of (5.4) for A = An which is unstable and the sequence

(vn)n>1 is bounded in Li (SI). We first claim that (vn)n>1 is unbounded in Ho (52). Otherwise, let w E Ho '(Q) be such that, up to a subsequence, vn -k w weakly in H01(9) and strongly in L2(1). Then

-Avn -* -Aw in D'(SI) as n -> oo and

f (vn) -+ f (w)

in L2(SI) as n -4 oo,

which yields -Aw = Al f (w)/a. Thus, w E C2(SI) fl C(SI) is a solution of (5.4) for A = Al/a. From Lemma 6.2.4 we also deduce that A1(A1 f'(w)/a) < 0-that is, w u,\,/a,, which contradicts (iv) in the statement of our theorem. Because of the nature of our problem, the fact that (vn)n> 1 is unbounded in L2(SI) implies that (vn)n>1 is also unbounded in Ho(SI). Let vn = knwn with 1Iwn I12 = 1, kn > 0 and, up to a subsequence, k, -4 oo as n -* oo. This yields

-Awn = " f (vn) - 0 n

in Lio(I) as n --+ oo.

Also remark that the previous convergence holds.in the distribution space D'(Q), and (wn)n>1 is bounded in H01(Q) with an already provided argument. If w is a limit point of (wn)n>1 in H0 1(Q), we obtain -Aw = 0 in SI and 11w112 = 1, the desired contradiction. (vii) As before, it is enough to prove that (VA)a
Supposing the contrary, there exist (A.)a oo as n -> oo. If we write again vn = kvi,, then -Awn = A f (un) in Q. (6.22)

The fact that the right-hand side of (6.22) is bounded in L2(l) implies that (wn)n>1 is bounded in Ha (Q). Let to be such that, up to a subsequence, wn --> w weakly in Ho (SI) and strongly in L2(1). A computation already done shows that

-Aw = A*aw, w > 0 in SI and 11w112 = 1, which forces to v1 and A* = A,/a. This is clearly a contradiction because O A* > Al/a. This concludes the proof of Theorem 6.3.1.

With the same arguments as in Section 5.4 we may describe the asymptotic behavior of the unstable solutions as A \ Al/a.

Theorem 6.3.4 Let f : [0, oo) -+ (0, oo) be a Cl convex function that satisfies hypotheses (5.5), (6.1), and f'(0) > 0.

154

Bifurcation and asymptotic analysis: The nonmonotone case

(i) For any unstable solution va of (5.4) we have lim (A1 -- aA) IIvx 112 = L.

a\,Aila

(ii) If va = k(A)w,\ with IIW)112 = 1 we have wA -> W in Cl(l2) as A \ Al/a, and the quotient Cpl /w,, is uniformly bounded when A approaches Al/a.

6.4 An example Example 6.1 Let f (t) = t + 2 - t -+I, t > 0. Then lim (A -AZ)ztlva1I2

Al

j3f2) z

X01

Proof Multiplying by W, in (5.4) and integrating over lZ we obtain (Al - A)

Jn

cplvadx =

Aj(2 -

v + 1)(pldx.

v,\co1dx to both sides of the previous equality and then multiplying by A - Al we obtain Adding A fn

f c,1(A - A1) v),(A - (A - Al) va)dx = 2A(A - Al) fo Wjdx (6.23) - A ( P1(A - A1)(

va + 1 -

va dx.

Jsl

Let vA = k(A)w,\, where k(A), wa are as usual. We first prove that limsupa\,\, (A - A1)2k(A) < oo. Supposing the contrary, up to a subsequence, we have (A-A1)zk(A) -+ oo as A \ A1. Then the right-hand

side of (6.23) tends to zero whereas the left-hand side goes to -oo as A \ A1, which is a contradiction. Furthermore, from (6.23) we have

Cpl v (A - (A - A1) v,)dx < 2A r p,dx.

(6.24)

If lim inf,NA, (A-A1)2k(A) = 0 then the left-hand side of (6.24) goes to oo; which

is again a contradiction. Now let c E (0, oo) be a limit point of (A - al)zk(A) as A \ A1. From (6.23) we deduce c = (A1 f cpi/zdx)z and the proof is now complete.

D

6.5 Comments and historical notes We have seen in Theorem 6.3.1 that A is a returning point from which unstable solutions va begin to emanate, which exist for any A E (Ai/a, A*). It has been proved that any such solution va has the same behavior, in the sense that

Comments and historical notes

lim

,\\,\I /a

va = oo

155

uniformly on compact subsets of S2

and

lim va = u*

uniformly in S2,

where u* is the (unstable) solution corresponding to A = A*. An interesting open problem is to establish whether there is a unique unstable solution vA, for any A E (a;/a,A*). Returning points exist in many bifurcation problems. An interesting phenomenon is related to the Liouville-Gelfand equation (5.32) corresponding to f(t) = et, provided f C IRN is a ball (see, for instance, [111]). In such a case the bifurcation diagram strongly depends on N, as depicted in Figure 6.3. These results can be described as follows:

CASE 1: 1 < N < 2. Then there exists a unique solution for A = )A* and exactly two solutions for any A E (0, A*). Moreover, the unstable solution v., tends to oo as A \ 0, uniformly on compact subsets of St. This is a result of the work by Liouville [132] (if N = 1) and Bratu [29] (if N = 2). CASE 2: 2 < N < 10. Then there exists a continuum of solutions that oscillate around the line A = 2(N - 2). This result was found by Gelfand [82]. CASE 3: N > 10. Then A* = 2(N - 2) and there is a unique solution for any A E (0, 2(N - 2)). This result is a work by Joseph and Lundgren [112]. If N < 9 then the extremal solution u* corresponding to A* is smooth. If N > 10 then u* is unbounded and, in this case, u*(x) = In (1/1x12) provided fl is the unit ball in RN. In this situation, u* fails to be a classical solution but it is the unique weak solution of the Liouville-Gelfand problem in the limiting case A = A*.

FIGURE 6.3. The bifurcation diagram for the Liouville-Gelfand problem.

7

SUPERLINEAR PERTURBATIONS OF SINGULAR ELLIPTIC PROBLEMS The art of doing mathematics consists in finding that special case which contains all the germs of generality.

David Hilbert (1862-1943)

7.1

Introduction

In this chapter we are interested in the study of singular elliptic problems in the presence of smooth nonlinearities having a superlinear growth at infinity. Our analysis, which includes existence, regularity, bifurcation, and asymptotic behavior of solutions with respect to the parameters, will concern the model problem

1-Au = u-a + AuP

u>0 u=0

in Q, in Q,

on00

in a smooth bounded domain 12 C RN, where A > 0, 0 < a < 1. As a result of the continuous (or even compact) embedding of HH(dl) into

the standard LP(1l) spaces, we will be concerned with the case N _> 3 and

1 < p < (N+ 2)/(N - 2). We have already seen in Theorem 4.3.2 that if 0 < p < 1, then (7.1) has a unique solution for all A > 0. Furthermore, if p = I then problem (7.1) has solutions if and only if A < A1, and in this case the solution is unique. Uniqueness

of the solution has been obtained so far via Theorem 1.3.17. In our setting the assumption (HI) in Theorem 1.3.17 is not fulfilled, so that the issue of multiplicity of the solution to (7.1) is raised. The first task in this chapter is to investigate the existence of weak solutions to problem (7.1) in the following sense.

st

Definition 7.1.1 We say that a E Ho (Q) is a weak solution of (7.1) if u > 0 in S2 and

J VuVodx =

j(u

+ AuP)idx

for all 0 E H(St).

(7.2)

Superlinear perturbations of singular elliptic problems

158

In contrast to the case 0 < p < 1, a different approach is needed here because the hypotheses of Theorems 1.2.5 and 1.3.17 are not satisfied. However, we are able to show that for small values of A, problem (7.1) has at least two weak solutions, and no solutions exist if A is large. As usual in this case, the existence is proved by considering the energy functional

JA(u) = 2 / IVnI2dx Jsi

1 1

a

f ui-adx - p + I

up+idx,

u E Ha (52).

1

The main difficulty in the study of (7.1) here consists not only of the fact that Ja is not differentiable, but also of the presence of the unbounded term u_a combined with the presence of the superlinear term ur. The existence of the first solution is obtained by the sub- and supersolution method in the weak sense presented in the next section. Moreover, a direct analysis in the Hi neighborhood of the solution reveals the fact that this solution is also a local minimizer with respect to the Hi topology. The second solution is then obtained by Ekeland's variational principle. More precisely, the following result holds true.

Theorem 7.1.2 Assume that 0 < a < I and 1 < p < (N + 2)/(N - 2). There exists A* > 0 such that (1) for all 0 < A < A*, problem (7.1) has at least two weak solutions ua and va such that UA < VA in 0; (ii) for A = A*, problem (7.1) has at least one weak solution ua; (iii) for all A > A*, problem (7.1) has no weak solutions. If a is small enough, then any weak solution of (7.1) is in fact a classical solution. This will be proved in Section 7.6. In Section 7.7, the problem of asymptotic

behavior as p \ 1 will be addressed both for the bifurcation point A* and for solutions of (7.1). This matter will point out new and interesting features of problem (7.1). For instance, we shall see that if a < 1/N, then (7.1) possesses a minimal solution (which is in fact uA in the statement of Theorem 7.1.2), and any other solution of (7.1) blows up in the LO3 norm as p \ 1. 7.2 The weak sub- and supersolution method In this section we present an existence method for weak solutions that is similar to that described in Theorem 1.2.2 for classical ones. To begin with, we introduce the concept of sub- and supersolution in the weak sense. For our convenience,

let us set f,\(t) = t-a + At for A, t > 0. Definition 7.2.1 We say that u E Ho (S1) is a weak subsolution of problem (7.1)

ifu>0 in52 and n

VuVOdx < f fa(u)¢dx, s1

for all 0 E Ho '(Q), §6 > 0.

Reversing the sign in the previous inequality, we obtain the definition of a supersolution in the weak sense. The main result of this section is the following.

The weak sub- and supersolution method

159

Theorem 7.2.2 Let u and u be a weak subsolution respectively a weak supersolution of problem (7.1) such that u < u in Q. Then there exists a weak solution

uE Ho(1) of (7.1) such that u
of Theorem 7.2.2 and throughout this chapter, the relation u < u should be understood that it holds almost everywhere in Q. Proof The solution of (7.1) will be obtained by minimizing the functional JA over the set

M:= fu EHH(1):u
JA (u) ? 2 IIuIIHi(o) - 1

1

c

f I iI -"dx -

p+ 1

f

t I uIpdx,

which implies that JA is coercive. To apply Theorem C.1.1 we need to show that J,, is weakly lower semicon-

tinuous on M. To this aim, let (un)n>1 C M be an arbitrary sequence that converges weakly to u in M. Then, u --+ u almost everywhere in S1 as it 00. Because u < u,, < u in 52, by Lebesgue's theorem on dominated convergence we obtain

lim inn ul,-adx = ju1_'dx 2

and lint

n-.oo in J

un+ldx = f uv+ldx in

It follows that lim inf JA(un) < JA(u),

n-oo Thus, J, verifies the hypotheses of Theorem C.1.1. According to this one, there exists a relative minimizer ua E M of J,\. We show in what follows that ur is a solution of (7.1). Let 0 E Ha (Sl), c > 0, and consider

Set vE := ua+e¢-wE+w,. Then, vE E M, which implies that ua+t(vE-ux) E M, for all 0 < t < 1. This yields

JA(ua+t(vE-ua))-JA(ua)>0 forall0
Superlinear perturbations of singular elliptic problems

160

VUAV(v£ - u),)dx - Jim in

+ Ot(v£-u))(v(7 3) .

r

- A fn u'(v£

- uA)dx > 0,

for some 0 < 0 < 1. Notice that I(ua + Ot(v£ - ua))-"(u£ - ua)I <- u-"Iv£ - ual

in S2.

Because u is a subsolution of (7.1), we have u-"Iv£ - ual E L1(52). Hence, by Lebesgue's theorem we derive ii a J (ua + 0t(vE n

In

- ua))-a(v£ - ua)dx = ju(v- ua)dx.

VuaV(v - ua)dx -

Jn

(7.4)

fa(UA)(v£ - uA)dx > 0-

That is, f(VuiVc5 - fa(u>,)5)dx >

AE - BE a

where A£ .- fn (VuAVTU£ - fA(uA)2U£)dx,

BE :=

fa(ua)w)dx.

J

Let us first evaluate AE. Because vw, = (u,. + e4 - 9)X{uA+E0>u), we have AE _

n

V(ua -u)Vw£dx+ J (viiv - fa(ua)wE)dx n

IV(ua-u)I2dx+e J

V(ua-u)VOdx

JJ{ua+E4>u)

+

f{tiA+E¢>u)

V(ua - u)V¢dx +

>_ s f {

(.fa(u) - fA(ua))w£dx

+el

iZ}

J{'uA+Eb>pt)

(u-" - u. ")wdx.

The last integral in the right-hand side can be evaluated as

7.5)

H' local minimizers

f

161

(u-° - u-')(u), - u)dx

(u-a - uaa)1P dx = f u

u

f+

u,+e 5>a}

(u-a - udx

(u-° - ua °)q5dx

a - a11011oo f

uA+Ek>U}

(u-

Thus, we obtain AE E

>

f

(u-° - I-a)dx.

V(ux - u)VOdx - 11011,, f

Because the Lebesgue measure of the set {u,\ + aO >- `u} tends to zero as a -# 07

the previous inequality leads us to Ae/e > o(1) as e -* 0. Similarly, we obtain Be/a < o(1) as a --+ 0. Therefore, from (7.5) we find

J (VuAVO

fa(ua)Q)dx > o(1)

as a - 0.

Replacing 0 with -0 in the previous relations and letting a -i 0, we deduce

/o (Vuxv¢ - fa(ua)4i) dx = 0

for all 0 E Ho (0).

Hence, ux is a weak solution of (7.1). The proof is now complete.

7.3 H' local minimizers An interesting property of the weak solutions obtained by the method described in Section 7.2 is that these are H1 local minimizers for the energy functional Jx. To be more specific, a function u E Ho (S2) is called a local minimizer for JA in the H1 topology if there exists r > 0 such that JA(u) < J,\ (v)

for all v E H01(9) with liv - ujjH1

)


The main result of this section is the following.

Theorem 7.3.1 If ux is the solution of (7.1) obtained in Theorem 7.2.2, then ux is a local minimizer of J, in the H1 topology.

Proof Supposing to the contrary, there exists (un)n>, C HO '(Q) such that un --i ux in Ho(52) as n -> oo and JA(un) < Jx(ux). Define

wn :_ (un - u)+,

wn :_ (un - u)

Let also En = supp wn and Fn = supp wn. We first need the following result.

Superlinear perturbations of singular elliptic problems

162

Lemma 7.3.2 lim£_o IEnI = lime,o IFnI = 0.

Proof Lets > 0. Then there exists J > 0 such that IQ \ Stag < E/2, where 5t6 = {x E St : dist(x, 80) > b}. By Corollary 1.3.8 we have

c dist(x, 8S2) > 2 > 0 for all x E Q8/2, Then,

2) ua) > -a rl Cb

-0(u - u,\) >

-1-a (u - u,\)

in Q6/2,

Again by Corollary 1.3.8 we obtain

u - ua > cldist(x, 8126/2) > C2 > 0

for all x E Sta.

Hence,

IEnI < ISt\S16l+IEnn9tal s

1

2

c2

E

1

< 2+

f f

>,nn6

fsz(un - ua)Zdx.

c2

Because un -a ua in Ho (St), for n sufficiently large, the previous estimate yields IEnI < s. Thus, limE.._,o IEnI = 0, and similarly we obtain lim,o IFFI = 0. This finishes the proof. 0 Because (wn)n>1 and (wn)n>1 are bounded in Ho (St), by Lemma 7.3.2 we have

and wn --> 0

7lln --> 0

in Ho (1)

as n -> oo.

(7.6)

Set vn := max{u, min{un, u}}. Then vn E M, vn = u in E, and vn = u in F. Furthermore,

Ja (un) = J,\(v.,,) + An + B,

(7.7)

where An

4.

`IvunI2

2 A

p+1 B.

2

J

1

aJ

E

-TLI-«)dx

(IunI1-a

(IunIP+1 -2LP+11dx, 1

E

(Ivunl2

P+1

- IVul2)dx - 1

- IDUI2)dx - 1

(IunIP+1

(lunI1-a

1

- uP+11dx.

a IF,.

-

u1-a}dx

Existence of the first solution

163

Because ua is a minimizer of J., restricted to M, it follows that

J(un) > Ja(ua) + An + Bn. To evaluate An, we use the fact that un = u + wn in En. By the mean value theorem we have

A,,='f (IV(u +wn)f2 2

p+l,fE

- IVu(2)dx

(Iu+wnI1-a-ut-a)dx

(I1L+'t11nir+1 -UP+1)dx

>2IlwnIIH1(S2) + L,. ViVwdx

fE,. f(u+0w)wdx,

for some 0 < 0 < 1. Because u is a weak supersolution of (7.1), we deduce 2I10nIIHo(c)

An >

-

L.

(u-a + AP)wndx

+

((V + Bwn)-a + A(u + Own)P) wndx

? 2IIWnIIHi

A

JE

> 2 IIwnIIH (sz} - cA

((u + 0wn)P - uP Jwndx ll

(uP-t +

-1)wndx

E

By (7.6), Lemma 7.3.2, and Sobolev's inequality, for n large enough we derive

An > -IIWfIIH,i(n) - (L,, uP+1dx)

l

IIwnIIH) -IIH)

= 2IIWiIIHi() - o(1)IIwnhIHi(f) >- 0,

for some positive constant C > 0. In the same way we obtain Bn > 0. Hence, by

(7.7) it follows that J(un) > JA(ua), which contradicts our assumption. This finishes the proof.

7.4 Existence of the first solution By means of the weak sub- and supersolution method described in Section 7.2 we establish the existence of the first solution of (7.1).

The core of this section is the following result, which is the first step in proving Theorem 7.1.2.

Theorem 7.4.1 There exists 0 < A* < oo such that for all 0 < A < A* problem (7.1) has at least one weak solution ua, and no solutions exist if A > A*.

Superlinear perturbations of singular elliptic problems

164

Because the embedding Hol (Sl)y L7(1) is compact for 1 < p < (N+2)/(N2), we consider only the critical case p = (N + 2)/(N - 2). As we did in previous chapters, let us define A := {A > 0: problem (7.1) has at least one weak solution} and set A* := sup A.

Proposition 7.4.2 The set A is nonempty and A* is finite. Proof By Sobolev's and Holder's inequality we derive JA(u) > 2IIUIIHo(si) - CIIUIIHO'(cI) -

for all u c Ho (S2). Thus, we can choose A > 0 small enough and r, 8 > 0 such

that A

2IIuIIHo(O) 1

2

p

+ 1 IIuIIP+i > 26 A

p+ I IIuIIP+i j 0 JA(u)>b

for all u E BB,.,

for all u e Br, for alluEc3B,.,

where B,.{u E Ho(1l) : IIuIIH, (ci) < r}. Set

C:= inf Ja(u). uE Br

Because 0 < 1 - a < 1, for all u E B,. \ {0} there exists t > 0 small enough such that JA(tu) < 0. This yields c < 0. Let (un)n>1 C B, be a minimizing sequence for c. Then there exists u E B, such that, up to a subsequence, we have

asn --goo

un - it

weakly in H,1(0)1

'ILn -> u strongly in L9 (Q) for all 2 < q < (N + 2)/(N - 2), un -> it almost everywhere in Q. Because JA(un) = JA(Iunl), replacing if necessary un by un we may assume that un > 0 in 0, for all fn > 1. By Holder's inequality we have as n - oo

J un-c'dx < f ul-`rdx + J Iun -

uIl-adx

< J ul-adx + CIIun - U1121-c' on.

=

In

-a

In the same manner as shown earlier we obtain

Existence of the first solution

J ul-adx < j un^adx + 0

z

Thus,

i

J in

165

Iun, - ull-adx = j u'-o'dx +O(1). t

ul-adx = 1 un-adx + o(1)

as n --4oo.

n

(7.10)

A similar relation holds in the Lp norm, even in a more general setting.

Lemma 7.4.3 (Brezis-Lieb) Let 1 < p < oo and (un)n>1 C LL(D) be a bounded sequence such that u" -+ u almost everywhere in SZ as n -> oo. Then u E LP(Q) and hull' = ;-lim (Ilunllp - Ilun - ullp)

Proof Let f > 0 and M := supn>l Ilunlip. Let us first notice that there exists CE > 0 such that

Ilt+l1'-Itl1'-1I <eltl"+Cf

for alit ER.

(7.11)

This comes from the fact that lim

It + lip - Itlp - 1

Fti-00

- 0.

Itlp

From (7.11) we derive

Ila+ blp - Ialp - IbiPI <_ Elalp +CElbI7

for all a, b r= R.

(7.12)

Let vn :=Ilunlp

- lun - ulp -

lulp

w" :=(vn - flu" - uIp)+. By (7.12) we have w" -* 0 almost everywhere in Il as n -+ oo and 0 < wn < CEIuIp in Q. Thus, by Lebesgue's theorem we obtain Ilwnlll --+ 0 as n - oo. On the other hand,

0 < vn < flun - ulp + w"

almost everywhere in Q,

which yields

Ilvnll1 < ellu" - ull' +Ilwnlll < f2'M' + llwn Il . Hence, Ilvn 111 -+ 0 as n -p oo, which concludes the proof.

Superlinear perturbations of singular elliptic problems

166

By Lemma 7.4.3 we have Ilun

IIp+1 - IIuIIp+l + Ilun -

uIIp+1 +

O(1).

(7.13)

Note also that IIunIIHo(n) = IIUIIHa(cl) + Ilun - uIIH.,(o) +0(1).

(7.14)

Because J,\ > b on SBr and c < 0, there exist Eo > 0 small enough and no > 1 such that for all n > no. IIunIIH,;(c) < r - EO Now from (7.14) we find un - u E Br for all n > no. Therefore, by (7.9) we have 11Iun - ullHo (0) - p + 1 Thin - uliP'+11 >- 0.

(7.15)

From (7.10) and (7.13) through (7.15) we obtain C = Ja(un) + O(1)

IIun - Uhf 1) - p + 1 Ilun - ullp+i + o(1)

= J'\ (U) +

> J'\ (U) + 0(1) > c + 0(1),

for all n > M. Therefore J,\(u) = c-that is, u is a local minimizer of J, in the Hl topology. 0. Because JA (u) < 0 and JA > 0 on BBr, it follows that

Let r¢ E Ho (St),

BullHi(n) < r. Hence, for t > 0 small enough we still have u + to E B. This implies Ja(u + to) - JA (u) > 0,

(7.16)

which in turn implies 1 2

J IV(u+to)I2dx -12 J IVul2dx > 0. n

Dividing by t > 0 in the last inequality and then passing to the limit we find

jVuvdx >0

forallE>0.

z

This means that -Du > 0 weakly in Q. Because u > 0 in S2, by Corollary 1.3.8 we obtain u > 0 in 1. We claim that u is a weak solution of (7.1). This assertion will be proved in three steps.

Existence of the first solution

167

0>0we have Step 1: For all 0EHo(Sl),j(VuV5 - fa(u)O)dx > 0.

(7.17)

From (7.16) we obtain 1 1

((u+to)1-« - ul-«)dx <2

a

f (IV(u+ to)12 -

IVu12)dx

-1+

((U + t¢)p+i _ up+l) dx. pJ Dividing by t > 0 and then passing to the limit we have 1

1-a

li miQif is,

dx < f (DuVq - AuPO)dx.

t

(7.18)

n

Notice that tO)1-«

1-a J (u +

t

1

-

ul-« IQ 0to)_«

for some 0 < 0 < t. Because (u + --* u_« almost everywhere in SZ, by virtue of Fatou's lemma and (7.18) we retrieve (7.17). Step 2: We have

I ul-«dx - A r uP+1dx = 0. n L IVul2dx - In

(7.19)

Let h : R -4R, h(t) = Ja (u + tu). Then t = 0 is a local minimum of h, which yields h'(O) = 0. This leads us to (7.19). Step 3: u verifies (7.2). Let E > 0 and 0 E Ho (St) be arbitrary. Replacing 0 with (u + Eo)+ in (7.17), in view of.(7.19) we obtain

0
(VuV(u+EO)

= e f (vu0 i

-

- fa(u)(u+Eq))dx

dx - f

(VuV(u + Eo) - fa{u}(u + eo)) dx u+eq5<0}}

< EJ (VUV - fa(u)q)dx - EJVuVodx. u+Eq5<0}

f3

Next, we divide by e in the previous estimates and then let E into account the fact that I {u+s0 < 011 --> 0 as a --b 0, we deduce

in

(VuVO - fa(u)O)dx > 0.

Replacing 4) with -0 in the last estimate we finally derive (7.2).

0. Taking

Superlinear perturbations of singular elliptic problems

168

To end the proof, it remains only to show that A* is finite. Because p > 1, there exists A > 0 such that fa(t) > alt, for all A > A and t > 0. We claim that (7.1) has no weak solutions for A > A. Indeed, if u would be such a solution. multiplying by cpl in /(7.1) we derive

Al j ucpidx = in fa(u)cidx > Al j ucidx, 2 S2 in which is a contradiction. This means that A* < A < oo, and the proof of Proposition 7.4.2 is now complete. The following result will end the proof of Theorem 7.4.1.

Proposition 7.4.4 For all 0 < A < A*, problem (7.1) has at least one weak solution.

Proof Assume first 0 < A < A*. By the definition of A, there exist p > A and uu E Ho (1), a weak solution of

1-Du = fp(u)

in S2,

u>0 u=0

in St,

on Off

Clearly, u := uu is a weak supersolution of (7.1). On the other hand, by Theorem 4.3.2 there exists v E C2(1) fl Cl (S2) such that

1-Ov=v-`r

in S2,

v>0

in S2,

V=0

on 8fl.

(7.20)

Obviously, u := v is a subsolution of (7.1). Moreover, because 0 < a < 1, multiplying by u in (7.20) we infer that JA(u) < 0. To prove that u < u in Sl, let 0 e Cl (]l$) be a nondecreasing function such that

0(t) > 1 for all t > 1 and 9(t) = 0 for all t < 0. Fore > 0 we set 9f(t) = 9(t/E). Using 0,(u - u) as a test function in (7.1) and (7.20), by subtraction we obtain

0>- J IV(u-v,)I29E(u-u)dx u-«)9E(u - u)dx

>-f(n_' )9(u - u)dx. Letting e -> 0 it follows that

(u-" - w «)dx < 0, flit-u>0) which means that the set {u - u > 0} has the Lebesgue measure zero. Hence, u < u in !l, and by Theorem 7.2.2 we obtain that problem (7.1) has at least one solution.

Existence of the second solution

169

For A = A*, let (an,)n>1 C A be an increasing sequence such that An / A* as n - oo and let un be a solution of (7.1) for A = An. Then 1 f un-°dx - An JA(un) =21 f Iounl2dx - 1 -aJJSa P+1

and

un -« dx - A

z

f

S2

j un+1dx < 0

un+ldx = 0.

S2

From the previous relations it is easy to see that (un)n>1 is bounded in HH(Sl). Thus, there exists u* E Ha (Sl) such that, passing to a subsequence, we have as

n-aoo

un - u* weakly in Ho (Q), un --b u*

almost everywhere in Sl.

As before, we derive that un > v in Q. Because un is a weak solution of (7.1) with A = A, we have

L

Vu, VCbdx =

J A(un)0dx

for all q5 E Ho (Sl).

Passing to the limit in the previous equality and using Lebesgue's theorem, we O deduce that u* is a weak solution of (7.1) for A = A*. This ends the proof. 7.5 Existence of the second solution We have seen so far that there exists A* > 0 such that problem (7.1) has at least one solution ua if 0 < A < A*, and no solutions exist if A > A*. As a result of the superlinear character of (7.1), we are able to show that there exists another solution va of (7.1) provided 0 < A < A*. Moreover, we have 0 < ua < va in Q. We use Ekeland's variational principle, as stated in Theorem C.2.1. Define

A:= fu EHa(Sl) : u>uA inSl}. Because ua is an H1 local minimizer of JA, there exists 0 < ro < IIuA IIH (O) such

that JA(u) > JA(UA)

for all u E H1()) with IIu - UAIIH1(o) < ro.

In our further analysis, the following two cases may occur: CASE 1: inf{JA(u) : u E A, IIu - ual1Ho(O) = r} = JA (u.\), for all 0 < r < ro. CASE 2: There exists 0 < r1 < ro such that inf{JA(u) : U E A, IIu - U,\ 11H'(0) = r1} > JA(ua).

We shall discuss these two situations separately.

Superlinear perturbations of singular elliptic problems

170 7.5,1

First case

Proposition 7.5.1 Let 0 < A < A' . Then, for all 0 < r < TO there exists a weak solution va of (7.1) such that 0 < uA < va in St and Ilea - va 11 Ho (n) = r.

As a consequence, in this case, problem (7.1) has infinitely many solutions.

Proof Fix0 < r < ro and let p > 0 be such that o < r - p < r + p < ro. Define

B:={uEA:0 1 be a minimizing sequence of ua. Then, by Ekeland's variational principle (see Theorem C.2.1), for all n > 1 there exists vn E 13 such that

JA(vn) < JA(un) < Ja(ua) + 1,

(7.21a)

Ilun - vnIIHI(S) < 1,

(7.21b)

n

JA(vn) < JA (v) + 1 IIV - vnhIHI(c) n

for all v E B.

(7.21c)

Because (vn)n>1 is bounded in Ho(1), there exists va E HH(Sl) such that, up to a subsequence, we have vn - va in Ho (0) and vn -+ va almost everywhere in 9 as n -+ oo. The definition of the set A also implies va > ua in Q.

Lemma 7.5.2 va is a weak solution of (7.1). Proof We first claim that for all w E B we have

--11w - vn11H1(0) < fn Vvn0(w - vn)dx - I fa(vn)(w - vn)dx.

(7.22)

Let w E B. Then, fore > 0 small enough we have vn+e(w-vn) E 13. Furthermore, by (7.21c) we derive 1 -nllw-vnllHo(n) < JA(yn+6(w Evn))-Ja(yn)

for all n> 1.

Passing to the limit with a -> 0 in the previous relation we obtain

Existence of the second solution

171

( Vvn0(w - vn)dx n IIW - VnIiR (n) < JJsz

-1

lim ionf fn ( v"

1

+ a(w - v ))1-a - vn-a dx

p

- A J vn (w - vn)dx x c

=

VvnV(w - vn)dx

f - lim ionf J (v + E9(w - vn))-Of(w - vn)dx, (0<0<1)

E-o

- A J vn(W - vn)dx.

.lit

As we have argued for (7.4), we deduce lim inf

J

(vn + E0(w - vm))-"(w - vn)dx =

J

vn'(w - vn)dx.

This proves the claim. Let 0 E Ho '(9) and e > 0. Set

wn,e := (vn + co - ua)-, and wE := (v), + co - u,\)Therefore, as n -> oo we have we weakly in Ho (St), we almost everywhere in Q.

wn,e wn,e

Obviously, vn + 0 + wn,E E B. Then, replacing w with vn + E0 + wn,e in (7.22) we obtain

J OvnV(eq+wn.e)dx- J f (vn)(0+wn,E)dx. 52

(7.23)

52

By Lebesgue's theorem on dominated convergence we obtain fa(vn)(Ec +wn,E)dx

J fa(va)(eq+w,)dx

in

as n -> oo.

(7.24)

Because the measure of the set {vn+E0 < ua < va+E0} tends to zero as n -> oo, we also have Ovn©wn,Edx =

J{uA>v,,+ect} {u >vA+et'}

+ Jua> rf

VvnV (ua - vn - Eq)dx

VvnV(ua - va - 0)dx (7.25)

Vvn0(v,\ - vn)dx + 0(1)

J VvVwdx + 0(1).

Superlinear perturbations of singular elliptic problems

172

Using now (7.24) and (7.25) in (7.23) and then letting n --- oo we obtain (VvAV 10

j

- fa(va)we)dx - fa(va)O)dx > -E J (VvAVwE n

f (h(VA) - fa(ua))wedx

=E J V(ua - va)VwEdx+

sa

sfa

>1

J{ua>va+eO}

+

{uA>va

f

V (uA - VA)V (ua - va - eo)dx (VA

uA -) (ua - vA - E(5)dx

V(va - ua)V dx + f

ua>va+eq5}

(v, ` - ua a)gdx. uA>va+eq5}

Using again the fact that the Lebesgue's measure of the set {u,\ > va +ej} tends to zero as E - 0, the last estimate yields

i

(Vv\V - fa(va))dx > o(1)

as E->0.

Letting e -+ 0 and then reversing the sign of 0 we derive that va is a weak solution of (7.1).

El

Lemma 7.5.3 The sequence (vn)n>1 converges strongly to VA in Ho (52).

Proof Arguing as in the proof of Proposition 7.4.2, as n -+ oo we have 2

2

IIvnhII.1(0) = IIvn - val1H o(n) + IIVAIIHo(s1) + 0(1),

Ilvnllp+11 = IIvn - vAIl pp+1 + IIva

(7.26)

(7.27)

IIp+1 + 0(1),

r vn-adx = J va-adx + 0(1),

(7.28)

Ivn - val1-`tdx = o(1).

(7.29)

n

st

and

Let us take w = VA in (7.22). We have

JIV(vn - v),)I2dx + J v,,iavadx < J va-adx +. vn(vn - va)dx + o(1) J in S2

< Java-adx+.\IIvn-vaIIP+1+0(1). By Lebesgue's theorem, the previous estimate produces IIvn - vallH1(-) -< AIIv, - vaIIp+i + o(1)

as n

oo.

(7.30)

Existence of the second solution

173

Letting now w = 2vn in (7.22) we have IIvn112

(r?) - AlIvnllp+1 - J vn-ad2 > o(1)

as n -r oo.

(7.31)

Because vA is a solution of (7.1) we also have IIvxllp+i

- Jn v

ads = 0.

(7.32)

From (7.26) through (7.28) and (7.31), (7.32) we find Ilvn - VAIIHo(c) -> ) I]vn - vAllp+i +o(1)

as n -> oo.

(7.33)

as n -> co.

(7.34)

Combining (7.30) and (7.33) it follows that llvn - VAIIHa(O) = AIlvn - vallp+i + o(1)

The last equality together with (7.26) through (7.29) provides

J,\(v,, - va) = JA(vn) - J,\ (v,\) + 0(1)

as n -> oo.

(7.35)

Recall that vn - va in HO '(Q) as n -9 oo. Hence, by the definition of B we deduce

fIIvn-ual1Hi(n) llva - u,\IIH(f)
r+p
Therefore, according to the hypotheses in Case 1 we have JA(uA) < JJ,(v.,). Furthermore, by (7.21a) it follows that

A (vn) - Ja(va) < A(ua) - Ja(va) + 1 < o(1) n

as n --. oo.

(7.36)

as n -g oo.

(7.37)

From (7.29), (7.35), and (7.36) we finally obtain 2llvn - VA llH;(sz) -

p

+ l llvn - v.Ilp+i < o(1)

Now, by (7.33) and (7.37) it follows that vn -} va in Ho (1l) as n -> oo.

Let us come back to the proof of Proposition 7.5.1. Because l lvn - ua II x (rt) = r and vn -> va in Ho (S2) as n -F oo, it follows that Ilea - ua ll H,l (c2) = r. This yields va 0 ua. Notice that ua < va in Q. Furthermore, applying Corollary 1.3.8 as we did in the proof of Lemma 7.3.2 we derive that

va - u,\ > 0 in any subdomain w CC Q. Hence, ua < va in ft This completes the proof.

Superlinear perturbations of singular elliptic problems

174

7.5.2

Second case

Similar to Proposition 7.5.1 we have the following.

Proposition 7.5.4 Let 0 < A < A*. Then there exists a solution va of problem (7.1) such that 0 < ua < va in Q.

Proof Consider the complete metric space

P=

E c([o, 11, fl)

77(o) = uA, Ja(rl(1)) < JA(ua) II71(1) - uAIIH;(Q) > r1

endowed with the distance max IIn'(t) -17(t) II

tE [0,1]

for all 77,77' E P.

Set

cP := inf max J,,(77(t)). nEP tEJ0,1]

We first prove that P is nonempty. When this claim is established, we may proceed to determine the second solution va of (7.1). This will be done by making use of Ekeland's variational principle, but in a quite different manner to that in the previous case.

To show that P is nonempty, let S be the best Sobolev constant for the embedding Ho (S2) y Lp+1(SZ)-that is,

S = inf{IIouII2 : u E H0(Il),

IIuIIp+1 =1}.

(7.38)

Then, S is independent of Si and depends only on N. Furthermore, the infimum in (7.38) is never achieved in bounded domains. If Si = R ' then the infimum is achieved by CN (1 + Ix12)(N-2)/2 ,

for some normalization constant CN > 0. Also we have S

where

A=

f

B

Up+1dx, N

Bf

Let us fix y c i and 0 E Co (SZ) such that and define

(7.39)

= A2/(p-F1) '

IVUI2dx.

= 1 in a neighborhood of y,

CNE(N-2)/2 Ue(x)

Lemma 7.5.5 We have

(e2 + Ix -

(7.40)

N

yI2)(N-2)/245{x}.

Existence of the second solution

175

B+O(EN-2) and IIUEII1+I = A+0(EN); (ii) for all v E HO '(Q) and p > 0, the following estimate holds: (l)

=IIvIIp+1+pp+1IIUeIIp+1+pp

IIv+pUellp+l

fn U vpdx (7.41)

+ ppp

Upvdx + o(,- (N-2)/2).

J

Proof We give here a complete proof of (i). Remark first that VUe = CNE(N-2)/2

IX l (E2 + Ix - y12) (N-2)/2 - ((E2 + yl )N2)

Because,O = 1 in a neighborhood of y we obtain CNEN-2

II2 =

(IIoUe -

(N -2) 2 f

(E2

N

=CN(N-2)2

f

J

I

+

Ix-yl2y 12)N dx + O(Ely-2)

2 121NdX +O(EN-2)

x (1+1 1

= B + O(EN-2).

We also have II Ue IIP+1

= C2Nl N (N-2)EN

I

,52N/(N-2)

(x) dx x (EY2+Ix-y12)N

- 1 d2 NJRN (E2 + Ix - yi2)N dx + f x (E202N/(N-2) + Ix - y12)N I

- C2N/(N-2) EN

=

Cr2N/(N-2) eN N

= C2N/(N-2) N

f

1

RN

1 dx + O (Ely) (E2 + IX - y12)N

1

N (1 + Izi2)N

dz + O(EN)

= A+O(EN).

Estimate (7.41) follows with similar arguments.

Lemma 7.5.6 There exist so > 0 and po > 1 such that for all 0 < E < Co we have

for all p > pi),

JJ(ua + pUe) <JA(ua) SN/2

JA(UA + pUe) <JA(ua) + N.)(N-2)/2

for all 0 < p < po.

Superlinear perturbations of singular elliptic problems

176

51

Proof A straightforward computation yields

f F JA(ua+pU6) =2 J 1VU I2dx+p ( Du.,VUEdx+ n

1

a f(uA + pU)1dx -

2

fIVUeI2dx (7.42)

l,lua +

Because uis a solution of (7.1) we have

Jn

Du,\VUEdx = j ua OUedx +A / uXUdx.

(7.43)

n

1

Next, from (7.41) through (7.43) we obtain

J,\(ua + pUe) =Ja(ua) +

PB

- p+

1

pP+1A

- app fn Upu,dx

(7.44)

Pp0(E(N-2)/2),

+ DE +

for some 0 < ,0
_

D£

_

(ua°PUE +

1

1

- au

1

a

1 - a(u,` +

pue)dx.

Let us estimate DE. In this sense we fix 0 < 7 < 1/4 and we decompose (u_apUE + 1

De = I

+

1 u a-1

+ pUE)1-")dx

1

a(uA

x-yI<E*

J Ix-y1>e'

(uA kpu. +

1 1

aua a -

1 1

a (U,\ + pU£)1-a)dx.

Denote by I and II the two integrals in the previous equality. Taking E > 0 small enough, I can be evaluated as CNE(N-2)/2

I
Udx = cip Ix-yi<E= J

Ix-yl<_Er

(e2 + Ix - y12)( N-2 )/2

d

rdr < c6 pe(N-2)/2+2r


To evaluate the second integral we use the mean value theorem. We have

II < f

(ua « - (UA + B1

pUE)-a)

pUEdx

x
JsuPP4fi{Ix-yI>e'} (ua + for some 0 < 01 i 02 < 1. Using the fact that

02PUE)-I-a(PUe)2dx,

(7.45)

Existence of the second solution p9Ue)-1-a

(uA + 92

177

< ua I-° < M in supp q5,

we deduce

II
-yl>eT

S(N-2)/2

r

U,, dx < c4p2

yl2)(N-2)/2dx

{s2 + Ix -

(7.46)

p2EN-2-2T(N-2).


Therefore, from (7.45) and (7.46) we finally obtain

De < c(p +

p2)o(e(N-2)/2)

ass->0.

(7.47)

Let us now evaluate the integral in the right-hand side of (7.44). For this purpose, we extend ua by zero outside Q. Then u., E C(IRN) and

f UPuadx = c(N+2)/2

CNua(x)oP(x)

fjN (C2 + Ix _ yl2)(N+2)/242

_ 6- (N+2)/2 f N ua(x)Y (2)Y'(x where V)(x) = CT,(1 + TN

f

N

C

Y) dx,

Ix12)-(N+2)/2 E L1(RN). Notice that

ua(x)O'(x)b(x

y)dx s

- ua(y) f

M

V,(x)dx

ass , 0.

Hence, we have obtained APP J U,, uAdx =

pDTe(N-2)/2 + pP0(6(N-2)/2),

(7.48)

sl

where T = )qua(y) f R, ?P(x)dx. Now, from (7.44), (7.47), and (7.48) we find JA (ua

+ PUe)
p B _ ApP+IA

- pPTe(N-2)/2 + pyo(c(N-2)/2),

P+1

(7.49)

for some 0 < ry < p + 1. Thus, we can take po > 1 large enough such that J.\ (u), + pUe) < J), (u,\) for all p > po. This establishes the first claim in the statement of Lemma 7.5.6. To prove the second one, let us consider

q : (0, oc) - (0, oo),

q(t) =

t2B 2

- t Ts (N-2)/2 - AtP+IA P+1 p

.

Then q achieves its maximum at a point to > 0. This obviously implies that

teB - AtPA = Define

pTe(N-2)/2tP-I

(7.50)

Superlinear perturbations of singular elliptic problems

178

B 1/(p-1) So :_ (AA) Then, (7.50) yields 0 < tE < So as E\-- 0. Let us set tE := SO(1 - J,). We need to find the rate at which bE -40 as e -+ 0. From (7.50) we derive 1/(p-1)

BP

Expanding for bE we obtain (P

A (1 - (fE)p-1E(N-2)l2

l (1 - be) - (1 - SE}p I =

A}

l/(p-1)

- \ l 1

//

BAP l

8E

-

PTAE(N-2)/2

+o(,- (N-2)/2).

(7.51)

Now, from (7.49) and (7.51) we deduce Ja(ua + pUE)

Ja(ua) +

B2

=Ja(ua)+S0

4P+1 A

p

- tpTE(N-2)/2 + 0(,5(N-2)/2)

I

B-aS(P) +1 A

(7.52)

p+1

+ (SOB - ASo+1A)6E -

SOTE(N-2)12

+

O(E(N-2)/2).

Remark that SoB = AS(P)+'A and SoB

ASo+1A

2

P+1

SN/2 NA(N-2)/2

Furthermore, taking e > 0 small enough, estimate (7.52) produces SN/2

JA(ua + pUE) < Ja(ua) + NA(N-2)/2

for all0
This completes the proof. From Lemma 7.5.6 it follows that ?70 (t) := ua +tpoUE, t E [0,1] is an element

of P. Thus, P is nonempty and SN/2

t cp < ma[ Ja(rlo(t)) < Ja(ua) + NA(N-2)/2

Notice that by the definition of P, for all rl E P there exists to E [0, 1] such that I[7l(to) - uA Il xo (fa) = r1. By virtue of our assumption in Case 2 we have

max JA(rl(t)) > J,\(rl(to)) > JJ(ua).

tE[0,11

This yields cp > JA(UA). Hence, we have obtained

Existence of the second solution

179

SN/2

Ja(ua) < cP < J.\ (U,\) + N\(N-2)/2

(7,53)

In view of Ekeland's variational principle applied to the functional

P3rlH maxJa(rl(t))ER, tE [0,11 there exists a sequence (71k)k>1 c P such that

Max JA(rlk(t)) < cP + 1

tE[o,11

-

for all k > 1,

(7.54)

-

k

and for all q E P there holds Max JA(nk(t)) < Max J,,(77(t)) +

tE10,11

tE10,11

1

(7.55)

Max IIi1(t) -

k tE[0,11

Set

Mk := It E (0,1) : JA(rlk(t)) = sma JA(nk(s))}. 1

The key result in this case is the following lemma.

Lemma 7.5.7 For all k > 1 there exists tk E Mk such that Vk := rlk(tk) E A satisfies

(VVkV(W - vk) - fa(Vk)(w - vk))dx >

- maxi , 11Wk- V kIlHj (n)}

,

(7.56)

for aliw EA. Proof Suppose to the contrary that the conclusion in Lemma 7.5.7 does not hold. Then there exists k > 1 such that for all t E Mk we can find t E A with the property

f

V77k(t)v(St - 7ik(t)) - fa(Tlk(t))(St -

rlk(t)))dx < -akk

(7.57)

where at.k := max{1, may be assumed locally , (s1) }. We claim constant on Mk. Indeed, by continuity arguments, for all t e Mk there exists a neighborhood Vt of t such that for all s E Vt fl Mk we have

J vr1k(S)v(bt - 77k(8)) - fa(rlk(S))(St - r?k(S)))dx < -bt, s)

(7.58)

where bt,k(s) := max{1, Ili;t - nk($)IIHi (n)}. We choose t1, t2, ... , t0 E Uk such that Mk C UQ 1 Vt, and set t := fit,, 1 < i < q. Let also {X1, X2, ... , Xq} be a partition of unity subordinated to the covering {Vt,, Vt27 ... , VtY }. Define now

Superlinear perturbations of singular elliptic problems

180

{s}:_

ti{s}

i-

ti

0<S<1.

Then, /e is continuous and C(s) E A, for all 0 < s < 1. From (7.58) we deduce

Jn

l v1]k(S)0(C(S) -?Tk(S)) - ff(rlk(S))(.(S) -?lk(S)))dx < -ck(s)

(7.59)

where ck(S) := max{1, [[C(s) Now let z : [0, 11 -' 10, 1] be a continuous function such that z(t) = 1 in a neighborhood of Mk and z(0) = z(1) = 0. For 0 < e < 1 define '!E(t)

77k(t) + Ez(t) C(t)

- 77k(t)

for all 0 < t _< 1.

Ck (t)

Then rlE(t) E A for all 0 < t < 1. By (7.55) we obtain max JA(rlk(t)) < maX JA(77e(t)) + E maX z(t)

tE[0,11

k tE[0,1]

tE[0,1]

IIC(t) - +7k(t)IIHo(sa) Ck(t)

(7.60)

Let tk,e E [0, 11 be such that JA(?)E(tk,E)) = Max JA(7)e(t)).

tE 0,11

Passing to a subsequence if necessary, we may assume that tk,E --+ tk as a - 0. Obviously tk E Mk, which yields z(tk,e) = 1 for small s > 0. Set 7lk(tk)

"0k

,

71k(tk,E)-

Vk,e

Then, for small e > 0, in view of (7.60) we have A(Vk,e) < JA,(vk) < A (7JE(tk,e)) + k This yields

JA(77e(tk,E))

- JA(vk,e) >

1

E

-

k

for all k > 1.

for allk>1.

(7.61)

On the other hand, 7IE

(t k,E ) = 'IX (t k,e ) + E (tk,E) - 7Ik(tk,e) Ck(tk,E)

= Vk,E +

Ee(tk) - 7Ik(tk)

(7.62)

Ck(tk) +E(S(tk,e) - rlk(tk,E) Ck(tk,E)

_ b(tk) COO -?1k(tk) Ck(tk)

)

Existence of the second solution

181

Notice that C(tk:e) -> 1;(tk) and C(tk,e) -F c(tk) as E -> 0. Thus, from (7.62) we obtain Wk) - ?7k(tk) + O(E) as E -> 0. ?7e(tk,e) = Vk,e + E

Ck(tk)

Using this relation in (7.61) and passing to the limit with E - 0, we deduce

\

tr

J

max{1,

(VvkV(C(tk) - Vk) - MVk)(b(tk) - Vk))dx >

k

VkIIH1(o)}

0

The previous estimate contradicts (7.59). This concludes the proof. By Lemma 7.5.7 and (7.54) we also have JA(vk) --+ cp

ask -* oo.

(7.63)

Furthermore, letting w = 2ua in (7.56) we deduce CP + 0(1) =

I

1 1

> \2

a J IvkI1-°dx -

p+llllvkllHn(SZ) - (1 1

cx

p+1

I

IVklp+1dX

p+ll f

Ivkll-adx

1

k(p+ 1) (1 + IIVkIIH(;(n)) Therefore, (Vk)k>1 is bounded in Ho (S2). Hence, there exists vA E Ho (Q) such

that, up to a subsequence, we have vk - va in Ho (S2) and vk - va almost everywhere in 9 as k -* oo. At this point, we can proceed as in the proof of Proposition 7.5.1 to derive that (vk)k>1 converges weakly to va in Ha (S2) and va is a weak solution of (7.1). Let us set Ok := vk - VA, k > 1. Then, as k --> oo we have (7.64)

IAIIH,(n) = AIIfikIIp}1 + 0(1), 2II0kIIH, (S)

p+ 1

IIOkIIp}1 - CP - Ja(va) +0(1).

(7.65)

By (7.53) and for a suitable eo we obtain

l

2

S'N12

A

p+1

11411p+1 <

N.(N_2)/2

-Ep+o(1}.

(7.66)

We claim that Ok -> 0 in Ho (S1) as k -> oo. Supposing, to the contrary, it follows that, up to a subsequence, (IIQikIIH,,I(o))k>1 is bounded away from zero. Furthermore, from (7.38) we have II0kIIH.(SL)

SIIIkkIIp+1

Superlinear perturbations of singular elliptic problems

182

Therefore, by (7.64) we deduce S 1kl{p+1

as k --oo.

+o(1)

(-i)

(7.67)

Combining (7.64), (7.66), and (7.67), it follows that SN/2 NA(N-2)/2

/1

- CO > 12 -

\

1

p+1 1

Afl4kitp+l + o(1)

SN/2

N-(N2)/2 + o(1)

as k --> 00,

which is a contradiction. Therefore, vk -> vA strongly in HO '(11) as k --> oo. By (7.63) this yields .JA(VA) = Cr > JA(uA).

uA in Q. Moreover, by Corollary 1.3.8 we have uA < vA in Q. This 0 completes the proof of Proposition 7.5.4 and Theorem 7.1.2. Hence, vA

.

7.6 C1 regularity of solution The C' regularity of solution is based upon standard procedure for elliptic equations related to this issue. However, a restriction on the exponent of the singular term u- is required. Before we start, let us point out that the C' regularity is a particular feature of positive weak solutions to (7.1) in the sense of Definition 7.1.1. More precisely, if a weak solution u E Ho (S2) is nonnegative and vanishes at xo E SZ, then u is not differentiable at xo. Indeed, supposing to the contrary it follows that Vu(xo) = 0

and for each e > 0 we can find r > 0 such that Br(xo) cc Sl and u(x) = u(x) - u(xo) < Elx - xoI

for all x E Br(xo).

Let S E CC(Br(xo)) be such that 0 < ¢ < 1 in Br(xo), AO < cr-1_01 in B,.(xo) for some constant c > 0.

= I on B,./2(xo), and

Then, by (7.2) we have

f

uzq5dx =

Note that

f

uLodx > (xo)

J

sfA(u)gdx.

(7.68)

f u-abdx (7.69)

B ,-/2 (xo)

e:--

f

Ix - xoi--dx r/2(xo)

> E-"clrN -a

Ct regularity of solution

183

where cl > 0 is independent of T. On the other hand,

J

uzaq dx <

cer-l-a

B. (yo)

f

Ix - x01 dx < Ec2rN-a.

(7.70)

B, (xo)

Thus, from (7.68) through (7.70) we obtain El+a > cifc2, which is a contradiction if e is small enough. Hence, u is not differentiable in x0_ The main result in this section is the following.

Theorem 7.6.1 Assume 0 < a < 1/N and 1 < p < (N - 2)/(N + 2). Then u is a weak solution of (7.1) if and only if u is a classical solution of (7.1). Moreover, there exist a positive integer m > 0, 0 < y < 1, and C > 0 that are independent of p such that any solution u of (7.1) satisfies u E C2(9) n and (7.71) IIUIICi,,(sz) < C(1 + }}

Proof Assume first that u E C2(Q) n C(St) is a classical solution of (7.1). To obtain u E Ho (9), it suffices to prove that for all 0 E Ho (5l) there holds L,

u a¢dx < oo.

Let v E C'(fi) be the unique solution of problem (7.20). With similar arguments to those in Proposition 7.4.4 and by virtue of Corollary 1.3.8 we have

u(x) > v(x) > cdist(x, 9Q)

in Q,

(7.72)

for some positive constant c > 0 that does not depend on p. Because 0 < a < 1/N, by Holder's inequality we obtain

f

(N+2)/(2N)

(

u-aq dx < (J u,-2Na/(N+2)dx)

110112.

\n

<

Jr dist(x, oc)-2Na/(N+2)dx1 )

N+2)/(2N) 111'112 < 00,

sa

where 2* = 2N/(N - 2) denotes the critical Sobolev exponent. Conversely, let u E Ho (S2) be a weak solution of (7.1). To achieve the regularity in the statement of Theorem 7.6.1, we use Moser's iteration technique. Without losing the generality we may assume that 2'a < p; if not, we replace p with another value close to (N + 2)/(N - 2). Let us first choose q0 > N such that c q0 < 1. In view of (7.72) this yields u-a E LQ0 (SZ).

By standard elliptic estimates, there exists cl > 0 such that Cl(IIu-aII2-/p

11u11W2.2'/P(n)

+ )1IuPII2`/p)

< <_ C2(1 +

(7.73)

Ilullz).

Superlinear perturbations of singular elliptic problems

184

Therefore, by Sobolev's inequality we find (7.74)

Ilullq, < C3(1 + IluIl2.),

for some c3 > 0, where

ql = NN 2 * 2p

= 2* bo,

'/p

bo = (p - 2- N

)-1

> 1.

If q1a < p, we replace 2* with q1 in (7.73) and (7.74). We obtain Ilullgz <- C4(1 + Ilullq,) < C4(1 +

where q2 =

NN

21/p > 2*ao-

q1/p

After a finite number of such iterations we find a positive integer m, which is independent of p, and q,,,, > 1 such that p and Iluh- <- c(1 + 11u112:').

(7.75)

Notice that qma > p produces qm/p > N. Using the fact that the Sobolev space W2,min{QR,9m/p}(cl) is continuously embedded in C1,7(52) for some 0 < y < 1, by (7.75) it follows that IIUIIC1,,,(?) <

Clllullw2.,n;n{9c,4m/p} (0)

C2(Ilu-allgo + )llupllq,, /p)

<

C2(Ily-«llq,,

+ allupllgm/p) C3(1 + llullq,,.)

< C4(1 + IIuIIHO(n))I

where the constants C1, ... , C4 are independent of p. Now, by standard regularity theory we derive u E C2(52). This completes the proof.

Remark 7.6.2 Let 0 < a < 2/N and 1 < p < (N - 2)/(N + 2). With the same method as that used earlier we can prove that any solution u of (7.1) satisfies u E C2(1) 11 CY(Sl) for some 0 < y < 1.

7.7 Asymptotic behavior of solutions In this section we study the asymptotic behavior of solutions to (7.1) as p \ 1.

According to Theorem 7.1.2, for all I < p _< (N - 2)/(N + 2) there exists 0 < A* < oo such that (7.1) has at least one solution if 0 < A < ap, and no solution exists for A > A. The asymptotic behavior of A is described in the following result.

Asymptotic behavior of solutions

185

Theorem 7.7.1 We have lim A* P = A1(-0).

P'\l

Proof Let 0 < A < Al := A,(-L1) and fix b > 0 such that A(l + b) < A1. By virtue of Theorem 5.6.1, there exists w E C2(Q) fl C1,1-a(SZ) such that

-dw = w-a + A(l + b)w w>0 w=0

in Q, in fl, on 8S1.

Hence,

-©w > w-' + Awp + A((' + b) - llw 1lo0 1) w >w_a+Awp

in0 asp\1.

Therefore, w is a supersolution of (7.1). Notice that v defined in (7.20) is a subsolution of (7.1). As in the proof of Proposition 7.4.4, we find v < w in fl. Thus, by Theorem 7.2.2, problem (7.1) has at least one solution. Therefore, lim infpNl Ap > A. Because A < Ai was arbitrary, we obtain lim infp\, 1 Ap > Al. Now let A > A1. Then we can find pp > 1 such that

t-«+Atp>Jilt forallt>Oand1
(7.76)

We Indeed, it suffices to take, for instance, po > 1 such that A > claim that problem (7.1) has no solutions for 1 < p < po. Indeed, if u would be a solution of (7.1), multiplying in (7.1) by the first eigenvalue cl of (-A) we have f (fx(u) - A,u)W 1dx = 0.

Using (7.76) and the fact that 0 in fl, the previous equality leads us to a

'

contradiction. Hence, AP < A for all I < p < po, which yields lim supp\1 Ap < A. Because A > Al was arbitrary, we deduce lim supp\l A* < A1. This completes the proof.

Let 0 < A < Ai and 0 < a < 1/N be fixed. By Theorem 7.7.1, for all I < p < (N + 2)/(N - 2), problem (7.1) has at least two solutions. Let up be

the solution of (7.1) obtained by Theorem 7.4.4. Recall that JA(up) < - 0, which means that 1 1 1

and

a J 1Lp-'dx - p+1 IluPllp+1

Hf upj 1IHI(o) =

2UP -adx+AlluPljp+l

0

(7.77)

Superlinear perturbations of singular elliptic problems

186

Combining these relations, by Holder's inequality we deduce

(2

p+i! f JuPll «dx

p+1/IIuPIIHo(O) < (1 la C1llupll2-«

<

C2IIupi1HI-or

ill (n)'

where c1, c2 > 0 do not depend on p. Thus, we may find c3 > 0 such that I1Up1IHa(S2)

C3 (P -

1)-1/(1+«).

(7.78)

Now, according to estimate (7.71), as p \ 1 we derive I1UPiICl.,(-a) < C4 (1

+ (p - 1)-P'"/(1+«)) S c5(p

(7.79)

where c4i c5 > 0 are independent of p. Furthermore, we have the following theorem.

Theorem 7.7.2 Assume that 0 < A < a1 and 0 < a < 1/N. Then, the sequence (up)p>1 is bounded in Ho (52) fl C1,7(Ti) asp " 1.

Proof We argue by contradiction. Assume that there exists a sequence (pk)k>>1 such that Pk \ 1 and 1lupkliHo(s1) -> oo as k -+ oo. Set Mk := IjupkiIH, (ft) and let Wk := upk/Mk, k > 1. Because (wk)k>1 is bounded in Ho(S2), there exists 0 < wo E Ho (9) such that, up to a subsequence, wk - wo weakly in Ho (&) and wk -} wo'strongly in L2(1) as k --> oo. We first claim that wo # 0. From (7.77) we have _

M2 =

I u1-«dx + A Pk

uPk+1dx. In,

Pk

Dividing by Mk in the previous equality and using the fact that Mk -3 co as k -1 oo, we have

1=o(1)+A

upk+1

in

Pk2 dx

Mk

= AIlwoII 22 + A 1

(7.80)

P' Mk

pk dx +

o(1) ask-' oo.

Let us evaluate the last integral in (7.80). By the mean value theorem and estimate (7.79), we have

L.

.upk+1 - u2 Pk Pk dx

Mk2

<

(P-

Mk 1) fft

unk ek In uPkdx (1 < 9k < Pk)

< (pk - 1)1ItPk1100-111uPkll2 M2k

f lnu Pkdx St

< C(pk -

1)2-ek In (C(pk

- 1)-')

as k -> oo,

(7.81)

Asymptotic behavior of solutions

187

where C > 0 does not depend on k. Therefore,

i

.uPPk +I - u k

M2 k

Pk dx

-0

as k - oo.

Passing to the limit in (7.80), by (7.81) we obtain )IIwol = 1. This proves the claim.

Now let 0 E Ha(0). Because upk is the solution of (7.1), we have

in

VwkVodx =

fk J

Vup,O¢dx

SZ

_

jf.,(upbdx

1

(7.82)

_ r -a up" up"- Odx. Odx + A f upk wk¢dx + J u''k Mk Mk

=Af

st

I

t2

Let v be the unique solution of problem (7.20). Because v < upk in 9 it follows that

-« uPk Odx

sl Mk

<J

k101dx,0

as k - oo.

(7.83)

On the other hand, with the same method as in (7.81) we obtain r uPk - upk Odx - 0 II

as k --> oo.

(7.84)

Mk

Passing to the limit in (7.8fn 2) with k -+ oo, byr (7.83) and (7.84) we deduce VwoVq dx = A.

wo4,dx.

In

That is, -Awo = Awo in Q. But this is a contradiction because A < A1. Hence (up)p>1 is bounded in Ho (SC) n C1,7(ul) as p \ 1.

D

Next, we shall be concerned with the asymptotic behavior of solutions to (7.1) as p \ 1. Let us consider the problem

1-Ow = w-a + Aw

w>0 w=0

in S2,

in 9,

(7.85)

on '9Q.

Recall that by Theorem 5.6.1, the previous problem has a solution if and only if A < Al. Moreover, for all A < Al, problem (7.85) has a unique solution w E C2(9) n Cl l-"(S2). The asymptotic behavior of solutions to (7.1) is described in the following result.

Superlinear perturbations of singular elliptic problems

188

Theorem 7.7.3 Assume 0
(ii) up -4 w in C'(T) as p \ 1, where w is the unique solution of problem (7.85).

(iii) For any other solution vp of (7.1) we have lIvII,,, -> oo as p

Proof (i) We first show that (7.1) has a minimal solution. To this aim, let v be the unique solution of (7.20) and consider the sequence (un)n,>o defined by uo - v, and for all n > 1, un satisfies

-Aun = una + Aun_1 un > 0 un = 0

in Q, in S2,

(7.86)

on 852.

By Theorem 4.3.2 it follows that un E C2(Q) fl C1,1-a(?) for all n > 0. If u is an arbitrary solution of (7.1), it is easy to check that un_1 < un < u in Q. As in Theorem 7.7.2 we prove that (un)n>o is bounded in Ho (S2) fl C1'?'(S2). Hence, (un)n>o converges to the minimal solution of (7.1). To show that up is the minimal solution of (7.1), it is enough to prove (ii) and (iii). (ii) Let (up,:)k>1 be a subsequence of (up)p>1 with Pk \ 1 as k --i oo. According to' Theorem 7.7.2 we have that (upk )k>1 is bounded in H_o (S2) fl C1"''(S2).

By the Arzela Ascoli theorem, there exists uo E HH(Sl) fl C1(S2) such that, up to a sequence, up,, -+ uo in C' (Il) as k --> oo. Let 0 E HH(1) be arbitrary. By (7.1) we have

f VupkV dx = j(uua + Aup-)Odx.

sz

Passing to the limit in the previous equality we deduce

f Vu0V dx =+ This means that uo is the unique weak solution of (7.85). By standard regularity

theory, we easily derive that uo is a classical solution of (7.85), so uo - w. Therefore, we have proved that any subsequence of (up)p>1 has a subsequence that converges in C1(11) to the unique solution w of problem (7.85). It follows that the whole sequence (up)p>1 converges to w in C'(-Q) as p \ 1. (iii) Let vp be an arbitrary solution of (7.1) such that vp 0 up, for all 1 < p < (N + 2)/(N - 2). Assume by contradiction that there exists a subsequence of (vp)p>1 still denoted by (vp)p>1 such that IIv Ik < M < oo as p \ 1. Note that by Corollary 1.3.8 there exists cp > 0 such that up, vp > cp dist (x, 8S1)

in Q.

Comments and historical notes

189

Because up,vp E C2(1l) fl C1,7(i ), it follows that up2

-2

2

yP up

UP

-

2

VP

E C2 (n) fl Ci 1

VP

Then, 2

0<

J,(0 S2 \

2

Vu,-upVvpi +IVvp-vPVup uP

d2'

UP

VP

aui)l(up - vp)dx vP

(7-87)

= f 1 (up 1-a +.Xup-1) - (vp 1-a + Avp-1))(U,2 - vP)d2

- I)Op-2 ! (up - vp)2(Up + vp)dx = fn (- (1 + a) + A(p - 1)9p+a) BF, 2-a(up - vp)2(up + vp)d2, =

J

- (1 + a)9p 2-a + A(p

for some Bp between up and vp. Notice that by our assumption and Theorem 7.7.2 we have IIUPIIoo, IIVPIIoo < C,

for some C > 0 independent of p > 1. Hence,

-(1 + a) + A(p - 1)ep+a < -(1 + a) + A(p - 1)CP+a < 0

as p\1.

This means that the right-hand side in (7.87) is negative, which is a contradiction. Therefore Ilvplloo -> oo as p \ 1 for any solution vp of problem (7.1) with vp # up.

As we have already argued in the first part of the proof, this also implies that O up is the minimal solution of (7.1). The proof is now complete. Remark 7.7.4 From estimate (7.71) in Theorem 7.6.1 and Theorem 7.7.3 (iii), it follows that any nonminimal solution vp of problem (7.1) satisfies IIvpIIH,,-(o)-9oo

asp\1.

7.8 Comments and historical notes Variational methods offer a deep insight into the nature of elliptic equations. The interested reader may find a valuable introduction in this sense in the book by Struwe [184]. These methods are a very useful tool when comparison principles fail to hold. This is the case, for instance, of elliptic problems involving smooth nonlinearities that have a superlinear growth at infinity. As we have seen so far in this book, this behavior is associated with the issue of multiplicity. In our case, the use of variational arguments provided us with the second solution, which was obtained by Ekeland's variational principle.

Superlinear perturbations of singular elliptic problems

190

The existence of weak solutions for singular elliptic equations was first investigated by Lair and Shaker [125] for the problem

I -Du = p(x) f (u)

in 92,

u > 0

in fl,

u=0

on 8S2

(7.88)

in a smooth bounded domain 52 C RN (N > 1), p E L2(Q) is nonnegative, and f is a positive decreasing function such that fo f (t)dt < oo. It was shown that (7.88) has a unique weak solution u E H01(92). The approach given here follows an idea of Haitao [101], but the sharp esti-

mates in proving the existence of the second solution originate in the works of Brezis and Nirenberg [33], Tarantello [187], and Badiale and Tarantello [10]. For a detailed proof of the estimate (7.41), the interested reader may consult Brezis and Nirenberg [33], (34].

This chapter completes the study of bifurcation and asymptotic analysis for the model problem

1-Du=u-°`+Aup

u>0

in9, in 92,

(7.89)

u=0 on 852, where 0 < p:< (N + 2)/(N - 2), 0 < a < 1, A > 0. The results obtained so far concerning (7.89) can be summarized as follows:

If 0 < p < 1, then problem (7.89) has a unique solution ua E C2(52) fl Cl,l-«(52) (see Theorem 4.3.2). If p = 1, then problem (7.89) has solutions if and only if A < A I. Moreover, for all 0 < A < Al there exists a unique solution ua E C2(Q) fl C1,1-«(S2) of (7.89) that satisfies limb, fa, uA = oo uniformly on compact subsets of SZ (see Theorem 5.6.1). If 1 < p < (N + 2)/(N - 2) and N > 3, then there exists A* > 0 such that (i) for all 0 < A < A*, problem (7.89) has at least two weak solutions ua and va such that uA < va in 0; (ii) for A = A*, problem (7.89) has at least one weak solution ua; (iii) for all A > A*, problem (7.89) has no weak solution (see Theorem 7.1.2).

Furthermore, if 0 < a < 1/N and 1 < p < (N+2)/(N - 2), then any weak solution of (7.89) is actually a C2(S2) n C1'7(52) (0 < ry < 1) solution and (iv) limp\,, A* = A1(-0) (see Theorem 7.7.1); (v) problem (7.89) has a minimal solution;

(vi) any other solution of (7.89) blows up in the L°° norm as p

(see

Theorem 7.7.3). As remarked by Meadows [138], the C1''Y regularity is a particular feature of the fact that the weak solutions of (7.89) are positive inside the domain.

8

STABILITY OF THE SOLUTION OF A SINGULAR PROBLEM All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. James C. Maxwell (1831-1879)

The issue of the stability of a solution was raised in Chapters 5 and 6 for elliptic equations involving smooth nonlinearities. In this chapter we provide more results related to this matter, being mainly interested in the case of singular nonlinearities. The analysis we develop here is carried out in a general framework that includes the classical singular elliptic problems already considered so far and for which we have discussed the existence, bifurcation, and regularity. This fact will be illustrated by some examples presented in the last part of this chapter. 8.1 Stability of the solution in a general singular setting We are mainly interested in the stability of solutions corresponding to the problem

1-Du = f (x, u)

in Il,

u>0 u=0

in Q,

on81,

where f C RN (N > 1) is a smooth bounded domain of class C3'7, 0 < ,y < 1. We assume that the nonlinearity f : fI x (0, oo) -* R satisfies the following: (f 1) There exists an integer m > 0 such that f,

a3f &f ate atj-lark

for all1<j<m+l and1
E

C(sz x (a ,

))

oo,

Stability of the solution of a singular problem

192

(f2) There exists -1 < a < 1 such that for all u : S2 -> R satisfying 0 < cid(x) < u(x) < cid(x) in 52, we have if (x, u(x))1 < C°(1 + d(x)")

in 12

and

.gjf(x, u(x))

8i f (x, U(x))

+

j

1

k

I


jda-? (x)

in 52,

k=1

for all 1 < j < m+1 and 1 < k < N, where the constants Co, C1,..., do not depend on u. Typical examples of nonlinearities that fulfill (f 1) and (f2) are those already encountered in the previous chapters. For instance, we may consider any combination of pure powers like

f (x, u) = a(x)u" + up (x, u) E S2 x (0, oo),

with -1 < a < 1, 0 < p < 1, and a E C1(S2) is such that d(x)2-"lDa(x)I is bounded in Q.

Throughout this chapter, by COk "(C2) we mean the set of all functions u E Ck,"(1l) such that u = 0 on 852. We assume that problem (8.1) has solutions in C2(92) nC1(Cl), and we study the stability of these solutions in the sense given by Definition 5.1.1. This means that we have to deal with the linearized problem

©v - ft (x, u(x))v = Av

v=0

in 52,

on 852.

The analysis of (8.2) will be carried out in a more general setting by considering the following linear eigenvalue problem

-Av - a(x)v = Ab(x)v V=0

in S2,

on 852.

Taking into account the hypotheses (f 1) and (f2) on f, a natural assumption on a is

(H)

a E C1(52) and d(x)2-"IVa(x)I is bounded in Q.

As a consequence, if a satisfies (H) then the mapping x 1-4 d(x)2a(x) belongs

to C°'O(fi) for all 0 < a < min{1 + a, y), and d(x)1-"a(x) is bounded in 1. Note that a(x) may change sign near the boundary. To control the sign of a near 852, we state the following result.

Stability of the solution in a general singular setting

193

Lemma 8.1.1 There exist two functions cp± E C2"3(?) with cp+ > 0 in St such that if V E C2(S2) n C'-p()) is a solution of (8.3) then the functions v± := cp+v belong to C2(Q) n C',O(-Q) and verify

G"v": = a: (x)v+ + Ab( x)v±

in Q,

vt = 0

(8.4)

on asp,

where Gt is defined as G±w = --Aw ±,D(x) - Vw

for all w E C"3(1l),

(8.5)

and RN is a C' function. Moreover, a± satisfies the assumption (H) and ±a± > 0 in a neighborhood of &Q.

Proof As a result of the regularity of the domain, we can find S > 0 such that d(x) E C2'7(f& ), where Q6 :_ {x E Q : d(x) < b}. Let us fix -1 < p < a and consider 0 E C3 [0, oo) such that

po(t)>0

forallt>0,

pi(t) = tp+i IP(t)=0

for all 0 < t < e < p,

forallt>

where s > 0 will be defined later in this proof. Consider now cp+(x) := e+O(d(x))

x E Ti

and

vt(x) := cp'(x)v(x) x Then

E SZ.

vi- verifies

vt = 0 av± an

_

av

an

on aQ, on 0Q,

and

-Av+ + 2Vv+ - V (O o d)(x) = a} (x)v± + Ab(x)v±

in Q,

where

a=':(x) := a(x) + ((,O'(d(x))2 ± "(d(x))) IVd(x)12 ± 0'(d(x))Md(x)

in U.

Setting 4)(x) := -2V(i o d)(x), x E Ti, we obtain (8.4). Now, to have fag > 0 in a neighborhood of an, it suffices to take c > 0 such that for all x E Il with d(x) < e there holds ±a(x)d(x)'-v

+ (p + 1) (1

- p+ 1 d(x)p+l) IVd(x) I2 + p + 1 d(x)Ad(x) > C.

Because' Vd(x) I is bounded away from zero in the neighborhood of 8SZ and the mapping x --* d(x)i-aIa(x)j is bounded in 11 according to assumption (H), we El easily can find e > 0 with the previous property. This concludes the proof.

Stability of the solution of a singular problem

194

Remark 8.1.2 Lemma 8.1.1 implies that the spectrum of (8.3) coincides with the spectrum of (8.4), provided that the eigenfunctions are required to be in C2(52) n Co (S2). Moreover, we have:

(i) v > 0 in SZ if and only if vt > 0 in Q. (ii) v = 0 in 1l if and only if v+ = 0 in Q. (iii) an < 0 on 852 if and only if an < 0 on 852. (iv) an = 0 on 852 if and only if an = 0 on 812. The following result states that the strong maximum principle holds for the operators of type L+ + a(x).

Proposition 8.1.3 Let a > 0 satisfy (H) and let f be a differential operator defined as

Lv :_ -Av + (D(x) Vv - for all v E C2(12), where -D E C1(SZ) (in particular, C may be any of two operators 1. (8.5)). Assume that v E C2(1) n C1"19(1) satisfies

in Il, in 0,

f Lv + a(x)v > 0

v>0 and let xo E U be such that v(xo) = 0. (i) If x0 E Ii then v 0. (ii) If xo E 852 and v > 0 in SZ then 8 0)

defined in

< 0.

Proof Because the coefficients of the linear operator L+a(x) are locally bounded in 52, the proof of (i) follows directly by the strong maximum principle as stated in Theorem 1.3.5.

(ii) Assume by contradiction that v > 0 in 52 and v(xo) = 8v(xo)/8n = 0. Because v E C1'p(52), there exists c > 0 such that

v(x) = Iv(x) - v(xo)I < clx - xol'+A

for all x E Q.

(8.6)

Note also that 52 satisfies the interior sphere condition at x0. Hence, there exists an open ball B C 52 with a center at yo and a radius r > 0 such that B n 852 = {xo}. Define w(x) := (r - Ix - y0l)1+0/2 for all x E B. Then,

Cw + a(x)w = - Aw +,t(x) Vw + a(x)w

_2+Q N-1 2

Ix-yol(r-

x - yoI)"/2

- P( 2 -r a) (r - Ix - yol)-1+0/2 2

2 +,6 (r - Ix - Y01) ,I/ 2 Vx) - (x - y)o + a(x)w 2 Ix - yol .

in B

Stability of the solution in a general singular setting

195

Because d(x)1-pa(x) is bounded, it is readily seen that there exists ro close to r such that Lw + a(x)w < 0 in the annulus

A := {xEc:ro < Ix - xci < r}. Let e > 0 and set vE := v - ew. Then CvE + a(x)vE > 0 in A. By the standard maximum principle it follows that vE attains its minimum at a point x1 E A.

Thus, if Ixl - yol = r, then

v(x1) > 0, and if Ixi - yoI = ro, then

v6(xl) = v(xl) -ew(xl). We can chooses > 0 small enough such that 0. Hence, in both cases we have vE > 0 in A. This property holds true in particular on the rectilinear segment S C A that joins yo with xo and for which we have r - Ix - yoI = Ix - xol. Therefore, v(x) > eIx - X0111,912 for all x E S n A. Because c > 0, the previous relation contradicts (8.6). This concludes the proof. For w E Co" (Ti), consider the linear problem

Av = a(x)w v=0

in 52,

on 852.

The following result provides a continuous dependence on data for solutions of problem (8.8) even when a is a singular potential. We omit the proof, which is lengthy and beyond the purposes of this chapter. Proposition 8.1.4 Let a satisfy (H). Then, for all w E 0-00,1 (-Q), there exists a unique solution v E C2(52) n C''(52), 0 <,O < min{1 + a, y} of problem (8.8). Moreover, there exists a positive constant C > 0 that is independent of w such that (8.9) llvllc1.o() < CllwllCO,l( ).

Let GQ be Green's operator of (8.8)-that is, for all w E C0'1(f), Qa(w) is the unique solution v E C2(Sl) fl C(Q) of the problem (8.8). Thus, Proposition 8.1.4 states that the Green operator !g,, : Co'1(52) - C1',3(S2) is continuous. Proposition 8.1.5 Assume that a and b satisfy the assumption (H) and b > 0 in f2. Then; there exists ko E R such that for all k > ko and all w E Co'1(52), the problem

-Av - a(x)v + kb(x)v = b(x)w

v=0

in n, (8.10)

on 852

has a unique solution v E C2(52) n C1"3 (S2) that fulfills

llvllcl,s(r)
(8.11)

for some positive constant C > 0 that does not depend on w. Moreover, if w > 0 in 52 and w is not identically zero, then 8v/8n < 0 on 852.

196

Stability of the solution of a singular problem

Proof We first choose ko which ensures the uniqueness. To this aim we rewrite problem (8.10) in the form

f G-v- - a-(x)v- + kb(x)v- = b(x){o w

in 52, (8.12)

V- = 0

on 852,

where G-,a-,v-, and cp are defined in Lemma 8.1.1 and v- = cp-v. Because a- > 0 in a neighborhood 525 of 852 and b > 0 in S2, we can define ko := sup xEn\IZ&

a (x) b(x)

This yields kb(x) - a- (x) > 0 in l for all k > ko. Furthermore, by Proposition 8.1.3 we derive the uniqueness of the solution to problem (8.12) and hence the uniqueness of (8.10). Moreover, 8v-/8n < 0 on 8S2 provided that w > 0 is not identically zero. Using Remark 8.1.2, this yields 8v/8n < 0 on 852. Let us prove now the existence for all k > ko. For this purpose let Cp,P(u)

9a,

be the Green's operators associated with problems (8.8) and Ov = b(x)w v=0

in S2,

on 852,

respectively. Then problem (8.10) may be formulated as fl(v) = Gb(w),

(8.13)

where

l(v) := v - qa(v) + kGb(v) v E Cl'p(S2). Because the embedding i : Cp'Q(?) -, Ca"(S2)

(8.14)

is compact, we derive that N := N o i is a compact perturbation of.the identity in CC'A(1l). Furthermore, N is injective for all k > ko. By standard Riesz theory on compact operators, we note that 7-l is a linear homeomorphism. Thus, if w E Co"(52), then problem (8.13) has a unique solution v E Cl"a(1). Moreover, according to Proposition 8.1.4 we also have C1 IIcb(w)IIcI.fl(-a)

c'211w11cO.1(),

where C1, C2 > 0 are independent of w. This finishes the proof.

D

Theorem 8.1.6 Assume that a and b satisfy the assumption (H) and b > 0 in Q. Then the spectrum of the linear eigenvalue problem (8.3) with v E C2(52) fl C0I'3 (0) has the following properties:

Stability of the solution in a general singular setting

197

(i) It consists of at most a countable set of eigenvalues which are isolated. (ii) The first eigenvalue is simple and the associated eigenfunction v1 satisfies v1 > 0 in n and 8v1/8n < 0 on 812. (iii) It does not change when the eigenfunctions are required to be in C2(Q) rl Ha (Sl).

Proof (i) Let ko be as in Proposition 8.1.5. Then, for all k > ko, problem (8.10) defines a Green's operator G : C00'1(12) -4 C011'3(- Q), which, in view of (8.11), is bounded. Furthermore, if i is the compact embedding defined in (8.14), then

G=goi

: CO, -6 (S') -4 C"(?) is compact. Thus, (i) follows by the spectral characterization of the compact operators and the fact that the eigenvalues of (8.3) and the eigenvalues it of 9 are related by

µ(A + k) = 1.

(8.15)

(ii) By Proposition 8.1.5, the compact operator G maps the positive cone of Co'a(S2) into its interior-that is, into the set of those functions u E C01'1(i) that fulfill u > 0 in 12 and 8u/8n < 0 on 81. Hence, by the Krein-Rutman theorem (see Theorem A.2.3 in Appendix A), it follows that the first eigenvalue µl of a is simple and positive. Taking into account the relation (8.15), we derive that A, = 1/µl - k is also simple. Moreover, the first eigenfunction vl is positive in 9 and satisfies 8v1/8n < 0 on 852. (iii) Let X be the completion of Ca(a) with respect to the norm IIvIIt :=

J5

b(x)v2dx.

It turns out that X is a Hilbert space with respect to the inner product (v, w) =

Jsi b(x)vwdx

v, w E X.

X is compact. Taking into account the standard compact embedding theorems into LP spaces with weights, we need only to prove that j is continuous. Indeed, by the assumption (H) it follows that d(x)2b(x) is bounded in Q. Hence, there exists c > 0 such that b(x) < d2(x) in Q. Furthermore, we claim that the embedding j : Ho (1Z)

By the Hardy-Sobolev inequality there exists co > 0, which is independent of v such that f51

'

dx < co fn

IVvj2dx.

(8.16)

Hence,

IIvIIb
(8.17)

Stability of the solution of a singular problem

198

Now let w E CC'1(S2) and v E C2(1) fl Co (Q) be a solution of (8.10). Multiplying by v in (8.10) and then integrating by parts, we obtain

r IvvI2dx n

J

n

b(x)v2dx = f b(x)vwdx.

a(x)v2dx + k l

(8.18)

z

On the other hand, by virtue of the assumption (H) there exists w cc St and ci > 0 such that d(x)2Ia(x)I < 21

in St \w

(8.19)

and Ia(x)I < cib(x)

in w,

where co is the constant from (8.16). This yields

j

Jra(x)v2dx< <

2

[dx+cijb(x)v2dx

f

(8.21)

Choosing k > lei 1, from (8.21) and (8.18) we obtain

Jsi

IOvl2dx < c2 f b(x)vwdx,

for some positive constant c2 > 0. Next, by Holder's inequality and (8.17) we derive IIvIIxo < ciiWIIb,

for some constant c > 0 independent of v and w. This means that Green's operator 9 : X -* H01(9) of problem (8.10) is bounded. Then 9 :_ 9 0 j :

Hp (Il) -+ Hp (Ii) is compact. From now on, we proceed as in the proof of (i) and use the fact that the eigenvalues of (8.3) and those of G are related by (8.15). 0 This finishes the proof of Theorem 8.1.6. Theorem 8.1.6 does not give a precise answer on the sign of the first eigenvalue of (8.2). In other words, we cannot decide whether the solution u of (8.1) is stable. In this sense, the following result provides some natural and simple conditions

to derive that the first eigenvalue of (8.2) is strictly positive. It is a very useful result when applying the implicit function theorem as we did in Theorem 5.2.1.

Proposition 8.1.7 Suppose that (8.1) has a solution u E C2(Sl) n C'(52) such that (i) an < 0 on 91l; (ii) f (x, u) - uft(x, u) > 0 in 0. Then the first eigenvalue Ai of (8.2) is positive-that is, u is a stable solution of problem (8.1).

A min-max characterization of the first eigenvalue for the linearized problem

199

8.2 A min-max characterization of the first eigenvalue for the linearized problem We start this section with an equivalent condition for the sign of the first eigenvalue of problem (8.3).

Lemma 8.2.1 The first eigenvalue A of the spectral problem (8.3) is positive if and only if the operator -A - a(x) satisfies the strong maximum principle-that is, for all v E C2(11) n C' (11) not identically zero such that

Av - a(x)v > 0

v>0

in ll, on an,

(8.22)

then v > 0 in S2 and 8v(x)/8n > 0 for all x E aQ with v(x) = 0.

Proof If -A - a(x) satisfies the strong maximum principle, then we clearly have Al > 0. To prove the converse, we transform (8.22) according to Lemma 8.1.1 in

,C-v- - a-(x)v- > 0

in 1, on an,

v->0

(8.23)

and there exists w CC S2 such that a- (x) < 0 in S2 \ w. Let v1 be the first eigenvalue of (8.3) corresponding to Al. Notice that v1 > 0 in Q. Set

k := sup

2la (x)l

xE1\w \Jb(x)vl(x)

and for e > 0 define

> Q,

(8.24)

w:=v-+e+ekvl.

Thus, by (8.24) we have

L-w - a-(x)w = (iv - - a-(x)v-) + ek(G-vi - a-(x)v1) - ea-(x) > E(kA1b(x)vi -a-(x) ) > 0

in Q.

Using the continuity of w, there exists S = 8(e) > 0 such that w > 0 in 128. Moreover, by the strong maximum principle (see Theorem 1.3.5) applied to w in St \ 128 we also have to > 0 in Q \ Q. Therefore, to > 0 in St and letting e -i 0, we obtain v- > 0 in Q. Thus, we have obtained

5 G-v- - a- (x)v- > 0

in St,

v->0

in Q.

Sl

Because v- is not identically zero, by Proposition 8.1.3 we find v- > 0 in S2 and

8v-(x)/8n > 0 for all x E an with v-(x) = 0. By virtue of Remark 8.1.2, the last conclusion also holds for v. This completes the proof.

Stability of the solution of a singular problem

200

A min-max characterization of the first eigenvalue to (8.3) is stated in the following proposition.

Proposition 8.2.2 Assume that a, b satisfy (H) and b > 0 in Q. Then, the first eigenvalue Al of (8.3) is stated by

Al =

sup

inf

-Av - a(x)v

(8.25)

b(x)v

vEC2(n)nco(ri) 2: En v>O in S1

Proof Denote by m the quantity in the right-hand side of (8.25) and let vl be the corresponding eigenfunction of the first eigenvalue Al to (8.3). Then vi E Cz(0)flCol (52),v>0in 92,and

_ Al

-Avl - a(x)vl b(x)vl

zESO

This means that Al < m. To prove that m < A1, we argue by contradiction and we assume that there exists e > 0 and w E C2(c) fl Co (St), w > 0 in S2, such

that i nf

-©w - a(x)w b(x)w

> A l + E.

( 8 . 26 )

Let us consider the spectral problem

Av - a(x)v - (Al - e)b(x)v = Ab(x)v V=0

in S2,

on 852.

(8.27)

Clearly, E > 0 is the first eigenvalue of (8.27). Furthermore, by Lemma 8.2.1 the

operator -Av - a(x) - (Al - E)b(x) satisfies the strong maximum principle. To raise a contradiction, let r := min{tl, tz}, where t1 := sup{t > 0 : w - tvl > O in S2}, t2 :=SUP {t > 0 :

8(w - tvl)

< 0 on on }

.

1JJ

Define now W := w - rv1. Clearly, W > 0 in 52. Moreover, if r = tl then W = 0

at some point in St and if r = t2 then W = 8W/an = 0 at some point on aSt. On the other hand, by (8.26) we have

-AW - a(x)W - (Al - s)b(x)W > sb(x)(w + W) > 0

in 52.

By Lemma 8.2.1 we find that W > 0 in S2 and aW/8n < 0 on aft, which is clearly a contradiction. Hence, Al = m and the proof is now complete.

Differentiability of some singular nonlinear problems

201

8.3 Differentiability of some singular nonlinear problems In this section we are concerned with the differentiability of the semilinear elliptic problem (8.1) around positive solutions u E C2(S2)f1C1(Sl) that satisfy au/an < 0 on aQ. To this aim, let 9 : CQ (St) - Co (S2) be Green's operator of AV =W

v=0

in 0, on ad,

defined as v = G(w). By virtue of Proposition 8.1.4, 9 is bounded and it can be extended as a bounded operator to C01,8 (Ti). Furthermore, u E C2(d) fl C1(12) is a solution of (8.1) if and only if 0 < u E C0(St) satisfies T(u) = 0,

(8.28)

where

T(u) := u - G(f(-, u)). Let C be the positive cone of Co (12)-that is, the set of all functions u E Ca (S2)

that fulfill u > 0 in Si and au/8n < 0 on 812. We are concerned here with the Frechet differentiability of the operator T : Co (d) - Co (Q).

Theorem 8.3.1 Under the assumptions (f 1) and (f2), the operator T is of class Cm on the positive cone C, and for all v E C the linear operator T'(v) : Co (12) - Co (Q) is given by

T'(v)(w) = w v)w). If m > I and I < j < m then the j linear operator

(8.29)

T('} (v) : [Co(d)]' -+ CO, M)

is given by

T(i)(v)(wi,w2,...,w?) _ -G(`f( - v) (w1,w2,...,w5)). ati

(8.30)

Proof Remark first that T = I - T, where I : Co (SZ) -+ CC (12) is the identity v)). Notice that I is linear and bounded; hence, I is of class C. Furthermore, its first derivative is I and its higher order derivatives are zero. Thus, it suffices to show that for all 1 < j < m the jth derivative of T

operator and T(v) = c(f is given by

T(j)(v)(w1,w2,...,wj)=JC( 2 and the mth derivative of T is continuous.

,

v)(w1,w2,...,wj)

(8.31)

Stability of the solution of a singular problem

202

We prove (8.31) by induction on j. Consider first the case j = 1 and let a E C2(SZ) be such that a > 1 in Sl and a(x) = d(x)'-1 in a neighborhood of 89. Then, a

and d(x)2-alVa(x)I are uniformly bounded in Q.

(8.32)

a(x) Let v E C and W E Col(U) with IIwIIco(cz) sufficiently small. Then there exist two

positive constants cl, c2 that are independent of w such that 0 < cld(x) < v + Ow < c2d(x)

in S2,

(8.33)

for all 0 < 9 < 1. Also define

W(x)

:= a (x) (f (x'v + w) - AX, v) - ft (X, v)w}.

By the mean value theorem, relation (8.33), and the hypotheses (f 1) and (f 2) we have

IW(x)1=

a Ift(x,v+91(x)w)-ft(x,v)IIwI

(0 < 91(x) < 1) (8.34)

2 Iftt(x, v + 02(x)w)Iw2 < CiIIwIICO(n) < a(x)

(0 < 02(x) < 1)

and

Wxk(x)I <

Iftxk(x,v +8(x)w) - ftxk(x,y)Ilwl

(0<0(x)
+ Iftt(x, v + 9(x)w) - ftt(x, v) IvxkwI a(x)

(8.35)

+ Iftt(x,v+9(x)w)Ilwwxkl + IW(x)axk(x)I a(x)

a(x)2

< C2IIwIICo(sl)T(IIwIICo (n)),

forall1
C3IIwIIc,(11)T(IIwIIco(1i)).

Taking into account the previous estimate and Proposition 8.1.4 we have

IIT(v + w) -T(V) -

v)w)lICi p) = I1c(aW) Il c, (1)

(8.36)

< C4IIwIlco(sl)T(Ilwllca(st)),

for all w E CC(Q) such that IIwIIeo(j1) is small and IIwIIC (si} <

T'(v) exists and

Hence,

Differentiability of some singular nonlinear problems

T'(v)(w) =

203

for all v E C,w E Co(S2).

Next we assume that (8.31) holds for jth derivative of T and let us prove that it also holds for its (j + 1)th derivative. Let w1, w2, ... , wj+1 E Co (0) be Remark that (8.33) holds for w = wj+l and such that Ilwj+111c, (iz) <_ we retake the previous arguments for 1

Wl(x)

a (x)

(x,v+wj+1)

(8,f

_ 3 f(x,v) _

+'f(x,v)

&j

8tj

tj+1

l

wj+1 J wl . . . rv

We obtain C511w1lIC,(sz)IIW211CC(n) ... IIwj+1

IICo(n)T(IIwj+1IIC;(-a)),

where C5 > 0 is independent of w1, w2i ... , wj+1. As in (8.36) we have

rT(j)(v+wj+l}-T{7)(v) (wl

,wj)-g

at+lv))W1---W

l

' co(w)

L

= I19(aW1)Ilc,;(u) C511w1IICS(n) 11w211ca, (s-1) ...

))l

with Cs > 0 independent of wi, w2, ... , wj+1. This estimate implies that T(j+1)(v) exists and is given by (8.31). This conipletes the induction argument. Finally, the same reasoning that led us to the previous estimate yields rT(m) (v + wj+1) - T(m) (v), (w1, w2, .... wj ) I

< CIIwi11q(si)

Co(w)

Ilwj+1llc,l(-n)T(Ilwj+1IICi(n)),

(8.37)

which implies that T(m) is continuous. This finishes the proof. Assume next that the nonlinearity f depends on a parameter A E A, where A C R is an interval, and the derivatives 8P f /aA , 1 < p < r, satisfy the assumption (f2). With the same arguments as used earlier we have the following corollary.

Corollary 8.3.2 Using the previous assumptions, the operator

T:QxCo(?)xA->Co(S2). defined as

T(u, A) := u - c(f (', u, A)), satisfies

(i) for all 1 < j < m and 1 < p < r, the derivative Bj+PT/8tj8Ar exists and is continuous at each point (v, A) E C x A;

Stability of the solution of a singular problem

204

(ii) the linear operator

aj+vT(v,

atiaap A) = [Co -)]' ->

Co (? )

is given by Oi+PT(v, A)

atiaAP

(WI, w2, ... ,

f aj+Pf (._ v_ A)

(WI, w2, ... , wj}),

for all w1i w2, ... , wj E Co (S2).

8.4 Examples The results obtained in this chapter concerning the stability and the differentiability in singular elliptic problems are illustrated by the following two examples.

Example 8.1 Let us first consider the problem

-Du = Aa(x)u'

u>0 u=0

in S2,

in Q,

(8.38)

ono)

in a smooth bounded domain SZ with -1 < a < 1, A > 0 and

(i) a E C1(9) is positive and there exists /3 E (-1 - a,1 - a) such that a(x) < cd(x)p in SZ for some c > 0; (ii) the mapping f (x, t) = a(x)ta satisfies (f 2).

If 0 < a < 1 then (8.38) arises in population dynamics when dealing with stationary solutions of the usual logistic equation with nonlinear diffusion. For

-1 < a < 0 and a E C(?i), a > 0 in Q, problem (8.38) was considered in Chapter 4, where we proved the existence and uniqueness of a solution u), E C2(1l) fl C1, (St), (0 < 7 < 1) for all A > 0. Similar results hold in the case 0 < a < 1. Concerning the stability and the differentiability of (8.38) we have the following.

Proposition 8.4.1 For all_A > 0, problem (8.38) has a unique solution ua E C2(9) fl C1 "(S2), 0 < ry < min{1, I + a + 0}, such that (i) a < 0 on c3S2; (ii) ua is stable; (iii) the mapping (0, co) D A'--4 ua E C'"-Y(52) is of class C.

The existence follows in a similar way as in the proof of Theorem 4.3.2. The conclusion in statements (ii) and (iii) follows by Proposition 8.1.7 and Theorem 8.3.1, respectively.

If a(x) changes sign in SZ, the analysis of (8.38) becomes more delicate. A detailed study in this sense was carried out by Bandle, Pozio, and Tesei [17].

Comments and historical notes

205

Because 0 < a < 1, the nonlinearity f is not locally Lipschitz near the origin, which may give rise to dead core solutions. Existence of this kind of solutions was obtained in [17] by the sub- and supersolution method. Also, the uniqueness is provided under some additional hypotheses.

Example 8.2 Our second example concerns the problem 5 -©u = a(x)u-°` + Au1'

u=0

in 52, on 852,

(8.39)

withO0, and (i) a E C1(52) is positive and there exists /3 E (a - 1, a + 1) such that a(x) < cd(x)a in 52 for some c > 0; (ii) the mapping f (x, t) = + AtP satisfies (f 2). If 0 < p < 1, in view of Theorem 4.3.2 we have the same conclusion as in Proposition 8.4.1. If p = 1, taking into account the bifurcation results in Theorem a(x)t_0

5.6.1 we obtain the following.

Proposition 8.4.2 Assume p = 1. Then, for all 0 < A < Al(-0), problem (14.71) has a unique solution ua E C2(52) n C1°"r(? ), 0 < -y < min{1,1 + a + Of such that (i) as < 0 on 852;

(ii) ua is stable; (iii) the mapping (0, A1) 9 A H ua E C1'"(52) is of class CO° and uA -+ 00 uniformly on compact subsets of 52.

8.5 Comments and historical notes In this chapter we pointed out new features of singular elliptic problems, being concerned with stability and differentiability of such types of problems. First we studied the spectrum of some linearized singular elliptic problems in a general setting involving singular weights. In this sense we considered the general problem (8.3), where the (possible singular) potential b is positive in Q. If the exponent a in the assumptions (f2) and (H) satisfies a > -1/N, then the standard regularity theory for elliptic equations applies to obtain eigenfunctions in the class Co(52). However, we preferred to state our problems in an integral form by means of Green's operator. Thus, Ca (S2) turns out to be a suitable space in our analysis for all exponents -1 < a < 1. The min-max characterization of the first eigenvalue in Theorem 8.2.2 was first introduced in Donsker and Varadhan [68] and then extended in Berestycki, Nirenberg, and Varadhan (24]. Next we have studied the Frechet differentiability of the associated integral problem with respect to u and parameters. The analysis presented in this chapter follows the general line in Hernandez and Mancebo [102] or Hernandez, Mancebo, and Vega [103], [104] for general

206

Stability of the solution of a singular problem

elliptic operators. We also mention here the work of Bertsch and Rostamian [25], where similar singular eigenvalue problems in divergence form have been studied. By means of Hardy-Sobolev-type inequalities, it is established in [25] that the eigenfunctions belong to the class C2(1) f1 Ha (1).

9

THE INFLUENCE OF A NONLINEAR CONVECTION TERM IN SINGULAR ELLIPTIC PROBLEMS To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.

Henri Poincare (1854-1912), Science et Hypothese, 1902

9.1

Introduction

In the previous chapters we discussed the existence, the bifurcation, and the stability of classical solutions for elliptic problems involving smooth or singular nonlinearities. Our aim in this chapter is to study the following class of singular elliptic problems Au = p(d(x))g(u) + AI VuIa + E.tf (x, u)

in S2,

u>0

in 52,

U=0

on 852,

(9.1)

in a smooth bounded domain 52 C II!N (N > 1). Here, d(x) = dist(x, 8Q), A E R,

µ>0,and0
Throughout this chapter we assume that g E C1(0, cc) verifies

(gl) g is a positive decreasing function such that limt\o g(t) = +oo. This hypothesis shows that g has a singular behavior at the origin. The standard example of such a nonlinearity is g(t) = t-', with a > 0. We also assume that f : Si x [0, oo) --> [0, oo) is a Holder continuous function

that is nondecreasing with respect to the second variable and such that f is positive in SZ x (0, oo). The analysis we develop in the sequel concerns the case in which f is either linear or sublinear with respect to its second variable. The main feature of this chapter is the presence of the convection term IVuIa combined with the possible singular weight p : (0, oo) - 1W, which is supposed to be Holder continuous. At this stage, two questions arise: 1. Does the gradient term IVula play a major role in the bifurcation results with respect to the parameters A and p?

208

1 'he influence of a nonlinear convection terns in singular elliptic problems

2. Is the presence of the possible singular weight p(d(x)) significant in the existence of classical solutions to (9.1)? To investigate these two questions, we first provide a general nonexistence result in terms of the decay rate of p near the origin. The existence part relies essentially on the sub- and supersolution method as described in Theorem 1.2.3. This method has the advantage of providing boundary estimates of solutions in case of some special nonlinearities. This matter will be emphasized in Section 9.6.

As a result of the presence of the gradient term and the lack of an adequate comparison result, we are not able to decide whether the solution of (9.1) (if exists) is unique. However, if f (x, t) depends only on x variable, a standard argument based on the maximum principle and the fact that the gradient vanishes in all the extremum points that lie inside the domain allow us to derive the uniqueness of the solution once the existence has been proved. The results obtained in this chapter allow us to extend the study of problem (9.1) in case 52 = RN, as we started in Section 4.7.

9.2 A general nonexistence result In this section we state a general nonexistence result related to problem (9.1). The following property strongly requires a quite strong decay rate of the variable potential at infinity. Theorem 9.2.1 Assume that g verifies (gl) and p : (0, oo) - (0, oo) is a Holder continuous function that is nonincreasing and

i

I

tp(t)dt = oo.

(9.2)

Then the problem

-Au + AIVuI2 > p(d(x))g(u) u>0

u=0

in S1,

in ),

(9.3)

on 852

has no classical solutions.

Proof Without losing the generality, it suffices to consider A > 0. We argue by contradiction and assume that there exists ua E C2(Q) n C(52) a solution of (9.3). Then, from (gl), we can find c > 0 small enough such that u := cco verifies

-Au +.IDi12 < p(d(x))g(u)

in Q.

Indeed, it suffices to consider c > 0 such that Al 1 +

Ac2IVVI I2 < p(d(x))g(cI PIIII)

in 52.

A general nonexistence result

209

This is obviously possible if we take into account the hypotheses on g and p(d(x)). Furthermore, we claim that (9.4) ua > u in St. -

Assuming the contrary, it follows that minj(ua - u) is achieved in R. At this point, say x0, we have i1(x0) > ua(xo), V(ua-u)(xo) = 0, and /..(u,\-u)(xo) > 0. Because g is decreasing, we obtain 0 > - A(ua - u)(xo)

_ ( - Aua + \I Vua l2\1 (xo) - (- Au + )tFViF2)\ (xo) ?p(d(xo))(g(ua(xo))/- g(u(xo))) >0,

which is a contradiction. Hence, ua > u in ft.

Let us perform the change of variable va = 1 - e- 'in (9.3). Then, 0 < VA

1 in Cl and

-Ova = all - va) (.IVual2 - ouA)

in Cl.

Furthermore, from (9.3) we have

- A(1 - v,\)p(d(x))g (_ln(1_VA))

-OVA >

va > 0 va = 0

l

in S2, in $Z,

on BQ.

To avoid the singularities in (9.5), let us consider the approximated problem

ln(1- v)) -Ov = all - v)p(d(x))g e -

v>0 v=0

in S2, in S2,

onc9Q,

with 0 < a < 1. Clearly, va is a supersolution of (9.6). Moreover, by (9.4) and the fact that limtNo(1 - e-)t)/t = A > 0, there exists c` > 0 such that vj,, > ccpr in Cl. On the other hand, there exists 0 < m < c" small enough such that mcpl is a subsolution of (9.6) and obviously mcpl < va in Cl. Then, problem (9.6) has a solution vE E C2(St) such that rnco < vE < vA Multiplying by cc Ai

f

in Cl.

in (9.6) and then integrating over Cl we find

AJ (1 - v£)Wip(d(x))g

(s - ln(1- ye)) dx.

(9.7)

The influence of a nonlinear convection term in singular elliptic problems

210

Using (9.7) we obtain

ai

(1 - va)cPip{d(x))91 -

colvadx > .1 si

ln(1

v,\)

dx

Jsa

> CJ pip(d(x))dx, na

where C > 0 and 08 = {x E 1

d(x) < 8}, for some b > 0 sufficiently small. Because cp1(x) behaves like d(x) in SZ6 and fo tp(t)dt = oo, by (9.8) we find a contradiction. It follows that problem (9.3) has no classical solutions and the proof is now complete. 0 :

A direct consequence of Theorem 9.2.1 is the following corollary.

Corollary 9.2.2 Assume that f' tp(t)dt = oo and conditions (gl), 0 < a < 2 are fulfilled. Then for all p > 0 and ) E R, problem (9.1) has no classical solutions.

9.3 A singular elliptic problem in one dimension We are concerned in this section with the following singular elliptic problem in one dimension -y"(t) = q(t)g(y(t)) 0 < t < 1, 0 < t < 1, y(t) > 0 (9.9) Y(O) = y(1) = 0, where g verifies (gl) and q : (0, 1) --+ (0, oo) is a Holder continuous function that

may be singular at 0 or 1. In the particular case q = 1 and g(t) = t-«, a > 1, we obtained in Section 4.1 some qualitative properties of the solution to (9.9). The main novelty here is the presence of the (possible singular) potential q. The existence results for problem (9.9) are of particular interest in the study of (9.1), because any solution of (9.9) may provide a supersolution in higher dimensions for problem (9.1). The main ingredient here is Theorem 1.1.2. To this aim we assume that q verifies 1

t(1 - t)q(t)dt < oo.

(9.10)

Theorem 9.3.1 Assume that (gl) and (9.10) hold. Then problem (9.9) has a unique solution y E C2(0, 1) fl C[0,1].

Proof Let us first remark that the mapping : [0, oo) -+ [0, ao),

I(t) = fat {s)

(9.11)

is bijective. Thus, by (9.10) we can choose M > 0 and e > 0 small enough such

that

A singular elliptic problem in one dimension

M ds > 2 1 1 t(1 - t)q(t)dt. 9(3)

211

(9.12)

o

We fix an integer ko > 1 with the property koe > 1, and for all k > ko consider T the problem -y"(t) = q(t)Gk(y(t)) 0
0
y(t)>0

(9.13)

y(0) = y(1) where

I

Gk(t)

9(t)

t>

,

(9.14)

1

1

1g( )

0
By means of Theorem 1.1.2 we deduce the following auxiliary result.

Lemma 9.3.2 For all k > ko, problem (9.13) has at least one solution yk E C2(0, 1) n C[0,11. Moreover, there exists m > 0, which is independent of k such that

mt(1 - t) < yk(t) < M for all 0 < t < 1.

(9.15)

Proof For fixed k > ko and 0 < A < 1, let y E C2(0, 1) fl C[0,1] be a solution of the auxiliary problem

-y"(t) = .iq(t)Gk(y(t))

0 < t < 1, 0 < t < 1,

y(t)>0 Y(O) = y(1) =

(9.16)

k

Then y is concave and y > 1/k in [0, 1]. Denote by tA,k E (0, 1) the maximum point of y. We also have y' _> 0 in (0, ta,k) and y' 0 in (ta,k,1). Because Gk < g in (0, oo), from (9.16) we derive

-

-yrr(t) < q(t)g(y(t))

forall0
Integrating over It, tA,k] (0 < t < tx,k) in (9.17) we obtain ptx,k

I

y'(t) < g(y(t)) /

q(s)ds

for all 0 < t < ta,k.

An integration over [0,tx,k] in the last estimate yields tAk

J That is,

y'(s)

k 9(y(s))ds <

q(s)ds.

o

t

(9.17)

212

The influence of a nonlinear convection term in singular elliptic problems

ds

fY(ta,k)

f

/'tA,k sq(s)ds. .

fo

g(S)

i1'k

Because kE > 1 for all k > ko, the last inequality also implies that

f

y(tA,k) ds

tA,k sq(s)ds.

g(s) - 10

Jf

Hence, tJ(tA,k)

ds

1

<

9(s)

f "" t(1

- t)q(t)dt.

(9.18)

1 - tax

In the same manner, integrating in (9.17) over [tak, t] (t,,k < t < 1), we have


1

fI

t 1 - t)q(t)dt.

{9.19}

A.k

Combining (9.18) with (9.19) we obtain y(tA.k) ds

9(s)

J£

<2

1

fo

t(1 - t)q(t)dt.

Using (9,12) we derive that IIy1Ioo = y(ta,k) < M. Hence, by Theorem 1.1.2, problem (9.13) has a solution yk E C2(0,1) n C[0,1] such that

k < yk(t) < M in (0, 1)-

(9.20)

Furthermore, by (9.20) and the definition of Gk it follows that Gk(yk) = 9(yk) in (0, 1). Hence, for all k > ko, yk satisfies -yk(t) = q(t)g(yx(t))

!

0 < t < 1,

0
yx(t)>0

(.921 )

Yk(O) = yk(1) = k

It remains to prove the first part of the inequality in (9.15). By Green's representation we have yk(t) = k + (1 - t) It sq(s)g(yk(s))ds + t I (1 - s)q(s)g(yk(s))ds, Jg

for all 0 < t < 1. Because

is decreasing, by (9.20) we obtain

yk(t) > Q(t) where

for all 0 < t < 1,

(9.22)

A singular elliptic problem in one dimension

I

213

t

Q(t) = g(M) ((1 - t) f sq(s)ds + t J (1 - s)q(s)ds/. t

Remark that Q(O) = Q(1) = 0 and

Q'(t) = g(1bI) (`

(t sq(s)ds + j (I - s)q(s)ds)

for all 0 < t < 1.

t

o

In view of (9.10) it follows that Q'(0) is finite. Also notice that Q(t) is positive in a neighborhood of zero. Hence, there exist cl > 0 and lh > 0 such that Q(t) > ctt

for all 0 < t < 771. Similarly, taking into account the fact that Q'(t) < 0 near t = 1, we can find c2 > 0 and 712 > 0 small enough such that Q(t) > c2(1 - t) for all 1 - 772 < t < 1. Because Q(t)/(t(1 - t)) is bounded away from zero on [r12, 1 - i 2], we can find m > 0 such that

Q(t) > mt(1 - t)

for all 0 < t < 1.

(9.23)

Now the second inequality in (9.15) follows from (9.22) and (9.23). This finishes the proof.

Lemma 9.3.3 The sequence (yk)k>k,, is bounded and equicontinuous.

Proof Let tk E (0,1) be the maximum point of yk. Because yk is concave, it follows that y' > 0 in (0, tk] and yk < 0 in [tk, 1). Integrating between t and tk in (9.21) we derive Iyk(t)I

<

jtk

9(yk(t)) -

forall0
q (s)ds

(9.24)

We claim that

0 < inf tk < sup tk < 1k> ka

(9.25)

k>k,,

Assume by contradiction that (9.25) does not hold. If infk>ko tk = 0, then, up to a subsequence, we have tk -+ 0 as k --> oo. Integrating over [0, tk] in (9.24) we have rtk Iyk(t)I dt < / tk J tk q(s)dsdt = t k sq(s)ds. fo t 'o g(yk(t)) fo

f

This yields yk(tk)

f Jo

('l/k ds t` f sq(s)ds + J g(s) < Jo ds

9(s),

Because tk - 0 as k -p oo, from the previous inequality we deduce [IykII',', yk(tk) -4 0 as k -* oo. This means that yk --+ 0 in C[0,1] as k -* oo, which contradicts (9.15). Therefore infk>k,, tk > 0, and similarly supk>ko tk > 1.

The influence of a nonlinear convection term in singular elliptic problems

214

Let C1, C2 > 0 be such that

0 < c1 < inf tk < Sup tk < c2 < 1. k>ko

k>ko

By virtue of (9.24) and (9.25) we deduce that Iyk(t)I

where

< z(t)

for all 0 < t < 1,

(9.26)

max{c2,t} q(s)ds

Z(t)

for all 0 < t < 1.

min{ct,t}

Let 4, be the mapping defined in (9.11). According to (9.26) we obtain

Il'(yk(t)) - 't(yk(s))I =

y,(T)

yk(t) dT Jy':(8) 9(T)

L

g(yk(T))dT

<

I

t

z(T)dT.

Now, z E L'(0,1) and the uniform continuity of fi-1 on [0, 4)(M)] implies that (Yk)k>ko is equicontinuous. This finishes the proof of our lemma.

0

The result in Lemma 9.3.3 enables us to apply the Arzela-Ascoli theorem. Therefore, there exists y E C[O,1] such that, up to a subsequence, we have Yk -> y

in C[O, 1] as k -} oo. From (9.21) and Lemma 9.3.2 we deduce y(O) = y(l) = 0 and mt(1 t) < y(t) < M for all 0 < t < 1. In particular, we have y > 0 in (0, 1). Fix t E (0, 1). Without losing the generality, we may assume that t # 1/2. Remark that yk satisfies the integral equation yk(x) = yk

2+ yk \2l (x - 2} +

/2(s - x)q(s)g(yk(s))ds,

(9.27)

for all 0 < x < 1. Taking, for instance, x = 2/3 in the previous relation and using (9.15) we deduce that the sequence (yk(1/2))k>ko is bounded. Let r E R be such that, up to a subsequence, yk(1/2) -k r as k - oo. Now, letting x = t in (9.27) and passing to the limit with k -* oo we have

y(t) = y

`2/

+ r(t - 2) +

jt(s - t)q(S)9(yk(s))ds.

q(t)g(y(t)) = 0 for all 0 < t < 1-that is, y E C2(0,1) n C[0,1] is a solution of (9.9). The proof 0 is now complete. Because t E (0, 1) was arbitrary, the last equality implies

Condition (9.10) was introduced by Taliaferro [186]. In our setting, this condition reads 1

tp(t)dt < oo.

(9.28)

We have seen so far that condition (9.28) is necessary to have a classical solution for (9.1). In one dimension, Theorem 9.3.1 shows that the previous

Existence results in the sublinear case

215

condition is also sufficient. In the next section we will argue that (9.28) suffices to have classical solutions in higher dimensions provided that the parameters A and it belong to a certain range.

9.4 Existence results in the sublinear case Our aim in this section is to supply existence results for problem (9.1) in case f is sublinear. Recall that this means that f satisfies the hypotheses

(fl) the mapping (0, oo) ? t - f (X't) is nonincreasing for all x E St;

(f2) do f (t' t) = oo and tliu f (x't) = 0 uniformly for x E St. We also assume that p : (0, oo) - (0, oo) is a Holder continuous function such that p is nonincreasing. Nevertheless, we prove that condition (9.28) suffices to guarantee the existence of a classical solution for A belonging to a -certain range. In this case the existence of a solution is strongly dependent on the exponent of the nonlinear gradient term IVuIQ. To understand this dependence better, we

assume µ = 1, but the same results hold for any it > 0 (note only that the bifurcation point A in the following results is dependent on u.).

Theorem 9.4.1 Assume that 0 < a < 1, p = 1, and conditions (f 1), (f 2), (gl), and (9.28) are fulfilled. Then problem (9.1) has at least one classical solution for

allAER. Proof CAsE A > 0. By Theorem 1.2.5 there exists a classical solution ( of the problem

-A( = f (x, ()

in Q,

>0 =0

in S2,

(9.29)

on &Q.

Using the regularity of f we have ( E C2(1). Because A > 0, it follows that (is a subsolution of (9.1). We now focus on finding a supersolution uk of (9.1) such

that (< ua in Q. According to Theorem 9.3.1, there exists H E C2(0, 1) fl C[0,1] such that

-H"(t) = p(t)g(H(t)) H'(t) > 0

0 < t < 1, 0 < t < 1,

(9.30)

H(0) = H(1) = 0.

Because H is concave, there exists H'(0+) E (0, oo]. We fix 0 < b < 1 such that H' > 0 in [0, b]. Then H is increasing on [0, b]. Multiplying by H' in (9.30) and integrating on [t, b), (0 < t < b) we find

216

The influence of a nonlinear convection term in singular elliptic problems

(H')2(t)

- (H')2(b) = 2 tf p(s)g(H(s))H'(s)ds b

H(b)

(9.31)

g(r)dr,

2p(t) Jx(t)

for all 0 < t < b. Using the monotonicity of g it follows that

(H')2(t) < 2H(b)p(t)g(H(t)) + (H')2(b)

for all 0 < t < b.

(9.32)

Hence, there exist C1, C2 > 0 such that

H'(t) < cip(t)g(H(t))

for all 0 < t < b

(9.33)

and

(H`)2(t) < c2p(t)g(H(t))

for all 0 < t < b.

(9.34)

The existence of a supersolution of (9.1) is obtained in the following result.

Lemma 9.4.2 There exist two positive constants M > 0 (which depends on A) and c > 0 such that v,), := MH(ccpl) is a supersolution of (9.1).

Proof Let us first consider c > 0 such that cWj < min{b, d(x)}

in ft

(9.35)

By the strong maximum principle we have &W1 /On < 0 on i9Q. Hence, there exist

w CC Q and 6 > 0 such that IVVI I > 6

in f2 \ w.

(9.36)

Moreover, because lim {c 2p(cpl)g(H(cco1))IV9pjI2 - 3f (x,H(cco1))} = oo,

d(z)\o

we can assume that c2p(ccPl)g(H(c'pl))IV lI2 > 3f (x, H(ccoi))

in St \ w.

(9.37)

Let M > 1 be such that Me 262 > 3.

(9.38)

Using the fact that H'(0+) > 0 and 0 < a < 1, we can choose M > 1 with M(cb) 2 cl

H'(ccPi) ? 3A(McH'(cW,)IvWiI)a

in 0 \w,

where cl is the constant from (9.33). By (9.33), (9.35), (9.36), and (9.38) we derive

Existence results in the sublinear case

217

Mc2p(cpi)g(H(cpi))IVcpi12 > 3A(McH'(ecpi)1Vcpi1)a

in fI \ w.

(9.39)

Because g is decreasing and H'(cpi) > 0 in -0, there exists M > 0 such that McAicpiH'(ccpi) > 3p(d(x))g(H(ccpi))

in w.

(9.40)

In the same manner, using (f2) and the fact that 0 in 0, we can choose M > 1 large enough such that in w

McA1w1H'(ccpi) > 3)(MH'(ccpi)1Vtpil)a

(9.41)

and

Mc)1 piH'(cwl) > 3f (x, MH(coi))

in w.

(9.42)

For M satisfying (9.38) through (9.42), we claim that

U,\(x) := MH(cpi(x))

for all x E 9

(9.43)

is a supersolution of (9.1). We have

-qua = Mc2p(cSoi),9(H(cpi))IV iI2 + McAicPiH'(cpi)

in Sl.

(9.44)

We first show that in SZ \ w there holds

f(x,ua).

Mc2p(ccoi)g(H(ccai))I©WiI2

(9.45)

Indeed, by (9.35), (9.36), and (9.38) we have

Mc2p(cWi)g(H(epi ))IVWi 12 > p(d(x))g(H(ccpi))

> (d(x))9(MH( -1

c

= p(d(x))g(ii)

in fl \ w.

Pi ))

(9.46)

The assumption (f 1) and (9.37) produce Mc2p(cpi)g(H(ccpi))l Vtpi12 > Mf (x, H(ccP1))

2 f (x, MH(ccpl)) = f(x,z9,)

(9.47)

in S2\w.

From (9.36) and (9.39) we obtain 3

c2p(coi)g(H(cpi))lVWi12> A(McH'(ccpi)lVcpil)a = AIVuala

in fl \ w.

Now, estimate (9.45) follows by (9.46), (9.47), and (9.48).

(9.48)

218

The influence of a nonlinear convection term in singular elliptic problems

Next we prove that f (x, U ,

McA1VjH'(c
in w.

(9.49)

From (9.40) and (9.41) we have 3

cAicoiH'(ccP1) ? p(d(x))g(H(c 1)) (9.50)

p(d(x))g(MH(crpl)) = p(d(x))g(ua) in w and

M

cAiW1H'(ccP1) ?A(McH'(ccP1)IDcP1j)a (9.51)

=aIDu,,!a

in w.

Finally, from (9.42) we derive 3

cA,(p,H'(c(p1) ? f(x,MH(wi)) = f(x,uA)

in w.

(9.52)

Now, relation (9.49) follows from (9.50), (9.51), and (9.52). Combining (9.44) with (9.45) and (9.49) we conclude that ua is a supersolution of (9.1). This ends the proof. Let us come back to the proof of Theorem 9.4.1. So far we have constructed a subsolution C and a supersolution zta such that

Dua + f (x, ua) < 0( + f (x, ()

U,\,( >0

ua=S=0

in S2,

inn, on 852.

Because A( E L1(S2) (note that ( E C2(S2)), by Theorem 1.3.17 we obtain < ua in Q. The conclusion in this case follows now by the sub- and supersolution

method for the ordered pair CASE A < 0, We fix v > 0 and let u E C2()) fl C(II) be a solution of (9.1) for A = v. Then u, is a supersolution of (9.1) for all A < 0. Set r n:=

inf

(p(d(x))g(t)

(x,t)ESZx (0,00)

+ f(x,t)).

(9.53)

Because g(0+) = oo and the mapping (0, oo) E) t 1--4 minxE? f (x, t) is positive and nondecreasing, we deduce that m is a positive. Consider the problem Av = m + AIDvI°

v=0

in S2,

on 911.

(9.54)

Existence results in the sublinear case

219

Clearly, zero is a subsolution of (9.54). Because A < 0, the solution w of the problem

-Aw=m W=0

in St,

on 99

is asupersolution of (9.54). Hence, (9.54) has at least one solution v E C2 (Q) n

C(fl). We claim that v > 0 in Q. Indeed, if not, we deduce that min.,, E-0 V is achieved at some point xo E Q. Then Vv(xo) = 0 and

-Av(xo) = m + AIVv(xo)[6 = m > 0,

which is a contradiction. Therefore, v > 0 in Q. It is easy to see that v is a subsolution of (9.1) and -Av < m < -Au in Sl, which yields v < u,, in 1 Again by the sub- and supersolution method we conclude that (9.1) has at least one classical solution ua E C2 (Q) n C(?!). This completes the proof.

In the case 1 < a < 2, the complete description is given in the following result.

Theorem 9.4.3 Assume that 1 < a < 2, u = 1, and conditions (f 1), (f 2), (gl), and (9.28) are fulfilled. Then there exists )1* > 0 such that (9.1) has at least one classical solution for all A < )1*, and no solutions exist if A > A*.

Proof We proceed in the same manner as in the proof of Theorem 9.4.1. The only difference is that (9.39) and (9.41) are no longer valid for any A > 0. The main difficulty when dealing with estimates like (9.39) is that H'(ccpi) may blow up at the boundary. However, combining the assumption 1 < a < 2 with (9.34), we can choose A > 0 small enough such that (9.39) and (9.41) hold. This implies that problem (9.1) has a classical solution provided A > 0 is sufficiently small. Set

A := {A > 0: problem (9.1) has at least one classical solution}.

From the previous arguments. A is nonempty. Let A* := sup A > 0. We first claim that if A E A, then (0, A) C A. To this aim, let Al E A and 0 < A2 < A1. If ua, is a solution of (9.1) with A = A1i then ual is a supersolution of (9.1) with A = A2, whereas c defined in (9.29) is a subsolution. Using Theorem 1.3.17 once more, we derive that (< ux> in 1 so that problem (9.1) has at least one classical solution for A = )12. This proves the claim. Because Al E A was arbitrary, we conclude that (0, A*) C_ A.

Next we prove that A* < oo. This claim will be achieved in a more general framework. We have the following proposition. Proposition 9.4.4 Let F : [0, oo) --+ [0, oo) be a C1 convex function such that

(Fl) (0)Ff
Fs

220

The influence of a nonlinear convection term in singular elliptic problems

Then there exists a positive number a' such that the problem

-Au =F(JDuJ)+a

in S, in 9,

u>0 v=0

(9.55)

on &Q

has no classical solutions for a > Q.

Proof Let v E C2(0) n C(Sl) be a solution of (9.55) and fix 0 E Cu-(St). Multiplying by 0 in (9.55) and then integrating over SZ we obtain v fn Odx

4F(IVu[))dx = 1. (vuVO - F(JVuJ) ) dx.

0 (ivui1

(9.56)

Consider now the convex conjugate F* of F defined as

F*(t) = sup{at - F(a)}

for all t E R.

aER

Now, from (9.56) we deduce

v J 4'dx < fn OF* (i.Y1) dx

for all 0 E Co (St).

(9.57)

holds

for any 0 E W1.OO(1l). Hence, if we By density, the previous estimate such that 0 > 0 in S2 and fa OF* ([04I /gi)dx < oo, then, construct 0 E by (9.57), it follows that or is finite. Without loss of generality, we may assume that St is the unit ball in RN. Let t

fi(t)-Jo F(s)+M

forallt>0.

Using the assumption (F2) we can choose M > 0 large enough such that t limt-,, ' (t) < 1. Then 4P : [0, oo) -> [0, e) is bijective and let r : [0, e) [0, co) be the inverse of 4. Define now 0 : [0, 1] - [0, oo) by 1

fi(t) =

F(r(t)) + M 10

0
We consider O(x) = t/i(Ixl), x E SZ. Then

/i'(t) = -- i(t)F'(r(t)),

I:=nOF*

e

l [/dx=WN

tN-1

(t)F*(F'(r(t)))dt.

Existence results in the sublinear case

221

On the other hand, a straightforward computation yields

F*(F'(r(t))) = r(t)F'(P(t)) - F(P(t))

for all t > 0.

Therefore, we have

I < WNNPN-1 f (- v(t)r(t) - V (t)F(r(t))) dt = WNeN-1

(_(t)r(t)r +

= WNeN-1M

f(t)(F'(t) fiF(P(t))) dt

rt 7,L (t)dt < oo.

J

0

This completes the proof.

Consider A E A and let ua be a classical solution of (9.1). Then

-Aua > .\IVuala + m

in Il,

where m > 0 is the infimum in (9.53). Because 1 < a < 2, it follows that va := Al/(a-1)u,\ verifies

-©v,\ > IVVAIa +

mAl/(a-1)

in Q.

Hence, vA is a supersolution of

-AV = IQvIa +

m)11/(a-1)

v=0

in R

on ac.

(9.58)

Because zero is a subsolution, it follows that' problem (9.58) has a classical solution that, in view of the maximum principle, is positive. According to Proposition 9.4.4 with F(t) = ta, we obtain mAlAa-1) < a--that is, A < This means that A* = inf A < < oo. Hence, )i* is finite. The existence of a solution in the case A _< 0 can be achieved exactly in the same way as in Theorem 9.4.1. This finishes the proof of Theorem 9.4.3. (Q/m)a-1

The case a = 2 is a special one, because by the change of variable (often called the Gelfand transform) v = eau - 1, we obtain a new singular problem without a gradient term. If f (x, u) depends on u, this change of variable does not preserve

either the sublinearity conditions (f 1) and (f2) on f , or the monotonicity of g. In turn, if f (x, u) does not depend on u, this approach can be successfully used. This may help us to understand better the dependence between A and it in problem (9.1). Let m := lim g(t) E [0, oo).

too

The influence of a nonlinear convection term in singular elliptic problems

222

Theorem 9.4.5 Assume that a = 2, A > 0, it > 0 and p m 1, f - 1. (i) Problem (9.1) has a solution if and only if A(m + µ) < \,. (ii) Assume it > 0 is fixed and let A* = Aj/(m+p). Then (9.1) has a unique solution ua for every 0 < A < A*, and the sequence (ua)o
(0) lima/A. ua = oo uniformly on compact subsets of Q. The situations described in Theorem 9.4.5 are depicted in the following bifurcation diagrams. Figure 9.1 corresponds to (i) and a = 0 (respectively a > 0), whereas Figure 9.2 is related to (ii), A > 0, and µ = fixed.

FIGURE 9.1. The bifurcation diagrams in Theorem 9.4.5 (i).

Proof If A = 0, the existence follows by Theorem 1.2.5. Next we assume that A > 0 and let us fix p > 0. With the change of variable v = eau - 1, problem (9.1) becomes

-Av =

in 52,

v>0

in Q,

v=0

on BSZ,

(9.59)

where

(t) = A(t + 1)g ( ln(t + 1) I + A (t + 1)

for all t > 0.

Obviously, 0a is not monotone, but we still have that the mapping (0, co) t 1--1 fa(t)/t is decreasing and

E)

Existence results in the sublinear case

223

U

(0,0)

A=

A

A,

m+N

FIGURE 9.2. The bifurcation diagram in Theorem 9.4.5 (ii). li

t

fi

t (t) = A(m + p) 00 m

and

li

o

fi(t) fi = 00,

for all A > 0. We first remark that fia satisfies the hypotheses of Theorem 1.2.5 provided that A(m + p) < A1. Hence, problem (9.59) has at least one solution. On the other hand, because g > m on (0, oo), we obtain

fia(t) > A(m + tc)(t + 1)

for all A, t e (0, oo).

(9.60)

This implies that (9.59) has no classical solutions if .(m+p) > al. Indeed, if ua,,, would be a solution of (9.59) with A(m+,u) > A1, then ua+, is a supersolution of the problem Au = A(u + 1) in c,

U=0

on all,

(9.61)

where A = A(m + µ). Clearly zero is a subsolution of (9.61), so there exists a classical solution u of (9.61) such that u < u,\, in Q. By the maximum principle and elliptic regularity, it follows that u is positive in 12 and u E C2(ci). To raise a contradiction, we multiply by c.p in (9.61) and then integrate over Q. We find

- I w1Oudx = AJ ucpldx + AJ coidx. J S2

in

S2

This implies Al ff uW,dx = A fn u A1. The proof of the first part in Theorem 9.4.5 is therefore complete. (ii) Follows exactly in the same manner as in Theorem 5.6.1. This concludes the O proof.

The influence of a nonlinear convection term in singular elliptic problems

224

In what follows we discuss the case a = 1. Note that the method used in Theorem 9.4.1 does not apply here for large values of A.

Assume that Il = BR for some R > 0, where BR denotes the open ball centered at the origin, with a radius R > 0. In this case, and with u = 1, problem (9.1) reads

-Au = p(R - Ixl)g(u) + AIVul + f (x, u) u>0

lxi < R, lxj < R,

u=0

(9.62)

jxl=R.

Theorem 9.4.6 Assume that 11 = BR for some R > 0, a = 1, p = 1, and conditions (f 1), (f2), and (gl) hold. Then problem (9.62) has at least one solution for all A E R.

Proof The proof in the case A < 0 is the same as in Theorem 9.4.1. In what follows we assume that A > 0. By virtue of Theorem 9.4.1, there exists u E C2(SZ) fl C(Sl) such that

-Au = p(R - lxj)9(u)

lxi < R,

u>0

ixl
u=0

jxj = R.

It is obvious that u is a subsolution of (9.62) for all A > 0. To provide a supersolution of (9.62) we consider the problem

-Au = p(R - lxl)g(u) + AIVuI + 1 u>0 U=0

lxi < R, lxi < R,

(9.63)

1xl = R.

We need the following auxiliary result.

Lemma 9.4.7 Problem (9.63) has at least one classical solution.

Proof We are looking for radially symmetric solutions u of (9.63)-that is,

u(x) = u(r)

for all 0 < r = jxj < R.

In this case, problem (9.63) reads

-u" - N - 1 u'(r) = p(R - r)g(u(r)) + Alu'(r)j + 1

r

u > 0

t

0 < r < R,

u(R) = 0.

The first equality in (9.64) implies

-(rN-'u (r))' > 0

0 < r < R,

for all 0 < r < R.

(9.64)

Existence results in the sublinear case

225

This yields u'(r) < 0 for all 0 < r < R. Again by (9.64) we find

- (U,, + N - 1 u(r) + Au'(r)) = p(R - r)g(u(r)) + 1 The previous relation may be written as -(earrN-lu

0 < r < R.

0 < r < R,

(r))' =

(9-65)

where z)(r,t) = p(R - r)g(t) + 1

(r, t) E 10, R) x (0, oo).

From (9.65) we have

u(r) = u(0) -

jr e-)tt-N+1

f t eAs8N-10(S, u(s))dsdt

0 < r < R.

(9.66)

0

On the other hand, in view of Theorem 9.4.1 and using the fact that g is decreasing, there exists a unique solution w e C2(BR) fl C(BR) of the problem

Aw = p(R - I a[)g(w) + 1

IxI < R,

w>0

IxI
W=0

IxI = R.

(9.67)

Clearly, w is a subsolution of (9.63). As a result of the uniqueness and the symmetry of the domain, w is radially symmetric, so w(x) = w(r) for all 0 < r = IxI < R. As noted earlier, we obtain w(r) = w(0) -

t-N+1

t 0

w(s))dsdt 0 < r < R. (9.68) Jo fo We claim that there exists a solution v E C2 [0, R) fl C[0, R] of (9.66) such that v > 0 in (0, R). Let A = w(0) and define the sequence (vk)k>o by vo = w and

vk(r) = A - j e-att-N+f eA$sN-1 /ls, vk-1(s))dsdt,

(9.69)

a

for all 0 < r < R and k > 1. Note that vk is decreasing in [0, R) for all k > 0. From (9.68) and (9.69) it is easy to see that v1 > vo in [0, R). A further induction argument yields vk > vk_ 1 in [0, R) for all k > 1. Hence,

w=vo
in BR.

Thus, for all 0 < r < R there exists v(r) := limk.o vk(r) and, clearly, v > 0 in [0, R). We now can pass to the limit in (9.69) to obtain t v(r) = A - or e-Att-N+1 r eASSN-1'p(s'v(s))dsdt, Jo

0

for all 0 < r < R. It follows that v E C2[0, R) fl C[0, R] is a solution of (9.66). This proves the claim.

226

The influence of a nonlinear convection term in singular elliptic problems

We have obtained a supersolution v of (9.63) such that v > w in BR. So, problem (9.63) has at least one solution and the proof of lemma is now complete.

Let u be a solution to problem (9.63). For M > 1 we have

-A(Mu) = Mp(R - IzI)g(u) + ) I V(Mu)I + M

p(R - IxI)g(Mu) + \IV(Mu)I + M in BR.

(9.70)

Because f is sublinear, we can choose M > 1 such that

M > f (x, MIIulI.)

in BR.

Then, ua := Mu satisfies -AU. x> p(R - IxI)g(ua) + AIVuaI + f(x,ua)

in BR.

It follows that ua is a supersolution of (9.62). Because g is decreasing, we easily deduce u < ua in BR. Hence, problem (9.1) has at least one solution. The proof of Theorem 9.4.6 is now complete. If p is bounded in the neighborhood of the origin, the previous procedure can be applied for general bounded domains. Let fl be a smooth bounded domain

and R > 0 be such that fl C BR. According to Theorem 9.4.6, there exists V E C2(BR) fl C(BR) such that

-Av = Mg(v) + ) I Vvl + f(x, v)

IxI < R,

v>0 v=0

IxI
IxI=R,

where M = max,,E-5 p(d(x)). Obviously v is a supersolution of (9.1) with a = 1, whereas w E C2(cl) fl C(f2) defined by

-Ow = p(d(x))g(w)

w>0 w=0

in fl, in ft, on 19Q

is a subsolution of (9.1). It is easy to see that to < v in ) and, by Theorem 1.2.3, we deduce that problem (9.1) has at least one classical solution.

9.5 Existence results in the linear case In this section we study problem (9.1) in which we drop the sublinearity assumptions (f 1) and (f2) on f , but we require, in turn, that f is linear. More precisely, we assume that f (x, t) = t for all (x, t) E ft x [0, co) and consider the problem

Existence results in the linear case

227

-Au = p(d(x))g(u) + AI Vula + pu

in 52,

u > 0

in 52,

u=0

on 852,

(9.71)

where A > 0, p > 0 and p, g verify (g1) and (9.28). We shall be concerned in this section with the case 0 < a < 1. Note that the existence result in Theorem 1.2.5 does not apply here because the mapping '(x,t) = p(d(x))g(t) + pt

(x, t) E SZ x (0,00)

is not defined on 852 x (0, oo).

Theorem 9.5.1 Assume that 0 < a < 1 and conditions (gl), (9.28) are fulfilled. Then for all A > 0, problem (9.71) has solutions if and only if p < Al. Proof Fix p E (0, Al) and A > 0. By Theorem 9.4.1 there exists u E C2(Q) n C(Q) such that in 9, -Au = p(d(x))g(u) +.1IVul'

u>0 u=0

in S2,

on8Sl.

Obviously, uap := u is a subsolution of (9.71). Because p < A1i there exists v E C2(S2) such that

1-Ov = pv + 2

in 92,

v>0 v=0

in 0,

(9.72)

on 852.

Using the fact that 0 < a < 1, we can choose M > 0 large enough such that (9.73) M > pljull,, and M > A(MIVvl)a in Q. From (9.72) and (9.73) we note that w := My satisfies -Aw > AIvwIa + p(u + w) in 9. We claim that uxu := u + w is a supersolution of (9.71). Indeed, we have

-Duaµ > p(d(x))g(u) + AjVuIa + ANVwia + piiaµ

in 52.

(9.74)

Using the assumption 0 < a < 1, we can easily deduce

t1 + t2 > (tl + t2)a Hence,

for all tl, t2 >0-

IVuia + IVwla > (IVul+ IVwl)a > IV( u + w)!a

in 52.

(9.75)

Combining (9.74) with (9.75) we obtain

in 9. is an ordered pair of sub- and supersolutions of (9.71), and

-Du), ,, > p(d(x))g(ua,) + AI VzAu is + puau Hence,

thus problem (9.71) has a classical solution uaµ provided A > 0 and 0 < p < A1.

228

The influence of a nonlinear convection term in singular elliptic problems

If it > A1, using the same method as in the proof of Theorem 9.4.5, we deduce that problem (9.71) has no classical solutions. This finishes the proof.

9.6 Boundary estimates of the solution In this section we are concerned with the asymptotic analysis of solutions of problem (9.1) in the special case p(t) = t-a, g(t) = t-p, where a, 6 > 0. Hence, we study the boundary behavior of classical solutions to

-Au = d(x)-au-0 + AIDula + f (x, u)

in 0,

u>0

in S2,

U=0

on 852,

(9.76)

where 0 < a < 2, A > 0, and f satisfies (f 1) and (f2). Recall that if fo tp(t)dt < oo and A belongs to a certain range, then Theorems 9.4.1 and 9.4.3 assert that (9.76) has at least one classical solution ua satisfying ua < MH(cpi) in fl, for some M, c > 0. Here, H is the solution of

-H"(t) = t-aH-a(t) H,H' > 0

for all 0 < t < b < 1, in (0, b],

(9.77)

H(0) = 0.

With the same idea as in the proof of Theorem 9.4.1, we can show that there exists m > 0 small enough such that v := mH(cpi) satisfies

-Av <

d(x)-av_ p

(9.78)

in 52.

Indeed, we have

-AV = m{c2-aIOcpjI2(pi aH-a(c(pi) + AictpiH'(cpi))

in Q.

Using the properties of cpi and (9.33), there exist two positive constants cl, c2 > 0

such that -AV < m(cL1V ,i!2 +

c2Wi)d(x)-aH-16(clpi)

in 0.

Clearly, (9.78) holds if we choose 0 < m < 1 small enough such that

m(ciIV ,12+c2Wl) <1

in 0.

Furthermore, from (9.78) we obtain that v is a subsolution of (9.76), for all A > 0. It is easy to see that v < ua in 52. Therefore the solution ua satisfies

mH(cpi) < u,\ < MH(cpi)

in fl.

(9.79)

Now, a careful analysis of (9.77) together with (9.79) provides boundary estimates for the solutions of problem (9.76), as stated in the following result.

Boundary estimates of the solution

229

Theorem 9.6.1 Assume that A > 0, 0 < a < 2 and conditions (f 1) and (f 2) hold.

(i) If a > 2, then problem (9.76) has no classical solutions. (ii) If a < 2, then there exists 0 < A* < oo (with A* = oo if 0 < a < 1) such that for all 0 < A < A*, problem (9.76) has at least one classical solution. Moreover, there exist 0 < b < 1 and C1, C2 > 0 such that uA satisfies the following:

(iil) If a +)3 > 1, then for all x E Il we have

C1d(x)(2-a)/(1+A) < U'\ (x) < C2d(x)(9.80) (ii2) If a + 13 = 1, then for all x E SZ with d(x) < 6 we have

Cid(x)(-Ind(x))1/(2-") < u, (x) <

C2d(x)(-ind(x))1/(2-a).

(9.81)

(ii3) If a +,6 < 1, then for all x c f we have Cid(x) < u .(x) < C2d(x).

(9.82)

Proof The existence and nonexistence parts follow directly from Theorems 9.2.1, 9.4.1, and 9.4.3. We next establish the boundary estimates (9.80) through (9.82).

(iii) Remark that 1/(1+0)

(1 + p0)2

(2 - a)(a +,8 - 1)) is a solution to (9.77) provided that a+,0 > 1. The conclusion in this case follows from (9.79).

(ii2) Note that in this case problem (9.77) becomes

-H"(t) = t-«H«-1(t) H>0

for all 0 < t < b < 1, in (0, b],

(9.83)

H(0) = 0. Because H is concave, it follows that

H(t) > tH'(t) for all 0 < t < b. Relations (9.83) and (9.84) yield (H,(t))«-1

-H"(t) < Hence,

for all 0 < t
(9.84)

The influence of a nonlinear convection term in singular elliptic problems

230

for all 0 < t < b.

-H"(t)(H'(t))1-a <

t

(9.85)

Integrating in (9.85) over [t, b] we obtain

(H' )2-a(t) - (H')2-"(b) < (2 - a) (in b - In t)

for all 0 < t < b.

Thus, there exist cl > 0 and 81 E (0, b) such that

H'(t) < cl(-

for all 0 < t < 81.

int)1/(2-«)

(9.86)

Fix t E (0, 61]. Integrating in (9.86) over [E, t], 0 < e < t, we have

H(t) - H(e) < clt(- In

t)1/(2-a)

+

2

cl

t

a J (- In

s)(a-1)/(2-a)ds.

(9.87)

E

Note that t (-Ins)(a-1)/(2-a)ds

< 00

s)(a-1)/(2-a)dS

Jp (- In

and

t\O

10

t(- In t) 1/(2-a)

= 0. (9.88)

Hence, taking e -* 0 in (9.87), there exist c2 > 0 and 62 E (0, 81) such that

H(t) <

c2t(-1nt)1/(2-a)

for all 0 < t < 82.

(9.89)

From (9.83) and (9.89) we then obtain

-H"(t)>c2 (-

lnt)(a-1)/(2-a)

t

for all0
Integrating over [t, 82] in the previous inequality we find

H'(t) > (2 -

a)c2-1 [(-

In t) 1/(2-a) - (- In

62)1/(2-a)j

for all 0 < t <62.

Therefore, there exist c3 > 0 and 83 E (0, 62) such that

H'(t) >

c3(-lnt)1/(2-a)

for all0
Using the same arguments as in (9.86) through (9.89) we obtain c4 > 0

and

84 E (0, 83) such that

H(t) >

CA-lnt)1/(2-a)

for all0
(9.90)

The conclusion of (ii) in Theorem 9.6.1 follows now from (9.89), (9.90), and (9.79).

The case of a negative singular potential

231

(ii3) Using the inequality (9.84) and the fact that H'(0+) > 0, there exists c > 0 such that

H(t)>ct

forall0
This yields

-H"(t) < c At-(c'+a)

for all 0 < t < b.

Because a + Q < 1, it follows that H'(0+) < oo-that is, H E C1 [0, b]. Thus, there exist c1iC2 > 0 such that

forall0
c1t
(9.91)

The conclusion in Theorem 9.6.1 (iii) follows directly from (9.91) and (9.79). o This completes the proof of Theorem 9.6.1.

9.7 The case of a negative singular potential This section is devoted to the study of (9.1) in case p is negative. More exactly, we shall be concerned with an equivalent form of (9.1)-namely, -Du + p(d(x))g(u) = A f (x, u) +,zIVuIQ u>0

u=0

in 52, in 52,

(9.92)

on 852,

where 0 < a < 2, A > 0, to E R, g satisfies (g1), and p : (0,oo) -> (0,oo) is a nonincreasing Holder continuous function possibly singular at the origin. In this framework, the existence of a classical solution for problem (9.92) depends on both singular terms p(d(x)) and unbounded nonlinearity g. More precisely, we show that a necessary condition to have a classical solution is 1p(t)g(t)dt < oo.

(9.93)

J

In case f is sublinear-that is, f fulfills the hypotheses (f 1) and (f2)-condition (9.93) is also sufficient for the existence of classical solutions provided that A and a belong to a certain range. Obviously, relation (9.93) implies the following Keller-Osserman-type condition around the origin that have been already encountered in Chapter 4: 1/2

s

(fo 45(t)dt 1 10,

where fi(t) = p(t)g(t), t > 0.

III

ds < oo,

(9.94)

232

1 'he influence of a nonlinear convection term in singular elliptic problems

A nonexistence result We establish here the following nonexistence result related to problem (9.92). 9.7.1

Theorem 9.7.1 Assume that f' p(t)g(t)dt = oo and let 4 : St x [0, oo) -> [0, oo) be a continuous function such thatP ;6 0. Then the problem

-Au + p(d(x))g(u) < 4)(x, u) u>0

u=0

in f2, in S2,

(9.95)

on on

has no classical solutions.

Proof Assume that problem (9.95) has a classical solution u and fix C > maxi D(x, u). Let v E C2 (cl) be the unique solution of -AV = C v>0 v=0

in Q,

in il,

(9.96)

on BSl.

Moreover, there exist cl, c2 > 0 such that

cld(x) < v < c2d(x)

for all x E Q.

(9.97)

By the weak maximum principle, it follows that u < v in 0. Next we consider the perturbed problem -Au + p(d(x) + e)g(u + c) = C u>0 U=0

in Q, in Sl,

(9.98)

on an.

Obviously, u and v are, respectively, sub- and supersolution of (9.98). By standard arguments, there exists uE E C2(?) a solution of (9.98) such that u < uE < v in Q. Integrating in (9.98) we obtain - in Au,dx + J p(d(x) + e)g(uE + e)dx = CjQj,

in

which yields

-1

u dv(x) + I p(d(x) + e)g(u + e)dx < M,

(9.99)

where M is a positive constant. Taking into account the fact that Sue/8n < 0 on 81Z, from (9.99) we find

fP(d(x)+e)9(u. + e)dx < M.

The case of a negative singular potential

233

Because g is decreasing and uE < v in S2, it follows that

j p(d(x) + e)g(v + -)dx < M, for any compact subset rv CC Q. Passing to the limit with e -> 0 in the previous estimate we obtain f p(d(x))g(v)dx < M, for all w CC Q. Therefore

II p(d(x))g(v)dx < M.

(9.100)

On the other hand, by (9.97) and fo p(t)g(t)ds = oo, it follows that M > f p(d(x))g(v)dx > r p(d(x))g(c2d(x))dx = oo, Jtz t

which is a contradiction. Hence, problem (9.95) has no classical solutions. This completes the proof.

Corollary 9.7.2 Assume that fo p(t)g(t)dt = oo. Then, for all ,a < 0, problem (9.92) has no classical solutions. 9.7.2

Existence result

Theorem 9.7.3 Assume that fa p(t)g(t)dt < oo and conditions (f 1), (f 2), and (gl) are satisfied.

(i) If ti = -1 and 0 < a < 2, then there exists A* > 0 such that problem (9.92) has at least one classical solution if A > A*, and no solutions exist if 0 < A < A*. (ii) If IL = 1 and 0 < a < 1, then there exists A* > 0 such that problem (9.92) has at least one classical solution for all A > A*, and no solutions exist if

0 0 the problem -DU = A f (x, U) in fl,

U>0 U=0

in 0,

(9.101)

on 49Q

has at least one classical solution U. Using the regularity of f it follows that Ua E C2(ii) and there exist CI, C2 > 0 depending on A such that

cid(x) < UA(x) < c2d(x)

in Q.

(9.102)

Fix A > 0 and observe that UA, is a supersolution of (9.92). The main point is to find a subsolution uA of (9.92) such that ua < U., in Q. For this purpose, let 41(t) = p(t)g(t), t > 0, and define

234

The influence of a nonlinear convection term in singular elliptic problems

W : [0, 00) -+ [0, 00),

4(t) =

ft (j8 (r)d'r f

1/2

ds.

j

.

As in the proof of Theorem 4.5.1, we obtain that ' is a bijective map. Let h : [0, oo) -+ [0, oo) be the inverse of T. Then h satisfies

in (0, oo), 2 f.h(t}

h'(t) h"(t} _ '(h(t))

in (0, oo), in (0, co),

(s)ds

(9.103)

h(0) = h'(0) = 0.

Thus, h E C2(0, 00) n C1 [0, 00).

The key result for this part of the proof is the following lemma.

Lemma 9.7.4 There exist two positive constants c > 0 and M > 0 such that ua := Mh(ccp1) is a subsolution of (9.92) provided A > 0 is large enough.

Proof Because h E C'[0, oo) and h(0) = 0, we can take c > 0 small enough such that h(ccpl) < d(x)

in Q.

(9.104)

By the strong maximum principle, there exist S > 0 and w CC fl such that

IV it>Sin 1\w. Let M = max{1, 2(cS)-2}.

(9.105)

Because

(Mch'(ccpl)IVwi[)a} = -oo,

lim I -p(d(x))9(h(cco1)) + d(x) \0

we can assume that in 0 \ w there holds -p(d(x))g(h(ccpl)) + McAlcplh'(ccpl) + (Mch'(ccpl)I owl I)a < 0.

(9.106)

We are now able to show that ua := Mh(ccpl) is a subsolution of (9.92), provided A > 0 is sufficiently large. Indeed, we have

-4ua =

-Mc2p(h(cp1))g(h(cp1))IVcp1I2 + Mc.lcp1h'(ccpl).

By the monotonicity of g and (9.104) we obtain

-oua +p(d(x))9(ua) + Iouala
+ (Mch'(ccpl)IVcpl J)a.

Next, by the definition of M and (9.106), for all x E St \ w we find

(9.107)

The case of a negative singular potential

235

-Du,, +p(d(x))g(ua) + Iouala < -p(d(x))g(h(cpi)) + McA1WI h'(coi )

(9. 108)

+ (Mch'(ccoi) I V j I)a

<0. On the other hand, from (9.107) and for all x E w we have

-Dua +p(d(x))g(u.) + IVUAla p(d(x))g(h(ap1)) +

(9. 109)

+ (Mch'(ccpi)[V(pi I)a.

Because cpl > 0 in w and f is positive on w x (0, oo), we may choose A > 0 large enough such that Af (x, Mh(cipj)) ? p(d(x))g(h(ccol )) + McAiVlh'(ccpz) + (Mch'(ctpi)IOWil )a for all x E Z1.

(9.110)

From (9.109) and (9.110) we deduce

-Aua +p(d(x))g(ua) + IVu,,Ia < Af (x, ua)

in w.

(9.111)

Now, relations (9.108) and (9.111) show that ua = Mh(api) is a subsolution of problem (9.92) provided that A > 0 satisfies (9.110). This finishes the proof of the lemma.

By virtue of Theorem 1.3.17, it follows that ua < Ua in 0 and, thus, from Theorem 1.2.3 we obtain a classical solution ua of (9.92) such that ua < ua < Ua in Q.

Step 2: Nonexistence for small A > 0. Let us remark that

liim(f (x, t) - p(d(x))g(t)) _ -oo t\O

uniformly for X E S2.

Hence, there exists to > 0 such that f(x,t) - p(d(x))g(t) < 0

for all (x,t) E S2 x (0,to).

(9.112)

On the other hand, assumption (f 1) yields f(x,t) - p(d(x))g(t)

t

<

-

f(x,t)

t

<

-

f(x,to)

to

(9.113)

'

for all (x, t) E S2 x [to, oo). Let m = maxxE?i f (x, to)/to. Combining (9.112) with (9.113) we find

f (x, t) - p(d(x))g(t) < mt

for all (x, t) E 11 x (0, oo).

(9.114)

Set Ao = min (1, Al/2m} . We claim that problem (9.92) has no classical solution for 0 < A < Ao. Indeed, assume by contradiction that uo is a classical solution of (9.92) with 0 < A < Ao. Then, according to (9.114), uo is a subsolution of

The influence of a nonlinear convection term in singular elliptic problems

236

-©u =

in ft,

21 u

u>0

in ft,

u=0

on M.

(9.115)

By Theorem 1.3.17 we have uo < UA in ft. Furthermore, from (9.102) we have cuo < o 1 in S2 for some positive constant 0 < c < 1. Note that cuo is still a subsolution of (9.115) whereas ipl is a supersolution of (9.115). Hence, problem (9.115) has a solution u E C2(ft). Multiplying by S°1 in (9.115) and then integrating over SZ we have

j

- J cp]4udx = nt

12 J A

uc1dx.

That is,

Al f uW,dx = -

in

Jo uA(ldx = Al2 J nuW,dx.

The previous equality yields fnucpldx = 0, but this is clearly a contradiction, because both u and W, are positive in Q. It follows that (9.92) has no classical solutions for 0 < A < Ao.

Step 3: Dependence on A > 0. Set

A:= {A > 0: problem (9.92) has at least one classical solution}. From the previous arguments we deduce that A is nonempty and A* := inf A is positive. We show that if A E A, then (A, oo) C A. To this aim, let Al E A and A2 > A1. If ua, is a solution of (9.92) with A = A1i then ua, is a subsolution of (9.92) with A = A2 whereas Ua2 defined by (9.101) for A = A2 is a supersolution. Furthermore, we have DUa2 + A2 f (x, Ua2) < 0 < Aua, + A2 f (x, uA,)

Ua2,u,\l>0 Ua2 = ua, = O

in Q,

in n, on Oft,

Ua2 E Ll (SZ).

Again by Theorem 1.3.17 we find ua, < Ua2 in Q. Therefore, problem (9.92) with A = A2 has at least one classical solution. Because A E A was arbitrary, we conclude that (A*, oo) C A. This completes the proof of (i). (ii) Step 1: Existence of a solution for large A. According to Lemma 9.7.4, there exists A* > 0 such that (9.92) has a subso-

lution ua for A > A* and p = -1. Then ua is also a subsolution in case p = 1,

The case of a negative singular potential

237

provided that A > A*. Let us now construct a supersolution. By Theorem 1.2.5, for all A > A* there exists v, E C2(St) a solution of

Av=Af(x,v)+1

in ),

v>0 v=0

ins),

onl3l.

Because 0 < a < 1, we can choose M = M(A) > 1 large enough such that

M > MaIVvaIa

in Q

Then, using (f 1) we obtain

-0(MvA) = AM! (x, v,\) + M > Af (x, Mva) + IV(Mva)I'

in Q.

Hence, uA := MVA E C2(S2) is a supersolution of (9.92) for all A > A*. On the other hand, because i 1i + X f (x, ua) _< 0 < Dua + A f (x, ua) in 11, by Theorem 1.3.17 we obtain uA < 9A, and finally problem (9.92) has at least one solution for all A > A*.

Step 2: Nonexistence for small A > 0. We first extend Theorem 1.3.17 in the following way.

Lemma 9.7.5 Let 0 < a < 1 and I : Sz x (0, oo)

]l8 be a Holder continuous

*(x, t)/t is decreasing for all function such that the mapping (0, oo) t x E 1. Assume that there exist v, w E C2 (I) fl C(f) such that 1

)

(a) Ow+41(x,w)+IVwla <0<-Av+T(x,v)+IVvla in Q; (b) v,w>Oin Sl andv<w on MI. Then v <w in f. Proof Assume by contradiction that the inequality v < w does not hold in S2 and let = v/w. Clearly, ( < 1 on 81 and

-V (w2VC) _ -wLv +viw

in Q.

Let xo E St denote a maximum point of . In particular,

V((xo) = 0,

-O((xo) > 0,

which yields

0 < (-wOv + vzw) (xo).

(9.116)

Because w(xo) < v(xo), from assumption (a), the property of ', and (9.116), it follows that 0 < (wIVvIa -vjVwIa)(xo). From Vr (xo) = 0 we also have w(xo)IVvI(xo) = v(xo)IVwI(xo).

238

The influence of a nonlinear convection term in singular elliptic problems

Using the previous two relations we find `a

0 < f (v J in - v] IOwla(xo) = va I wl-a - vl-a) I0wla(xo), LL W

which contradicts w(xo) < v(xo). This concludes the proof of our lemma. Assume by contradiction that there exists a sequence of solutions (un,),,>1 of problem (9.92) associated with a sequence of parameters 1 C (0, oo) such that A . -* 0 as n -+ oo. A simple computation shows that w(x) := A(R2 - 1x12) is positive and satisfies the inequality

Aw + f (x, w) + lvwla < 0

in S2,

where A, R > 0 are large constants. In particular, it follows from Lemma 9.7.5 that 0 < u, < in whenever A < 1. Let x,, E ) be a maximum point of um. Then Vu,,(x,,) = 0 and 0. Letting d,, = d(x ), M. = u.(xn), it follows from (9.92) that p(dn)g(Mn) < A2f (xn, Mn) : A,,, max f (x, w(x)) < CA, xEO

which yields a contradiction as n --+ oo. Hence, problem (9.92) has no classical solutions when a > 0 is small. The third step concerning the dependence on A follows in exactly the same way as in the case p = -1. This finishes the proof of Theorem 9.7.3.

9.8

Ground-state solutions of singular elliptic problems with gradient term

We continue here the study of singular elliptic problems in unbounded domains started in Section 4.7. We shall be concerned in the sequel with the corresponding problem to (9.1) in the case K = RN, N > 3. More precisely, we consider

-Au = p(x)(g(u) + f (u) + Ivu'la) u>0

I u(x) - 0

in RN,

in RN,

(9.117)

as lxl -+ oo,

where p : RN -+ (0, oo) is a Holder continuous function and 0 < a < 1. We also assume that f fulfills (f 1) and (f2), and g E C'(0, oo) satisfies g > 0, g' < 0 in (0,oo) and (gl). As we have argued in Section 4.7, solutions of such problems are called ground-state solutions, especially for the prescribed condition at infinity. For f =_ 0 and a = 0, we saw in Section 4.7 that a necessary condition have a solution is 00 I 1

tzp(t)dt < oo,

(9.118)

Ground-state solutions of singular elliptic problems with gradient term

239

where 1b(r) = minjxj=r p(x), r > r0. As in Section 4.7 we require that p satisfies 00

J

(9.119)

to(t)dt < oo, 1

where O(r) = maxjz1=r p(x), r > 0. In this framework we have the following theorem.

Theorem 9.8.1 Assume that (f 1), (f2), (gl), and (9.119) are fulfilled. Then, problem (9.117) has at least one solution.

Proof The approach is similar to that in Theorem 4.7.1. The construction of the monotone sequence with a pointwise limit that is the solution of (9.117) relies on Theorem 9.4.1. More precisely, let B,, := {x E RN : Ixl < n}. According to Theorem 9.4.1, for all n > 1 there exists un E C2(B,,,) fl C(B,,) such that

in B,,, in B,,, on 8B,,.

Dun = p(x)(9(un) + f(un) + Ivunla)

un>0 un=0 Set 1(t) := g(t) + f (t), t > 0. Then we have

Dun+1+p(x)(F(un+l)+IVun+lla) :5 0 < Dun+p(x)(W(un)+IVu,Ia)

un=0
in Bn*

on 49B,,.

With the same proof as in Lemma 9.7.5 (note that the only difference here is the presence of the positive potential p(x)), we find un < un+1 in B,,, for all n > 1. Extending un by zero outside of Bn, this means that

in RN. The main point is to find an upper bound for the sequence (u,,),,>1.

Lemma 9.8.2 There exists v E C2(RN) such that

-Ov > p(x)(g(v) + f (v) + Ivvla)

v>0

in RN,

inRN,

v(x)-+0

(9.120)

as 1XI -->00.

Proof Let C be defined by (4.56) and fix k > 2 such that kl-a > 2map ca(r). In view of Lemma 2.2.4 we can de(x)

:= k

J

xl

Then 6 satisfies

c(t)dt

for all x E RN.

(9.121)

The influence of a nonlinear convection term in singular elliptic problems

240

-A = kO(Ixl)

in RN,

>0

in-RN ,

C(x)-0

as IxI-400.

To proceed further, we implicitly define the mapping w : RN -* (0, oo) by w(x)

J

dt

_

g(t)+1 =fi(x)

forallxERN.

It is easy to see that w E C2(RN) and w(x) - 0 as IxI -+ oc. Moreover, we have IVwl =

1) = k<(Ixi)(g(w) + 1)

in RN

(9.122)

and

-Ow = -(g(w) + 1)At - g'(w)(g(w) + 1) Iv I2 kc(IxI)(g(w) + 1) O(Ixi)(g(w) + 1) + 2O(Ixl)(g(w) + 1)By (9.121) and (9.122) we deduce 2.O(IxI)(g(w) + 1) >- O(Ixl)ka(g(w) + 1)a(,a(Ixl) > p(x)Ivwla

in RN

Hence,

-ow > p(x)(g(w) + 1 + Ivwla) w>0

in RN, in RN, aslxl ---goo.

w(x)-}0

(9.123)

Using the assumption (fl) and the fact that w is bounded in RN, we can find M > 1 large enough such that M > f (Mw). in RN. Multiplying by M in (9.123) we deduce that v := Mw satisfies (9.120), and the proof of Lemma 9.8.2 is now complete.

Let us come back to the proof of Theorem 9.8.1. With the same arguments used earlier, we obtain u,, < v in B, for all n > 1. This implies

<...
in RN

Thus, there exists u(x) := llmn_ un(x), for all x E RN and u,i < u < v in RN. Because v(x) - 0 as IxI -* oo, we deduce that u(x) -> 0 as IxI -+ oo. A standard bootstrap argument implies that u,,, -+ u in C(RN) for some 0 < y < 1 and that u is a solution of problem (9.117). This ends the proof of Theorem 9.8.1.

Comments and historical notes

241

9.9 Comments and historical notes We have discussed in this chapter the influence of a gradient term in singular elliptic problems. As remarked by many authors (see, for instance, Serrin [181], Choquet-Bruhat and Leray [45], Kazdan and Warner (114]), the requirement that the nonlinearity I Vu[Q grows at most quadratically is natural to apply the maximum principle. Elliptic equations involving a gradient term appear in many fields. For instance, Bellman's dynamic programming principle arising in optimal stochastic control problems indicates that the value function (also called Bellman function) u, which minimizes the cost functional, is also a solution of the nonlinear elliptic equation 2 Du + Q Ivuv' + Au = f (x, u)

where Il C RN is a bounded domain, 0 < a < 2, A > 0 denotes the discount factor, and f is a smooth or singular nonlinearity. Special attention has been paid to the case when p > 0 in problem (9.1). In this sense, we saw that if the exponent a is inferior to 1, then the presence of the gradient term does not change in a significant way the results presented in previous chapters. More precisely, if the nonlinear gradient term has a sublinear growth, then it does not perturb the structure of the problems encountered so far in this book. A different situation occurs when 1 < a < 2-that is, the gradient term has a superlinear behavior. A key result in the study of problem (9.1) is played by Proposition 9.4.4, which is a work by Alaa and Pierre [3]. Another major role in our analysis was played by the potential p(d(x)). We emphasize the different levels at which the presence of the (possibly) singular

potential p affects the study of existence in this chapter. First, we saw that the existence of a classical solution to problem (9.1) is related to the growth of tp(t) near zero. In turn, the existence of a classical solution to problem (9.92), is closely related to the behavior near the origin of both singular terms p and g. The competition between all these regular or singular nonlinearities or potential terms, combined with the presence of the bifurcation parameters A and p, decides whether a solution exists. The study of problems (9.1) and (9.92) in smooth bounded domains Sl C IRN, N > 2, is a result of the work by Dupaigne, Ghergu, and Radulescu [72] (see also [88], [89]). Problem (9.117) was considered in [90]. We emphasized the importance

of the growth condition (9.28) supplied first by Taliaferro [186] (see also [2]) for similar one-dimensional problems. The nonexistence result in Theorem 9.2.1 generalizes a result by Zhang and Cheng [202].

10

SINGULAR GIER.ER-MEINHARDT SYSTEMS To myself I am only a child playing on the beach, while vast oceans of truth lie undiscovered before me. Sir Isaac Newton (1692-1727)

10.1

Introduction

In this chapter we extend the study of single singular elliptic equations to the following singular elliptic system:

-Du + au =

v + 9( u)

p (x ) ,

u>

f_v+flv=1v>0 k(v) u=0, v=0

0

in

S2 ,

in S2,

(S)

on On.

Here, Sl C RN, (N > 1) is a smooth bounded domain, a,,3 > 0, p E (0 < y < 1), p > 0, p 0, and f, g, h, k E C°'" [0, oo) are nonnegative and nondecreasing functions such that g(0) = k,(0) = 0. This last assumption on g and k, together with the Dirichlet boundary conditions on aci, makes the system singular at the boundary. In the particular case of pure powers in nonlinearities, system (S) reads p

v

-AV + ,3v =

U'.

8v , v > 0

u = 0, v = 0 {_u+au=!L+p(x),u>o

in S2, in S2,

(10.1)

on asi.

The associated parabolic system to (10.1) subject to Neumann boundary conditions is known as the Gierer-Meinhardt system and it occurs in morphogenesis and cellular differentiation. These problems arise in the study of biological pattern formation by auto- and cross-catalysis, and are related to known biochemical processes and cellular properties. The unknowns u and v represent the concentrations of two morphogens called activator and inhibitor, respectively.

Singular Cierer-Meinhanit systems

244

The main difficulties in solving system (S) are the result of the presence of singular nonlinearities on the one hand and the noncooperative (that is, nonquasimonotone) character of our system on the other. We are mainly interested in the case when the activator and inhibitor have different source terms-that is, the mappings t H f (t)/h(t) and t 1--+ g(t)/k(t) are not constant on (0, oo).

10.2 A nonexistence result In this section we present a nonexistence result for classical solutions to (S). The main idea is to speculate the asymptotic behavior of v from the second equation of (S). This will be then used in the first equation of the system and, by a similar argument to that in Theorem 9.2.1, we obtain the desired nonexistence result. Special attention will be paid to the case of pure powers in nonlinearities. In this sense we obtain some relations between the exponents p, q, r, and s for which the system (10.1) has no classical solutions. Several times in this chapter we shall apply the following comparison result, which is a direct consequence of the weak maximum principle stated in Theorem 1.3.2.

Lemma 10.2.1 Let k E C(0, oo) be a positive nondecreasing function and let a1, a2 E C(S2) be such that 0 < a2 < al in SZ. Assume there exist # > 0 and v1, v2 E C2(cl) n C(St) such that

(i) Avl - 3v1 + a' (x) <- 0 < AV2 - Qv2 + a2(vx) in SL; (ii) v1i v2 > 0 in 0, v1 > v2 on sf1. Then v,>v2 in Q. Let ' : [0,1) -* [0, oo) defined by t

'P(t) =

`

1

(21

1/k(8)dO)/ ]

1/ 2

dr,

0
(10.12)

Set a,:= limt/, 4)(t) and let xP : [0, a) --+ [0, 1) be the inverse of C The main result of this section is the following nonexistence property.

Theorem 10.2.2 Assume that tf (mt) dt = 00, g(M'I'(t)) for all 0 < m < 1 < M. Then system (S) has no classical solutions. I°

(10.3)

Proof Assume by contradiction that there exists a classical solution (u, v) of system (S) and let W1 be the normalized first eigenfunction of (-A) in Ho (fl). Also let S be the unique solution of the problem

Q,(+ a(= p(x) (=0

in S2, on 852.

(10.4)

A nonexistence result

245

By standard elliptic arguments and the strong maximum principle, we deduce that S E C2(St) and C > 0 in SZ. Furthermore, taking into account the regularity of the domain, there exist c1, c2 > 0 such that cid(x) < cpl, c < c2d(x)

(10.5)

in 51,

where d(x) = dist(x, 851). Because

A(u-()+a(u-()>0

in f2,

u-S=0

on 852,

by the weak maximum principle (see Theorem 1.3.2) we have is > c in I. Hence, by (10.5), it follows that u(x) > md(x) in 51, (10.6)

for some m > 0 small enough. Set C := max. h(u(x)) > 0. Then v satisfies

-Av +)3v < k(

in n,

v)

v>0 v=0

(10.7)

in Q,

on on.

Let c > 0 be such that cW1 < min{a, d(x)}

in 9.

(10.8)

We need the following auxiliary result.

Lemma 10.2.3 There exists M > 1 large enough such that 1 := MT (ccpl ) satisfies

-Ov +,V ? kC

in Q.

(10.9)

Proof Because '4 is the inverse of 1D defined in (10.2), we have t'(0) = 0 and 4+ E C'(0, a) with

IF'(t) =' 2

1

JJ(

d7

for all 0
(10.10)

t ) k(T)

This yields

-"(t)= k('I(t))

forall0
'(t),P(t) > 0

for all 0 < t < a,

1

( 10 . 11)

'F(0) = 0.

By the strong maximum principle, there exist w cc 9 and b > 0 such that

IV iI>S in S1\w

and cpl>S

in w.

(10.12)

Singular Gierer-Meinhardt systems

246

Fix M > 1 large enough such that M(cb)2 > C

and McA18W'(cIIw1II... ) >

minzE ((cal))

(10.13)

We have -ATV =

k(W j)) I°cpil2 +

in 52.

By (10.12) and (10.13) we obtain C -AU > McAi
-ov >

in w,

2

°cP1I2

k(

Cpl))

?

in Q \w.

v)

The last two inequalities imply that v satisfies

-Ov > k(v)

in n.

Hence, v fulfills (10.9). This ends the proof of the lemma.

U

By virtue of Lemma 10.2.1, from (10.7) and (10.9) we find v <'F in Q. Using (10.6) we then obtain

f (md(x))

f (u} >

g(v)

g(M`1'(c(Pi))

in 52.

Furthermore, u satisfies

-Du + au > f (md(x))

in 52,

9(M'1'(ccoi))

u>0 u=0

in 9, on an.

(10.14)

To avoid the singularities in (10.14) near the boundary, we consider the approximated problem

f (md(x)) - A w + aw - g(Ml'(ca1)) +

w=0

in 52 , (10.15)

on an.

Clearly, U7:= u is a supersolution of (10.15) whereas w = 0 is a subsolution. By the sub- and supersolution method and the strong maximum principle, problem (10.15) has a unique solution wE E C2(52) such that 0 < wf < u in Q. To raise

A nonexistence result

247

a contradiction, we multiply by cci in (10.15) and then we integrate over Q. We obtain f (md(x)) dx. (a + A,) wEcPldx = W1 ftt in g(M'1'(ccci))+e Because wE < u in 1, we find (

f and x a+ , ) fo u tdx > f (Pi g(M'(co ))+ 6 dx cP

for all w CC Q.

Let C := (a + )q) fn ucojdx. Passing to the limit with E -+ 0 in the previous inequality we obtain f (md(x))

J

dx < C < oo

for all w CC Q.

Hence,

f Vi g(MIF(c i)) dx < C < 00. Now let go :_ {x E Il : d(x) < a}. The previous estimate combined with (10.8) produces p

j

f (md(x))

dx < oo,

d(x).9(M`F(d(x)))

but this clearly contradicts (10.3). Hence, system (S) has no positive classical solutions. This finishes the proof.

If k(t) = ts, s > 0, condition (10.3) can be written more explicitly by describing the asymptotic behavior of T. We have the following corollary.

Corollary 10.2.4 Assume that k(t) = t8, s > 0, and for all 0 < m < 1 < M one of the following conditions hold:

(i) s > 1 and fo tf (mt)/g(Mt2/(1+s))dt = oo. (ii) s = 1 and fo '°{a,l/2} t f (mt)/g(Mt - In t)dt = oo. (iii) 0 < s < 1 and fo t f (mt)/g(Mt)dt = oo. Then, system (S) has no positive classical solutions.

Proof The main idea is to describe the asymptotic behavior of ' near the origin. Notice that in our setting the mapping [ : (0, a) --- [0, 1) satisfies

-W"(t) = W-s(t) 'F'(t), '(t) > 0 I

for all 0 < t < a, for all 0 < t < a,

(10.16)

'2 (0) = 0.

The asymptotic behavior of T was studied in a more general framework concerning problem (9.77) in Chapter 9. Using the arguments in the proof of Theorem 9,6.1, there exist cl, c2 > 0 such that

Singular Gierer-Meinhardt systems

248

c1t2/(1+s)

< qi(t) < c2t2/(1+s)

cit - In t < 9(t)

c2t - In t

clt<9(t)
ifs > 1, 0 < t < a, ifs 1, 0 < t < min{1/2, a}, if 0 < s < 1, 0 < t < a.

This completes the proof.

In case of pure powers in the nonlinearities, we have the following nonexistence result for system (10.1).

Corollary 10.2.5 Let p, q, r, s > 0 be such that one of the following conditions hold:

(i) s > 1 and 2q > (s + 1)(p + 2). (ii) s = 1 andq > p + 2. (iii) 0 < s < 1 andq > p + 2. Then, system (10.1) has no positive classical solutions.

Proof The proofs of (i) and (iii) are simple exercices of calculus. For (ii), by Corollary 10.2.4 we have that (10.1) has no classical solutions provided s = 1 and 1/2

fo

tl+p-q(-lnt)-q/2dt = oo.

(10.17)

On the other hand, for a, b E R we have fo /2 to (- In t)bdt < oo if and only if

a > -1 or a = -1 and b < -1. Now condition (10.17) reads q > p + 2. This concludes the proof.

10.3 Existence results In this section we provide existence results for classical solutions to (S) under the additional hypothesis /3 < a. The existence is obtained without assuming any growth condition on p near the boundary, because we are able to establish general bounds for the regularized system associated with (S). In particular, we

obtain that problem (10.1) has solutions provided that r - p = s - q > 0 and

q>p-1.

For all t1, t2 > 0 define

A(t1it2) := f(t1) h(tl) In this section we suppose that A fulfills (A1)

- g(t2) k(t2)

A(t1,t2) < 0 for all t1 > t2 > 0.

We also assume that k E C1(0, oo) is a nonnegative and nondecreasing function such that '") = oo for all c > 0, where K(t) = f0t k(T)dT. (A2) limt-,,. h(t+c)

Here are some examples of nonlinearities that fulfill (Al) and (A2):

Existence results

249

(i) f (t) = tp; g(t) = tQ; h(t) = t1 ; k(t) = t3; t > 0; p, q, r, s > 0; r - p =

s-q>0;andp-q<1.

(ii) f (t) = ln(1 + t5); g(t) = ez' - 1; h(t) = tp; and k(t) = 0, t > 0, p, q > 0,

p-q<1.

(iii) f (t) = log(1 + at); g(t) = log(1 + t); h(t) = at; and k(t) = t; t > 0, a > 1. In the following we provide a general method to construct nonlinearities f, g, h, k that verify hypotheses (Al) and (A2). Let f, g, h, k : 10, oo) -4 [0, oo) be nondecreasing functions such that k and h verify (A2) and one of the following assumptions hold:

(a) f k = gh and the mapping (0, oo) a t H f (t)/h(t) is nonincreasing. (b) There exists m > 0 such that f (t)/h(t) < m < g(t)/k(t), for all t > 0. Then mapping A verifies (Al). For instance, the mappings in example (i) satisfy condition (a) whereas the mappings in (ii) verify condition (b). The first result of this section concerns the existence of classical solutions for the general system (S).

Theorem 10.3.1 Assume that the hypotheses (Al) and (A2) are fulfilled. Then system (S) has at least one classical solution. The existence of a solution to (S) is obtained by considering the regularized system

-Au + au = g(v + e) + p(x) -AV + /3v =

in 0,

h(u + E) k(v + e)

(S)E

in 0,

u = 0, v = 0

on an.

First, we need to provide some a priori estimates for a classical solution of (S)E. We start with the following auxiliary result.

Lemma 10.3.2 There exists M > 0 that is independent of e such that any solution (uE, vE) of system (S)E with uf, vE E C2(1l) n C(St) satisfies

M.

(10.18)

Proof Let wE := uE - v£ and w := {x E 1 : wE > 0}. To establish (10.18), it suffices to provide a uniform upper bound for v, and WE. Subtracting the two equations in (S)E we have -AwE + awE _ (/3 - a)wE + f (uE +

e)

g(vE + e)

a)vE +

h(uE + e)

(u,,

+ e)

h(uE +,F)

gk(vE +,F)

+ p(X)

A(uE + r,vE + e) + p(x)

Notice that A(uE + e, vE + e) > 0 in w and wE = 0 on 8w. This yields

in Q.

Singular Gierer-Meinhardt systems

250

-Owc + aw, < p(x)

in w.

Let c E C2(Q) be the unique solution of (10.4). Then,

A(w,-()+a(w., ->0 we - < 0

in w,

on 8w.

in w. Because w, < 0 in

Furthermore, by the weak maximum principle, wE 9 \ w, it follows that We < C

in Q.

(10.19)

We next multiply by k(v,) in the second equation of (S)E and obtain -k(vE)Av, +,3v,k(vE) = k(v(v+)$) h(uf + a)

in ft

(10.20)

in ft

(10.21)

On the other hand, -AK(ve) + k'(vf)IVV'12 Because k is nondecreasing, we also have

K(v£) =

J

vE

k(t)dt < vEk(v,,)

in St.

(10.22)

Using (10.21) and (10.22) in (10.20), we deduce -AK(v.) + k'(ve) J Vv. I2 + /3K(vE) < k(v(+)e) h(uE + E)

in ft

Hence,

-AK(vE) + QK(vE) < h(uf +,-)

in 0.

(10.23)

By Theorem 1.3.3, there exists a positive constant C > 1 that depends only on n such that sup K(ve) < C sup h(u, + e) < C sup h(v, + 11511. + 1). sa

5

N

Using assumption (A2), we deduce that (v,)E>o is uniformly bounded-that is, 11v, 11,, < m for some m > 0 independent of E. In view of (10.19), this yields uE = vE + wE < m + I]( I],,.

in Q,

and the proof of Lemma 10.3.2 is now complete.

Lemma 10.3.3 For all 0 < C2(a) of system (S)E.

< 1 there exists a solution (uE, v,) E C2(?) X

Existence results

251

Proof We use topological degree arguments. Consider the set

U:=

{(v)EC2()xC2():

Hull.,Ilvll-<M+11 u,v > 0 in fl,ulast= vlast= 0

where M > 0 is the constant in (10.18). Define 4it(u,V) = ((p t (u, v), (1>t (u, v)),

U _4 U, by

4)'(U, v) = u

- t(-0 +a)-'

9(v + E)

+P

'1(u,v)=v-t(-d+p)-L fh(u+E) l k(v + e) Using Lemma 10.3.2, we have bt(u, v) # (0,0) on BU, for all 0 < t < 1. Therefore, by the invariance at homotopy of the topological degree we have deg (4?1, U.

(0, 0)) =deg (%, U, (0, 0)) = 1.

Hence, there exists (u, v) E U such that 1 i (u, v) = (0, 0). This means that system (S)E has at least one classical solution. The proof is now complete. Let us come back to the proof of Theorem 10.3.1. Let (u6, v6) E C2(SI) x C2(SI) be a solution of (S)E. Then,

d(uE - ) - a(u6 - 0 < 0

uE-(=0

in Q, on BSI,

where ( is the unique solution of (10.4). Hence t; < uE in 0. By (10.19) it follows

that Let

w6<{
in n.

(10.24)

E C'(-a) be the unique positive solution of the boundary value problem

At; +)3 =

h(S) k( + 1)

=0 In view of Lemma 10.2.1 we have following estimates hold:

in 12, (10.25)

on BSI.

vE in Q, so that, by Lemma 10.3.2, the

fi(x) < u6(x) < M in Q, fi(x)

v5(x) < M in 9.

(10.26)

Now, standard Holder and Schauder estimates imply that {(u6,VE)}o<E<1 converges (up to a subsequence) in C1 ,(SI) x C, ,,,,(Q) to (u, v). Passing to the limit in (10.26) we find

Singular Gierer-Meinhardt systems

252

((x) < u(x) < M in 0,

(10.27) fi(x) < v(x) < M in Q. It remains only to obtain an upper bound near OD for (ui, vs), which will lead us to the continuity up to the boundary of the solution (u, v). This will be done by combining standard arguments with the estimate (10.24). First, by

(10.23) we have

AK(vE) + h(M + 1) > 0 in Q. (10.28) Fix xo E asp. Because 80 is smooth (more precisely, f satisfies the exterior sphere condition), there exist y E RN \ Q and a ball B centered at y, having radius R > 0 such that f fl B = aIZ fl B = {xo}. Let 5(x) Ix - yj - R and ]o :_ {x E 9 : 4(N - 1)5(x) < R}. Consider zJi E C2(0, oo) such that ' > 0 and Eli" < 0 on (0, oo), and set iJi(5(x)), x E Do. Then

AO(x) _

'/1(5(x))M5(x)

+ 0"(5(x))IV5(x) I2

y10'(E(x)) + O"(5(x))

N < NR O'(5(x)) + 0"(6(x))

in Do.

Let us now choose O(t) = Cvl-t, t > 0, where C > 0. Therefore, LO(x) < 4 5-3/2(x) [2(N -R1)5(x)

- i]

< _C6-312(X) 5-3/2(x) < 0

in Do.

We choose C > 0 large enough such that

Qt (x) < -h(M + 1)

in Do

(10.29)

and

i(x) > K(M) > supK(vE)

on aQo \ asp.

(10.30)

50

Furthermore, by relations (10.28), (10.29), and (10.30), we obtain

0(q(x) - K(vE)) > 0 O(x) - K(vE) > 0

in Do,

on Mo.

This implies O(x) > K(ve) in po-that is, 0 < vE(x) < K-1(O(x))

in Do.

Passing to the limit with s --40 in the last inequality, we have 0 < v < K-1(O(x) ) in Do. Hence,

0 < lim v(x) < lim K(qS(x)) = 0. X-XO

X-.Xo

Because xo E ffl was arbitrarily chosen, it follows that v e C(1i). Using the fact that ue = wE + vE < c + vE in 1, in the same manner we conclude u c C(U). This finishes the proof of Theorem 10.3.1. 0

Existence results

253

The next result concerns the following singular system:

-Du + an = -AV +)9V =

uvqP

+ p(x)

uP+C

in 1Z,

in D'

v4+°

u=v=0

(10.31)

on 090,

where u > 0 is a nonnegative real number.

Theorem 10.3.4 Assume that p, q > 0 satisfy p - q < 1. Then the following properties are valid: (i) System (10.31) has solutions for all v > 0. (ii) For any solution (u, v) of (10.31), there exist c1, c2 > 0 such that in Q.

cid(x) < U, V < c2d(x)

(10.32)

Moreover, the following properties hold:

(iil) If -1 < p - q < 0, then u, v E C2(Q) fl C1,1+P-Q(S2). (ii2) If0 (> ZWj in 12. From the second equation in (10.31) we deduce

-

ipn+cr

-Av +'3v > iF+° vQ+Q

in 52.

Because p - q < 1, we deduce that vv := c satisfies p+a

-4v-+Oii> cP+v`pl

in SL ,

v4

for some c > 0 small enough. Therefore, by virtue of Lemma 10.2.1, we obtain

v>c(pl in Q. Let us now prove the second inequality in (10.32). To this aim, set w := u-v. With the same idea as in the proof of Lemma 10.3.2, we find

-Ow + aw < p(x)

in the set {x E St : w(x) > 0},

which yields

w<<
in Q.

(10.33)

Let w+ := max{w, 0}. Then v satisfies

4v+J3v<

(w+ + v)P+or

v=0

vq+o

on 312.

Singular Gierer-Meinhardt systems

254

Consider now the problem

-Az + Oz = z>0

in cl, in cl,

2P+azP-q

z=0

(10.34)

on 00.

The existence of a classical solution to (10.34) follows from Theorem 1.2.5. More-

over, if 0 < p - q < 1, then Z r C2(Sl), and if -1 < p - q < 0 by Theorem 4.3.2, we have z E C2(cl) n Cl,l+p-q(1l) and Az E L'(cl). Furthermore, z < mcpl in fl for some m > 0. On the other hand, cj is a subsolution of (10.34) provided that i > 0 is small enough. Therefore, by Lemma 1.3.17 we derive z > c`cpl in Q. This last inequality, together with (10.33), allows us to choose M > I large enough such that Mz > w+ in Q. Hence,

-0(Mz) +,Q(Mz) = M(-Oz +,3z) = 2P+°MzP-q >

2P+°(Mz)P-q

Mz)P+o

(w

in Q.

+ This means that -,U:= Mz verifies

-0v + /3v > (+ v)

P+o

in fl and

v = 0 on t3cl.

Remark now that w+

P+o (x,t) E Sl x (0,00)

tq+a

satisfies the hypotheses in Lemma 1.3.17, because p - q < 1. Furthermore, we have

Ov + T(x, v) < 0 < Av +'I'(x, v) in Q, v, v > 0 in g, v = v = 0 on On, and iv E L' (cl). Hence, by Lemma 1.3.17, we obtain

v
in fl.

(10.35)

Combining (10.33) and (10.35) we derive u = w +v < Ccpi in ct, for some C > 0. This completes the proof of (ii). As a consequence, there exists M > 1 such that up+o

< Mcpp-q 0 < up vq , v4+a

in Q.

If 0 < p - q < 1, then by classical regularity arguments we have u, v E C2(? ). If -1 < p - q < 0, then the same method as in the proof of Theorem 4.3.2 can be used to obtain u, v E C2(cl) fl C',1+P-q(cl). This finishes the proof of Theorem 10.3.4.

Uniqueness of the solution in one dimension

255

10.4 Uniqueness of the solution in one dimension The uniqueness of the solution is a delicate matter. Actually, this issue seems to be a particular feature of the Dirichlet boundary conditions because in the case of Neumann boundary conditions, the Gierer-Meinhardt system does not have a unique solution. In this section we are concerned with the uniqueness of the solution associated with the one-dimensional system-

u" - au + V" -'6V +

UP

e

+ p(v) = 0

vq+a

= 0

in (0,1), in (0,1),

(10.36)

u(0) = u(1) = 0, v(0) = v(1) = 0.

Our approach originates in the methods developed by Choi and McKenna [43], where a C2 regularity of the solution up to the boundary is needed. So, we restrict

our attention to the case 0 < q < p < 1. Thus, by virtue of Theorem 10.3.4. any solution of (10.36) belongs to C2[0,1] X C2[0,1]. By the strong maximum

principle we also have u'(0) > 0, v'(0) > 0, u'(1) < 0, and v'(1) < 0 for any solution (u, v) of system (10.36). The main result of this section is the following.

Theorem 10.4.1 Assume that 0 < q < p < 1, a > 0. Then system (10.36) has a unique solution (u, v) E C2(Q) X C2(S2).

Proof The existence part follows from Theorem 10.3.4. We prove here only the uniqueness. Suppose that there exist two distinct solutions (u1, vi), (u2, v2) E C2[0,1] X CZ[0,1] of system (10.36). First we claim that we cannot have u2 > u1 or v2 > v1 in [0,1]. Indeed, let us assume that u2 > u1 in (0,1]. Then, V2'.'

-

uply 16V2 +

q+Q

u+Q P+°

= 0 = of - five +L,

v2

v1

in (0,1)

and, by Lemma 10.2.1, we further obtain v2 > v1 in 10, 1]. On the other hand, uP

uP

v2

v2

ui - au1 + e + p(x) < 0 = u2 - au2 + 4 } p(x)

in (0, 1).

(10.37)

Because p < 1, the mapping

1'(x, t) = -at +

tv

+ p(x) V2(x)q

(

- T '

( 0

,

satisfies the hypotheses in Lemma 1.3.17. Hence, u2 < u1 in [0, 1]-that is ul U2. This also implies v1 _= v2i which is a contradiction. Replacing ul with u2 and

Singular Gierer-Meinhardt systems

256

v1 with v2, we also deduce that the situation ul > u2 or vl > v2 in (0,1) is not possible.

Set U := U2 - u1 and V := v2 - vl. From the previous arguments, both U and V change sign in (0, 1). Moreover, we have the following proposition.

Proposition 10.4.2 The functions U and V vanish only finitely many times in the interval [0, 1].

Proof We write system (10.36) as

5 W"(x) + A(x)W(x) = 0 W(0) = W(1) = o,

in (0,1),

{10.38}

where W = (U,V) and A(x) _ (Atj(x))1
1

-

u 2 {x)

v42 (x)

A 11(x) =LY +

up

P

-1 (

u2(x),

ul(x) =

I

lx) -

ul(xy2W

,

x)

4

vl(x)

vq(x)vI(x)

u2 (x)

u 1 (x)

u l (x)

V, (-T)

v1(x)7 v2(x),

V2 (X) - VI (X)

A12(x) =

-q {W

vi(x) = v2(x),

vp+a

u2 1

(x) -

u2(x)

v2+a(x)

ulp+a

A21(x) =

up+a(x)

_ -Q -

ul(x) 54 u2(x),

,

+a-1 (p + a)uvl+v((x)

A22 ( x )

(x)

- u1(x)

u1(x) = U2(4

,

v2+a(x)

-

yr °'(x)

v1(x)

vi+l(x)v2+a(x)

v2(x) - vl(x)

(q + CO

ui Q (x) + q+a+1(x)

v2(x),

'

1

vi(x) = v2(x)-

Therefore, A E C(0, 1) and A12(x) # 0A21(x) 0 for all x E (0, 1). Moreover, xA(x), (1 - x)A(x) are bounded in L-(0,1). Indeed, let us first notice that by (10.32) in Theorem 10.3.4, there exist cl, c2 > 0 such that 0,

uz Vi cl < < c2i (i = 1, 2) in (0,1). - min{x,1 - x} min{x,1 - x}

Then, by the mean value theorem we have xIA12(x)I

ql(xq) max{vi-1(x), v2-1(x)}

y

4+1

p

max

I ( vl(x))

for all 0<x<1f2.

' \v2(x)

F1

Uniqueness of the solution in one dimension

257

We obtain similar estimates for xA11, xA21, and xA22. Basic to our approach are the following two general results.

Lemma 10.4.3 Let 0 < a < b and W = (U, V) E C2 [a, b] x C2 [a, b] be such that

W 0 0 and W"(x) + A(x)W(x) = 0 in

[a, b],

(10.39)

where A = (Ai3)1
(i) A21 E C[a, b], for all 1 <_ i, j < 2; (ii) A12 (x) 34 0 and A21(x) # 0, for all x E [a, b].

Then neither U nor V can have infinitely many zeros in [a, b].

Proof Supposing the contrary, let us assume that U has an infinite number of zeros in [a, b]. Hence, we can find a monotone sequence (xn)n>1 C [a, b] such that

U(xn) = 0, for all n > 1. Without losing the generality, we may assume that (xn)n>1 is decreasing and let xO := lim,,-,, xn. Then U(xo) = 0 and, by Rolle's theorem, it follows that U' and U" have infinitely many zeros in (xo, xo + 6) for some 5 > 0 small enough. By continuity, we find U'(xo) = U"(xo) = 0. From (10.39) it follows that A12(xo)V(xo) = 0. Because A12(xo) # 0, we have

V(xo)=0. If V'(xo) = 0, then the uniqueness of the Cauchy problem

W"(x) + A(x)W(x) = 0 in (xo, b], 1 W(xo) = W'(xo) = 0 implies W = 0 on [xo, b] and, similarly, W = 0 on [a, xo], which is a contradiction. Therefore, V'(xo) # 0. Because A12 is bounded away from zero in [a, 6], we can find r) > 0 small enough such that IA12(x)V(x)I > IA11(x)U(x)I

for a1l x E (xo,xo+r7).

(10.40)

Combining now (10.39) with (10.40), we deduce that U" has a constant sign on the interval (xo, xo + r1). This clearly contradicts the fact that U vanishes infinitely many times in (xo, xo + 71).

Similarly, if we assume that V has infinitely many zeros in [a, b], we arrive at a contradiction using the fact that A21(x) # 0, for all x E [a, b]. This finishes the proof of our lemma. 0

Lemma 10.4.4 Let W E C2[0, a], a > 0, be such that

W"(x) + A(x)W(x) = 0 1 W(0) = W'(0) = 0. Assume that A = (A23)1<,1<2 satisfies

(i) A25 E C(0, a], for all 1 < i, j < 2; (ii) xA23(x) E L' (0, a), for all 1 < i, j < 2.

in [0, a],

Singular Gierer-Meinhardt systems

258

Then W - 0 in [0, a]

.

Proof Using the regularity of W, a straightforward argument based on integration by parts leads us to

W(x) =

f(t - x)A(t)W(t)dt

M:= max ]jB2j 1
for all 0 < x < a.

(10.41)

!W2(x) :0 <X
W(x) = fx(t - x)B(t)W (t)dt

t.

Jo

for all 0 < t < a.

(10.42)

It follows that

f2: IW(x)I < Mk

(x - t)dt =

2k x2

Using (10.43) in (10.42) we obtain 1W(x)I <

22k foX (x - t)tdt <_ M2 kx3

for all 0 < x < a.

Byinduction we deduce that for any n > 1 we have 1W(xW <

Mnk xn+1,

- (n+1)!

for all 0 < x < a.

-

Now, we can pass to the limit with n -+ oo in the last inequality to obtain the conclusion.

o

Let us come back to the proof of Proposition 10.4.2. By Lemma 10.4.3, it follows that W vanishes for finitely many times in any interval [a, b] C (0, 1). We next assume that U or V has infinitely many zeros near x = 0, the situation where U or V vanishes for infinitely many times near x = 1 being similar. Without losing the generality, we assume that U has infinitely many zeros in a neighborhood of x = 0. Because U E C2[0,1], by Rolle's theorem it follows that both U' and U" have infinitely many zeros near x = 0. As a consequence,

we obtain U'(0) = 0-that is, ui(0) = u2(0).

Uniqueness of the solution in one dimension

259

If V'(0) = 0, then W(0) = W'(0) = 0, and by Lemma 10.4.4 we deduce W - 0 in [0, a] for any 0 < a < 1. By continuity, we derive W = 0 in [0, 11, which is a contradiction. Hence, V'(0) # 0. From (10.36) we have uq

ui - our +

+ p(x) = 0

in (0,1),

u'2'-au2+ V22 +p(x)=0

in (0, 1).

vi

P 47

Subtracting the previous relations we obtain

U"(x) = all(x) + ui{x)

u2(x)

vi(x)

v'2(x)

=

XP-9

aU(x) xP-9

ur()P

4

x

x)

+

/

x

vr (x)

u2(x))P

/

x

9

x/

V2(x)

J

On the other hand, because 0 < p- q < 1, u1(0) = u2(0), and v'(0) # v2(0), we have

lQ

(U2(X))p(

la yxq + x (U,(X))p 1

v q{0}

q

-

1

v2(x)yj x

v,(0) # 0

Therefore, U" has a constant sign in a small neighborhood of x = 0, which contradicts the previous arguments. This finishes the proof of Proposition 10.4.2.

0

Let us complete the proof of Theorem 10.4.1. Set + := {X E [0,1J : U(x) > 0},

j+:_{xE{0,1]:V(x)>0},

1 ` := {x E [0,1] : U(x) < 0}, ..T

:_{xE[0,1]:V(x)<0}.

According to Proposition 10.4.2, the previous sets consist of finitely many disjoint

closed intervals. Therefore, I+ = UT ,111a . For simplicity, let I+ denote any interval It, and we will use similar notations for I-, J+, and J- as well. We then have the following lemma.

Lemma 10.4.5 For any intervals 1+, I-, J+, and J- defined earlier, the following situations cannot occur:

(i) 1+ C J+.

(ii) I- C J-. (iii) J+CI-.

(iv) J- C It

Singular Gierer-Meinhardt systems

260

Proof (i) Assume that I+ C J+. Because v2 > v1 in I+, we deduce that inequality (10.37) holds in I+. Using the fact that u2 = u1 on al+, by virtue of Lemma 1.3.17 we obtain u2 < ul in I+. Hence, u2 - u1 in 1+, which contradicts Proposition 10.4.2. Similarly we prove statement (ii).

(iii) Assume that J+ C I-. Then uPl+°/v1+° > u2+°/v2+° in J+. Notice that V = v2 - vl satisfies

1+4°

91P+°

2LP+°

v2

U1

-V'f + 3V = 22 0

V=0

in J+ ,

on aJ+.

By the weak maximum principle, it follows that V < 0 in J+-that is, v2 < V1 in J+. This yields v2 - v1 in J+, which again contradicts Proposition 10.4.2. The proof of (iv) follows in the same manner. With no loss of generality, we may assume that U > 0 in a neighborhood of x = 0. We may also assume that U > 0 on [0, a1], [a2, a3], ... , [a2n, a2n+1], and U < 0 on [al, a2], [a3, a4], ..., where a2,,+1 = 1 or a2n = 1 depending on the sign of U on the last interval on which U does not vanish. Let us analyze the following distinct situations. CASE 1: V > 0 in a neighborhood of x = 0. If V does not change the sign in (0, al), then we arrive at a contradiction by Lemma 10.4.5 (i). Furthermore, if V changes sign more than once in (0, a1), then we again find a contradiction, by Lemma 10.4.5 (iv). Therefore V changes sign exactly once in (0, a1)-that is,

V(a,) <0. CASE 2: V < 0 in a neighborhood of x = 0. If V changes sign in (0, all, then we contradict Lemma 10.4.5 (iv). Hence, V has a constant sign on (0, al]-that is, V(al) < 0. In both cases we have obtained V(al) < 0. A similar argument yields V(a2) > 0 and, more generally, V(ak) < 0 if k is odd and V(ak) > 0 for all even values of k.

If the last interval is [a2ni 1], then V (a2,,) > 0. Using Lemma 10.4.5 (i), it follows that V vanishes on (a2n, 1), but this will contradict case Lemma 10.4.5 (iv).

The case in which the last interval where U < 0 is [a2.}1, 1], follows in the same manner. This finishes the proof of Theorem 10.4.1. As a consequence of Theorem 10.3.1, solution (u, v) of system (10.31) can be approximated by the solutions of (S)e. Furthermore, the shooting method combined with the Broyden method (to avoid the derivatives) are suitable to approximate the solution of (10.31) numerically. We have considered a = 1, d = 0.5,

p = q = 1, s = 10-2, and p(x) = p1(x) = sin(rrx). In the following figures, we have plotted solution (u,v) of (Se) for a = 0 (Figure 10.1) and Q = 2 (Figure 10.2), respectively.

Comments and historical notes

261

FIGURE 10.1. Solution (u,v) of system (SE) for or = 0.

0.2

04

08

08

t

FIGURE 10.2. Solution (u, v) of system (SE) for a = 2. 10.5 Comments and historical notes Activator-inhibitor systems account for many important types of pattern formation and morphogenesis that rely on cell differentiation. A central question is how the cells, which carry identical genetic code, become different from each other. Spontaneous pattern formation is also common in inorganic systems. For instance, large sand dunes are formed despite the fact that the wind permanently redistributes the sand; sharply contoured and branching river systems (which are in fact quite similar to the branching patterns of a nerve) are formed as a result to erosion, despite the fact that the rain falls more or less homogeneously over

Singular Gierer-Meinhardt systems

262 the ground.

The main feature of these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. Pattern formation requires, in addition, a longer ranging confinement of the locally selfenhancing process. Turing [189] suggested in 1952 that, under certain conditions, chemicals can react and diffuse in such a way to produce steady-state heterogeneous spatial patterns of chemical or morphogen concentration. In 1972 Gierer and Meinhardt [94] proposed a mathematical model for pattern formation of spatial tissue structures in morphogenesis, a biological phenomenon discovered by Trembley [188] in 1744. The influential activator-inhibitor mechanism suggested by Gierer and Meinhardt [94], [140] may be written as P

ut = d1Lu - cru + cpu

v4

+ pop

vs Vt=d2L1v-,Qv+c'p'u

au av

0,

au 49V

=0

in 52 x (0, T),

inI x(0,T),

(10.44)

on tQ

in a smooth bounded region n C IRN (N > 1). In system (10.44), the unknown u represents the concentration of a short-range autocatalytic substance (that is, activator), and v is the concentration of its long-range antagonist (that is, inhibitor). Also p and p' stand for the source distributions of the activator and inhibitor, respectively; dl, d2 are diffusion coefficients with dl << d2i and cr, Q, c, c', po are positive constants. The exponents p, q, r, s > 0 satisfy the relation qr > (p - 1)(s + 1) > 0. System (10.44) is of a reaction-diffusion type and involves the determination of an activator and inhibitor concentration field. The basic idea behind CiererMeinhardt's system is the so-called diffusion-driven instability, originally the result of the work by Turing [189], which asserts that different diffusion rates could lead to nonhomogeneous distributions of the reactants. Also, the model introduced in [94] takes into account the classification between the concentration of activators and inhibitors on the one hand, and the densities of their sources on the other. The model introduced by Gierer and Meinhardt has been used with satisfactory quantitative results for modeling the head regeneration process of hydra, an animal a few millimeters in length, consisting of 100,000 cells of about 15 different types and having a polar structure. A similar model was proposed by Meinhardt [139] in 1976 that is applied both in geomorphology and in hydrology. For instance, the formation of leaf veins, blood vessels, and the fibers of the nervous system, but also the channels in which streams flow, are described by Meinhardt's model (see also Willgoose, Bras, and Rodriguez-Iturbe [1961). We refer to the recent monograph by Murray [145] for further details and references.

Comments and historical notes

263

It has been shown that dynamics of system (10.44) exhibit various interesting behaviors such as periodic solutions, unbounded oscillating global solutions, and finite time blow-up solutions. We refer the interested reader to the paper by Ni, Suzuki, and Takagi [150] for the entire description of dynamics concerning system (10.44). Also a global existence results for a more general system than (10.44) is given in the recent work by Jiang [109]. Many recent works have been devoted to the study of the steady-states solu-

tions of (10.44)-that is, solutions of the stationary system UP djAu-au+cp-4 +pop=0 in n,

d2Av - Qv + c'p' u8 =0 V8

I

Yv

0, 8Y

0

in S2 ,

(10.45)

on BSl.

System (10.45) is quite difficult to solve, because it does not have a variational

structure. An attempt in solving (10.45) is to consider the shadow system, an idea due to Keener [116]. More exactly, dividing the second equation of (10.45) by d2 and then letting d2 -* oo, we reduce system (10.45) to a single equation. The nonconstant solutions of such equation present interior or boundary peaks or spikes-that is, they exhibit a point concentration phenomenon. Among the great number of works in this direction, we refer the reader to [151], [192], [193], [194] and the references therein, as well as to the survey papers of Ni [148], [149]. For the study of instability of solutions to (10.45), we also mention here the works of Miyamoto [144] and Yanagida [198]. In the case SZ = RN, N = 1, 2, it has been shown in del Pino et al. [63], [64] that there exist ground-state solutions of (10.1) with single or multiple bumps in the activator that, after a rescaling of u, approach a universal profile. The study of system (S) presented in this chapter is a result of the work by Ghergu and Radulescu [91], [92] and is motivated by some questions addressed in recent research papers by Choi and McKenna [43], [44] and Kim [119], [120] concerning existence and nonexistence or even uniqueness of the classical solutions for model system (10.1). The existence of classical solutions for the general system (S) was obtained in this chapter under the additional hypothesis /3 < a.

In fact, this assumption is quite natural if we look at the steady-state system (10.45). We have only to divide the first equation by dl, the second one by d2, and then take into account the fact that d1 << d2. In [43], [119] it is assumed that the activator and inhibitor have common sources and the approach relies on Schauder's fixed point theorem through a decouplization of the system. More precisely, if p = r and q = s, then subtracting the two equations of (10.1) we obtain the following equivalent form:

264

Singular Gierer-Meinhardt systems

Ow - aw + (0 - a)wv + p(x) = 0, Ov-Qv+(v+w P

i v=w=0

Z

=0

in S2,

inn,

(10.46)

ont9Q.

Because the first equation in (10.46) is linear, we can easily obtain a priori estimates in order to control the map whose fixed points are solutions of (10.1). In Choi and McKenna [44] it is obtained the existence of radially symmetric

solutions of (S) in the case 12 = (0,1) or SZ = Bl C R2 and p = r > 1, q = 1, s = 0. In [44] a priori bounds are obtained via sharp estimates of the associated Green's function. The method we used in proving the uniqueness of the solution to system (10.36) is a result of the work by Choi and McKenna [43], where it

was initially obtained for p = q = r = s = 1 and p - 0.

APPENDIX A SPECTRAL THEORY FOR DIFFERENTIAL OPERATORS All intelligent thoughts have already been thought; what is necessary is only to try to think them again. Johann Wolfgang von Goethe (1749-1832)

A.1 Eigenvalues and eigenfunctions for the Laplace operator Recall first the following theorem. Theorem A.1.1 (Poincare) Let S2 C l[tN be a bounded domain (or, more generally, such that one of its projections to the coordinate axes is bounded). Then there exists a constant c > 0 such that for all u e HQ (SZ), IIUIIH1(n) <- CIIVuIIL2(n)

(A.1)

(-i.)-1

: L2(SZ) --> HH(SZ) defined as follows: For all Consider the operator f E L2(Q), (-A)-'f is the unique u e Ho (S2) such that

Du = f

u=0

in SZ,

onr9ll.

(A.2)

Remark that the operator (-©)-1 is continuous. Indeed, by (A.2) we have fudx <_ IIf1IL2(n) IIUIIL2(n).

IIuIIHUI (0)

CIIfIIL2(n)IIVuIIL2(o), where C By Poincare's inequality, we have is the constant from (A.1). Then lIuIIH;(n) < cIIfIIL2(st) for some c > 0 not dependent on f and u.

Theorem A. 1.2 The following properties hold true: (i) The eigenvalue set of the operator -/A subject to the Dirichlet boundary condition consists of a sequence

0 oo as n -> oo.

266

Spectral theory for differential operators

(ii) The associated eigenfunctions belong to C2 (?I).

Proof Let i : H0(Q) -> L2(1l) be the canonical injection. By the RellichKondrachov embedding theorem, i is a compact operator. Set T = -A o i

:

Ho (Sl) - Ho (S1). Then T is a compact, self-adjoint, and positive operator. The first part follows from the Riesz theory for compact operators in Hilbert spaces.

The last part is a direct. consequence of the Schauder and Holder regularity theories for elliptic equations. Taking into account that the first eigenvalue cpi belongs to C2(S1), we deduce the following asymptotic behavior of the first eigenfunction near the boundary.

Corollary A.1.3 The following properties hold true: (i) There exist positive constants cl and c2 such that cl d(x) < cpl (x) < c2 d(x) in Q. (ii) Bpi E L' (Q) if and only if k > -1.

We point out that the existence of a countable family of eigenvalues for the Laplace operator under the Dirichlet boundary condition was established in 1894 by Poincare [165]. This pioneering result is the beginning of the spectral theory, which has played a crucial role in the development of theoretical physics and functional analysis. Linear eigenvalue problems with nonconstant potential of the type in 51, Du = AV(x)u

u=0

on 8S1

(A.3)

were studied by Bocher [28], Minakshisundaran and Pleijel [141], Pleijel [162], and Hess and Kato [107]. For instance, Minakshisundaran and Pleijel [141], [162]

were concerned with the case when V is a variable potential such that V E L°'(11), V > 0 in 11, and V > 0 in no C Sl with 19101 > 0.

A.2 Krein-Rutman theorem Definition A.2.1 Let X be a real Banach space. A nonempty set C is said to be a cone if C is a closed convex set and C fl (-C) = {0}. Definition A.2.2 A real Banach space X is called ordered if there exists a cone C and a partial order relation " < " such that x < y if and only if x - y E C. A very useful tool in the spectral theory of differential operators is the following result.

Theorem A.2.3 (Krein-Rutman) Let X be a real Banach space with an order cone C having a nonempty interior. Assume that T : X -' X is a linear compact operator such that T(C) C C. Then the following properties hold true: (i) T has exactly one eigenfunction x E C. The corresponding eigenvalue equals the spectral radius r(T) of T.

Krein-Rutman theorem

267

(ii) For all complex numbers A in the spectrum of T that are different from r(T) we have JAI < r(T). Instead of using the Krein-Rutman theorem, we can establish by means of elementary arguments that the least eigenvalue of the Laplace operator in Ho(Q) is simple. For this purpose, because eigenfunctions corresponding to higher eigenvalues Ak (k > 2) do necessarily change sign in S2, it is enough to show that the solutions of the problem

-Au = Au

in 0,

u>0

(A.4)

in S2,

u=0 on 8d are unique, up to a multiplicative constant. For this purpose, we apply an idea introduced by Benguria, Brezis, and Lieb [21], which relies on the following facts:

(i) Any solution of problem (A.4) is a minimizer of the Dirichlet energy functional

E(v) := if VvI2dx [

on the manifold

M:={vEHH(d):v>0in0and f v2 dx1 JJJ}. 111

(ii) The energy functional E(v) is convex in v2 on the cone

C:= {vE HH(d):v>Oin0}. Assuming that ul and u2 are arbitrary solutions of (A.4), then both ul and u2 minimize E on M. Set v:= (u2+u2)/2 and w := . Thus, by (ii), we deduce that w E M. A straightforward computation yields Vw = 1L,Vu1 +U2Vu2 2,v/v-

in Q.

Hence,

E(w)

= fn 2v lu1Vu1 +u2Vu212dx VU, + u1 ui + u2 u2 Vu2 2

IVw12dx

I

n

2

< f ui+u2 2 fln

_

2

dx

ui + u2 u2 1Vui12 + ui u2 IVu2I2 dx u1 + u2 u2 ) u2 + u2 ui

u 2 + u2 u1

f (IVuil2 + JVu212) dx E(ul) + E(u2) 2

2

Because ul and u2 are minimizers of E, the previous relation shows that w is a

minimizer of E, too. Thus, E(w) = E(ul) = E(u2). Hence, Vul/ul = Vii2/u2 in Q. Therefore, V(ul/u2) = 0 in St, which implies that ul/u2 is constant.

APPENDIX B IMPLICIT FUNCTION THEOREM Mediocrity knows nothing higher

than itself, but talent instantly recognizes genius.

Arthur Conan Doyle (1859-1930)

The germs of the implicit function theorem appeared in the works of Newton, Leibniz, and Lagrange. In its simplest form, this theorem involves an equation of the form F(u, A) = 0 (we can assume without loss of generality that F(0, 0) = 0). where A is a parameter and u is an unknown function. The basic assumption is that F(u, A) is invertible for u and A "close" to their respective origins, but with some loss of derivatives when computing the inverse. In his celebrated proof of the result that any compact Riemannian manifold may be isometrically embedded in some Euclidean space, Nash [147] developed a deep extension of the implicit function theorem. His ideas have been extended by various people to a technique that is now called the Nash-Moser theory. We

recall that John F. Nash Jr. is an outstanding scientist who received the 1994 Nobel Prize in economics for his pioneering analysis of equilibria in the theory of noncooperative games. Throughout these lecture notes we have used the following "standard" version of the implicit function theorem. This property is an important tool in nonlinear

analysis and, roughly speaking, it asserts that the behavior of a nonlinear system is qualitatively determined by its linearized system around the zeros of the nonlinear system.

Theorem B.1 Let X and Y be real Banach spaces and let (uo, Ao) E X x R. Consider a mapping F = F(u, A) : X x IR -+ Y of class C1 such that (i) F(uo,A0) = 0; (ii) the linear mapping F,d(uo, A0) : X -4 Y is bijective. Then there exists a neighborhood U0 of uo and a neighborhood Vo of \0 such that for every .\ E Vo, there is a unique element u(A) E Uo so that F(u(A), A) = 0. Moreover, the mapping V0 3 A'--f u(A) is of class C1.

Proof Consider the mapping 4)(u, A) : X x R --> Y x R defined by 4(u, A) (F(u, A), A). By our hypotheses, 4i is a mapping of class C1. We apply to 4) the inverse function theorem. To conclude the proof, it remains to verify that 4?`(uo, A0) : X x ll8 -> Y x R is bijective. Indeed, we have

Implicit function theorem

270 it (uo

+ tu, AO + tA) = (F(uo + tu, AO + tA), AO + tA) =

(F(uo, Ao) + Fu(uo, Ao) - (tu) + FA(uo, Ao) - (tA) + o(i), Ao + tA).

It follows that AO) _

\

F.(uo, Ao) F.(uo, Ao) I 0

which is a bijective operator, by our hypotheses. Thus, by the inverse function theorem, there exist a neighborhood U of the point (uo, AO) and a neighborhood V of (0, A) such that the equation -11(u, A) = (f, Ao)

has a unique solution, for every (f, A) E V. Now, it is sufficient to take here f = 0 0 and our conclusion follows.

With a similar proof we can establish the following global version of the implicit function theorem.

Theorem B.2 Assume F : X x R --> Y is a function of class Cl such that (i) F(0, 0) = 0; (ii) the linear mapping F (0, 0) : X -+ Y is bijective. Then there exist an open neighborhood I of 0 and a mapping I a A -- u(A) of class Cl such that u(0) = 0 and F(u(A), A) = 0. The following result has been of particular importance in our arguments in the study of bifurcation problems. Theorem B.3 Assume the same hypotheses on F as in Theorem B.2. Then there exists an open maximal interval I containing the origin and there exists a unique mapping I E) A i--> u(A) of class C' such that the following hold: (i) F(u(A), A) = 0 for every A E I. (ii) The linear mapping Fu(u(A), A) is bijective, for any A E I.

(iii) u(0) = 0.

Proof Let ul, u2 be solutions and consider the corresponding open intervals I, and I2 on which these solutions exist, respectively. It follows that ul(0) _ U2(0) = 0 and F(ul (A), A) = 0

for every A E Il,

F(u2(A), A) = 0

for every A E I2.

Moreover, the mappings F, (ul (A), A) and Fu(U2(A), A) are bijective on I,, respectively I2. But, for A sufficiently close to 0, we have ul(A) = u2(A). We wish to show that we have global uniqueness. For this purpose, let

i:={AEI1n12:u1(A)=u2(A)}. Our aim is to show that I = Il n i2. We first observe that 0 E I, so I # 0. A standard argument implies that I is closed in Il n I2. To prove that I = Il n I2, it

Implicit function theorem

271

is sufficient to argue that I is an open set in Il fl I2. The proof of this statement follows by applying Theorem B.1 for A instead of 0. Thus, I = Il fl I2. Next, to justify the existence of a maximal interval I, we consider the C' curves u,,(A) defined on the corresponding open intervals I, such that 0 E In, A) = 0 and F,,(u7,(.\), A) is an isomorphism, for any A E I,,. u.(0) = uo, A standard argument enables us to construct a maximal solution on the set

U,, I,,. This concludes the proof. We refer to a recent book [121] for various applications of the implicit function theorem.

APPENDIX C EKELAND'S VARIATIONAL PRINCIPLE It is not enough that we do our best; sometimes we have to do what is required. Sir Winston Churchill (1874-1965)

C.1

Minimization of weak lower semicontinuous coercive functionals

An important topology in which many arguments can be carried out in reflexive Banach spaces (such as LP(S2) or Wk-p(St) for 1 < p < oo) is the weak topology, in which the unit ball is weakly compact. The weak lower semicontinuity is a major tool in reflexive Banach spaces. We recall that if E is a Banach space, then a functional 4i : E -+ R is said to be weak lower semicontinuous if 4)(u) < lim inf,,- 0 4)(u,,) for any sequence (un,) in E converging weakly to u. A basic sufficient condition for a continuous functional 4i : E - R to be weakly lower semicontinuous is that 4) is convex. One of the deepest consequences is that weak lower semicontinuous coercive functionals attain their infimum on some suitable sets, as stated in the following basic result.

Theorem C.1.1 Let E be a reflexive Banach space, M C E be a weakly closed subset of E, and 4) : M -+ R such that (i) 4) is coercive on M with respect to E-that is

4)(u) - oo

as jhulI -+ oo.

(ii) 4i is weakly lower semi continuous on M.

Then 4i is bounded from below and attains its infimum on M. The proof of Theorem C.1.1 uses elementary arguments based on the definition of the weak lower semicontinuity. We refer to [27] for complete details and further comments.

C.2 - Ekeland's variational principle Ekeland's variational principle [75] was established in 1974 and is the nonlinear version of the Bishop-Phelps- theorem [161], with its main feature of how to

Ekeland's variational principle

274

_

use the norm completeness and a partial ordering to obtain a point where a linear functional achieves its supremum on a closed bounded convex set. A major

consequence of Ekeland's variational principle is that even if it is not always possible to minimize a nonnegative C1 functional 4) on a Banach space; however, there is always a minimizing sequence (u7,),>, such that 4)'(u,,) --+ 0 as n -+ oo.

We first state the original version of Ekeland's variational principle, which is valid in the general framework of complete metric spaces. Theorem C.2.1 (Ekeland's variational principle) Let (M, d) be a complete metric space and assume that 4) : M (-oo, oo], 4) $ oo, is a lower semicontinuous functional that is bounded from below. Then, for every e > 0 and for any zo E M, there exists z e M such that (i) b(z) 4)(zo) - e d(z, zo); (ii) 4)(x) > D(z) - c d(x, z), for any x E M. Proof We may assume without loss of generality that e = 1. Define the following binary relation on M:

y<x

if and only if

4)(y) - 4)(x) + d(x, y) < 0.

Then "<" is a partial order relation-that is, (a) x < x, for any x E M;

(b) ifx < y and y<x thenx=y;

(c) ifx
S(x) := {y E M : y < x}. Let be a sequence of positive numbers such that en --> 0 and fix zo E M. For any n > 0, let zn+1 E S(zn) be such that

4)(zn+1) < inf 4) + c,

1

The existence of zn+1 follows from the definition of S(x). We prove that the sequence (zn)n>1 converges to some element z, which satisfies (i) and (ii). Let us first remark that S(y) C S(x), provided that y:5 x. Hence, S(zn+1) C S(zn). It follows that for any n > 0, $(zn+1) - D(zn) + d(zn, zn+1) < 0,

which implies 4)(zn+1) < $(z,,). Because D is bounded from below, we deduce that the sequence converges. We prove in what follows that (zn)n>1 is a Cauchy sequence. Indeed, for any n and p we have D(z.) + d(zn+v, z,) < 0. (C.1) Therefore,

Ekeland's variational principle

d(zn.+p, zn) < '(zn) - $(zn+p) -' 0

275

as n -+ oo,

which shows that (zn)n>1 is a Cauchy sequence, so it converges to some z E M. Now, taking n = 0 in (C.1), we find D (zp) - O(zo) + d(zp, zo) < 0.

So, as p -- oo, we find (i). To prove (ii), let us choose arbitrarily x E M. We distinguish the following situations. CASE 1: x E S(zn), for any n _> 0. It follows that -t(zn+1) < I'(x) + E,,,+1, which implies that 4>(x). CASE 2: There exists an integer N > 1 such that x ¢ S(zn), for any n > N or, equivalently,

fi(x) - 4'(z,,) + d(x, zn) > 0

for every n > N.

Passing at the limit in this inequality as n -, oo we find (ii).

0

The following consequence of Ekeland's variational principle has been of par-

ticular interest in our arguments. Roughly speaking, this property establishes the existence of almost critical points for bounded from below C' functionals defined on Banach spaces. In order words, Ekeland's variational principle can be viewed as a generalization of the Fermat theorem, which establishes that interior extrema points of a smooth functional are, necessarily, critical points of this functional.

Corollary C.2.2 Let E be a Banach space and let 4D : E -> R be a C' functional that is bounded from below. Then, for any e > 0, there exists z E E such that and

4)(z) < inf -D + E E

II4)'(z)IlE < ---

Proof The first part of the conclusion follows directly from Theorem C.2.1. For the second part we have

IIV(z)IlE = sup (V(z),u). II1,1I=1

But,

(z), u) = lim

8-.o

4i(z + Su) dflull

So, by Ekeland's variational principle,

(V (z), u) > -E. Replacing now u with -u we find (V (z), u) < E,

which concludes our proof.

0

276

Ekeland's variational principle

Sullivan [185] observed that Ekeland's variational principle characterizes complete metric spaces in the following sense.

Theorem C.2.3 Let (M, d) be a metric space. Then M is complete if and only if the following holds: For every application I> : M - (-oo, oo], 4P 0- oo, which is bounded from below, and for every E > 0, there exists zE E M such that (i) -t(zz) < infM 4) +e; (ii) -D(x) > 4D(z£) - e d(x, zz), for any x E M \ {zf}.

APPENDIX D MOUNTAIN PASS THEOREM There are more things in heaven and earth, Horatio, than have been dreamt of in our philosophy. William Shakespeare (1564-1616) Hamlet, Prince of Denmark. Act I

D.1 Ambrosetti-Rabinowitz theorem The mountain pass theorem was established by Ambrosetti and Rabinowitz in [8]. Their original proof relies on some deep deformation techniques developed

by Palais and Smale [1571, [158], who put the main ideas of the Morse theory into the framework of differential topology on infinite dimensional manifolds. In this way, Palais and Smale replaced the finite dimensionality assumption with an appropriate compactness hypothesis, which is stated in the following.

Definition D.1.1 (Palais-Smale condition) Let E be a real Banach space. A functional J : E -+ R of class C1 satisfies the Palais-Smale condition if any sequence (un)n>1 in E is relatively compact, provided

lim IIJ'(u,,)jIE. = 0. sup IJ(un)I Goo and n-.oo n>1

(D.1)

A simple consequence of Ekeland's variational principle, in connection with the Palais-Smale condition is stated next.

Proposition D.1.2 Let E be a real Banach space and assume that 4) : E - R is a functional of class C1 that is bounded from below, and satisfies the PalaisSmale condition. Then the following properties hold true: (i) 4> is coercive. (ii) Any minimizing sequence of I has a convergent subsequence.

The mountain pass theorem is a powerful tool for proving the existence of critical points of energy functionals, hence of weak solutions to wide classes of nonlinear problems. The precise statement of this result is the following.

Theorem D.1.3 (Mountain pass theorem) Let E be a real Banach space and assume that J : E -r R is a C' functional that satisfies the following conditions: There exist positive constants a and R such that (i) J(O) = 0 and J(v) > a for all v E E with jlvil = R;

Mountain pass theorem

278

(ii) J(vo) < 0, for some vo E E with ]lvoll > R. Set

r:_{pEC([0,1];E):p(0)=0 andp(1)=vo} and

c := inf max J(p(t)). pEr tE[0,1j

Then there exists a sequence (un)n>i in E such that J(un) -+ c and J'(un) -+

0 as n --' oo. Moreover, if J satisfies the Palais-Srnale condition, then c is a nontrivial critical value of J-that is, there exists u E E such that J(u) = c and J'(u) = 0. The elementary meaning of the mountain pass theorem is the following. The

set r denotes the family of all continuous paths that join the "villages" 0 and vo. Hypotheses (i) and (ii) imply c > a > max{J(0), J(vo)}. In fact, in the proof of the mountain pass theorem, it is used only the basic assumption c > max{J(0), J(vo)}. We also observe that the geometric conditions of the mountain pass theorem imply the existence of two valleys around these villages, but also of a mountain range represented by the sphere centered at the origin and of radius

R. Under these hypotheses, Theorem D.1.3 states that there exists a lowest mountain pass at the level c.

Proof Our arguments are the same as those developed by Brezis and Nirenberg [35], who provided a more elementary proof with respect to the original one of Ambrosetti and Rabinowitz. The arguments we give in what follows combine two basic tools: Ekeland's variational principle and the following pseudogradient lemma.

Lemma D.1.4 Let F : [0,1] -+ E* be a continuous function. Then, for any e > 0, there exists a continuous map v : [0,1] 11v(t)II < 1

and

E such that, for all t E [0, 1],

lJF(t)llE. - E.

Proof Fix to E [0, 1]. Then there exists w(to) E E such that 11w(to)II < 1

and

IIF(to)IIE -e.

By continuity, there exists an open neighborhood N(to) of to such that for any t E Ar(to),

IIF(t)IIE' - E. The set of all these neighborhoods .Af(t) (for t (=- [0, 1]) covers [0, 1]. So, there exist t1, ... , tk E [0,1] such that [0,1] C U;,1 M(t?). Next, we consider the partition

Ambresetti-Rabinowitz theorem

279

of unity associated with this finite covering. So, for any 1 < j < k, we define the continuous map

dist(t, [0,1] \N(tj)) Ii-1 dist (t, (0,11 \ N(ti))

j (t)

for any t E [0, 1]-

k

Set

k

v(t) :=

cbj(t)w(tj)

j

for any t E [0,1].

j=1

We first observe that k

k

< E oj(t) IIw(tj)II < E oj(t) =1.

IIv(t)II

j=1

j=1

To conclude the proof, it remains to argue that for any t E [0,1], k

(F(t),V(t))E',E _ EOj(t)(F(t),W(tj))E',E > IIF(t)IIE' -E'. j=1

This follows after observing that for any 1 < j < k (t being assumed to be fixed), only one of the following situations may occur:

(i) If t E N(tj), then (F(t),w(tj))E',E >- IIF(t)11E. - e. (ii) If t V N(tj), then 4(tj) = 0. This concludes the proof of the lemma. We are now in the position to give the proof of the mountain pass theorem. Consider the complete metric space I' endowed with the metric d(p1,p2)

for any p1, p2 E t.

ax IIpi(t) -p2(t)II

t

,11

Define the continuous mapping 4P(p) :

tjo 11

J(p(t))

for any p CE

'.

Then 4' is bounded from below, because 4)(p) > max{J(0), J(vo)}. Moreover, infVEr 4'(p) = c. Fix a positive integer n. Thus, by Ekeland's variational principle,

there exists pn E I such that

c<4'(pn)
(D.2)

and

4'(p) - C(pn) + 1 d(p, pn) > 0

for any p E I'.

(D.3)

Because 4' is continuous, it follows that

Mn:={tE[0,11:J(pn(t))=4(pn)) #0. Thus, to conclude the proof, it is enough to show that there exists to E M,such that IIJ'(pn(tn))IIE* < 2/n. To this end, we apply Lemma D.1.4 to F(t) =

Mountain pass theorem

280

J'(pn(t)). Hence, there exists a continuous function v : [0,1] -i E such that for any t E [0, 1],

and

11v(t)J[ < 1

(J'(pn(t)), v(t))

IIJ'(pn(t))II E. -

1

n

.

(D.4)

On the other hand, because c > max{J(0), J(v0)}, there exists a continuous function a : [0, 1] --' [0, 1] such that a(0) = a(1) = 0 and a 1 on an open set U D Mn. As admissible paths p E F in (D.3), we take pE(t) = pn(t) - ea(t)v(t), for E > 0 sufficiently small. We also observe that d(p,,,, pE) < e. Our choice of a

guarantees that p,(0) = p,,(0) = 0 and pe(1) = pn(l) = v0i hence, pE E r. Let tE E [0, 1] such that

JE(t.) = max J(p5(t)). tEI0,11

Setting p = pe in (D.3), we obtain J (pa(te) - ec (tE)v(tE)) - tEmO J(pn(t)) + 1) n

>_ 0.

(D.5)

On the other hand, we have the asymptotic estimate J (pa(te) - Ea(tE)v(tE))

J (pn(tE)) - E(J'(p.(te)), a(t6)v(te)) + 0(6) , (D.6)

as 6 - 0. Relations (D.5) and (D.6), combined with the observation that J(pn(tE)) < tma l yield

(J' (Pn(te)) , a(tE)v(te)) <

+0(1)

as 6

0.

But a(te) = 1, provided e > 0 is small enough. Hence,

(J' (pn(te)) , v(tE)) < n +00)

ash -r 0.

Using now (D.4), we deduce that IIJ'(pn(te))IIE+ <-

n +o(1)

as e -* 0.

(D.7)

We can assume, passing eventually at a subsequence, that tE -> to as E -40. Because to E Mn, relation (D.7) yields IIJ'(pn(tn))IIE. <_ 2/n. Now, it is enough to take un := pn(tn) and we obtain a sequence of "almost critical points" satisfying the conclusion. In the case where J satisfies the Palais-Smale condition, a standard compactness argument implies that c is a critical value of J corresponding to a nontrivial critical point.

Application to the Emden-Fowler equation

281

We point out that the conclusion in the mountain pass theorem remains true if assumption (D.1) in the Palais-Smale condition is replaced with sup IJ(u,ti)I < oo and

lim (1 + I[un[I) IIJ'(un)IIE = 0. n-oo

n>1

The previous weaker hypothesis corresponds to the Cerami compactness condition. More generally, Theorem D.1.3 is still valid if assumption (D.1) becomes

sup IJ(un)I < oo

and

lim g(IIunU) II J'(un)II Ex = 0, n-oo

n>1

provided g : [0, oo) -> [1, oo) is a continuous function such that f loo dt/g(t) = oc.

We left this property as an exercise to the reader.

D.2 Application to the Emden-Fowler equation A direct consequence of the mountain pass theorem is related to the existence of solutions for the subcritical Emden-Fowler boundary value problem

1-Au=up u>0

in 1, (D.8)

in S2,

u=0

on 099,

where S2 C RN is a bounded domain with smooth boundary and 1 < p < (N + 2)/(N - 2) if N > 3 or 1 < p < oo if N E {1, 2}. This equation was introduced by Emden [76] and Fowler [77]. Existence results for problem (D.8) are related not only to the values of p, but also to the geometry of I. For instance,

problem (D.8) has no solution if p > (N + 2)/(N - 2) (if N > 3) and if 9 is starshaped with respect to some point (say, with respect to the origin). This property was observed by Rellich [176] in 1940 and refound by Pohozaev [163] in 1965, and its proof relies on the Rellich-Pohozaev identity. More precisely, after

multiplication by x Vu in equation (D.8) and integration by parts we find

(L.uv+1 - N2 2 sz

a

un+1l

y dx

=2

fn (±)(x.n)thT(x).

(D.9)

Because Sl is starshaped with respect to the origin, then the right-hand side of (D.9) is positive; hence, problem (D.8) has no solution if N/(p+1)-(N-2)/2 < 0, which is equivalent to p > (N + 2)/(N - 2). We point out that a very general variational identity was discovered in 1986 by Pucci and Serrin [168]. The situation is different if we are looking for entire solutions (that is, solutions on the whole space) of the Emden-Fowler equation either in the critical or in the supercritical case. For instance, the equation

-Au = u(N F2)1(N-2) admits the family of solutions

in RN, (N > 3),

Mountain pass theorem

282

[CN(N -

2)](N-2)l4

(C2 +

IxI2)(N-2)/2

for any A > 0, where C = A2/(N-2)N-2(N - 2)-2. If 1 is not starshaped, Kazdan and Warner [114] showed in 1975 that problem (D.8) has a solution for any p > 1, provided ) is an annulus in RN. We also point out that if fl C RN is an arbitrary bounded domain with smooth boundary, then the perturbed Emden-Fowler problem

-Au = IuIP-'u + Au u

0

u=0

in fl, in SZ,

on Oft

has a solution, provided A is an arbitrary real number and 1 < p < (N+2)/(N-2)

if N > 3 or I < p < oo if N E {1, 2}. The proof combines the mountain pass theorem (if A < A1) and the dual variational method (if A > A1). As usually, Al denotes the first eigenvalue of the Laplace operator (-A) in Ho (Sl).

D.3 Mountains of zero altitude The limiting case c = max{J(0),J(vo)} in the statement of the mountain pass theorem corresponds to mountains of zero altitude. This degenerate situation was studied by Pucci and Serrin [167]. They considered a functional J : E -+ R of class Ci that satisfies the following geometric conditions:

(a) There exist real numbers a, r, R such that 0 < r < R and J(v) > a for

every vEEwith r R. Under these hypotheses, combined with the standard Palais-Smale compactness

condition, Pucci and Serrin established the existence of a critical point u E E \ {0, vo} of J with corresponding critical value c _> a. Moreover, if c = a, then the critical point can be chosen with r < IIuII < R. Roughly speaking, the mountain pass theorem continues to hold for a mountain of zero altitude, provided it also has nonzero thickness. In addition, if c = a, then the pass itself occurs precisely on the mountain, in the sense that it satisfies r < lull < R.

REFERENCES [1] Aftalion, A. and Reichel, W. (1997). Existence of two boundary blow-up solutions for semilinear elliptic equations. J. Differential Equations, 141, 400-421. [2] Agarwal, R. and O'Reagan, D. (2001). Existence theory for single and multiple solutions to singular positone boundary value problems, J. Differential Equations, 175, 393-414.

[3] Alaa, N. E. and Pierre, M. (1993). Weak solutions of some quasilinear elliptic equations with data measures. SIAM J. Math. Anal., 24, 23-35. [4] Alama, S. and Tarantello, G. (1996). On the solvability of a semilinear elliptic equation via an associated eigenvalue problem. Math. Z., 221, 467493.

[5] Amann, H. (1971). On the existence of positive solutions of nonlinear elliptic boundary value problems. Indiana Univ. Math. J., 21, 125-146. [6] Amann, H. (1976). Existence and multiplicity for semilinear elliptic boundary value problems. Math. Z., 150, 567-597. [7) Ambrosetti, A. and Gamez, J. L. (1997). Branches of positive solutions for some semilinear Schrodinger equations. Math. Z., 224, 347-362. [8) Ambrosetti, A. and Rabinowitz, P. H. (1973). Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349-381. [9] Aris, R. (1975). The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Clarendon Press, Oxford. [10] Badiale, M. and Tarantello, G. (1997). Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities. Nonlinear Anal., T.M.A., 29, 639-677.' [11] Ball, J. M. (1977). Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63, 337-403. [12] Ball, J. M. (1998). The calculus of variations and materials science. Current and future challenges in the applications of mathematics. Quart. Appl. Math., 56, 719-740. [13] Bandle, C., Diaz, G., and Diaz, I. J. (1994). Solutions d'equations de reaction-diffusion non lineaires explosant au bord parabolique. C. R. Acad. Sci. Paris, Ser. I Math., 318, 455-460. [14] Bandle, C., Greco, A., and Porru, G. (1997). Large solutions of quasilinear elliptic equations: Existence and qualitative properties. Boll. Unione Matematica Italiana, 7, 227-252. [15] Bandle, C. and Marcus, M. (1992). `Large' solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behavior. J. Anal. Math., 58, 9-24. [16] Bandle, C. and Porru, G. (1994). Asymptotic behavior and convexity of

References

284

large solutions to nonlinear equations. World Sc. Series in Appl. Anal., 3, 59-71.

[17] Bandle, C., Pozio, A., and Tesei, A. (1987). The asymptotic behavior of the solutions of degenerate parabolic equations. Trans. Amer. Math. Soc., 303, 487-501. [18] Bandle, C. and Stakgold, 1. (1984). The formation of the dead core in parabolic reaction-diffusion problems. Trans. Amer. Math. Soc., 286, 275293.

[19] Barbu, V. (1998). Partial Differential Equations and Boundary Value Problems. Kluwer Academic Publishers, Dordrecht. (20] Benci, V., Fortunato, D., and Pisani, L. (1998). Solitons like solutions of a Lorentz invariant equation in dimension 3. Rev. Math. Phys., 10, 315-344. [21] Benguria, R., Brezis, H., and Lieb, E. H. (1981). The Thomas-Fermi-von Weizsacker theory of atoms and molecules. Comm. Math. Phys., 79, 167180.

[22] Beznea, L. and Boboc N. (2004). Potential Theory and Right Processes. Kluwer Academic Publishers, Dordrecht. [23] Benilan, P., Brezis, H., and Crandall, M. (1975). A semilinear equation in EV(RN). Ann. Scuola Norm. Sup. Pisa, 4, 523-555. [24] Berestycki, H., Nirenberg, L., and Varadhan, S. R. S. (1994). The principal eigenvalue and maximum principle for second order elliptic operators in general domains. Comm. Pure Appl. Math., 47, 47-92. [25] Bertsch, M. and Rostamian, R. (1985). The principle of linearized stability for a class of degenerate diffusion equations. J. Differential Equations, 57, 373-405.

and die automorphen Funktionen. Math. [26] Bieberbach, L. (1916). Du = Ann., 77, 173-212. [27] Blanchard, P. and Briining, E. (2003). Mathematical Methods in Physics. Distributions, Hilbert Space Operators, and Variational Methods. Progress in Mathematical Physics, Birkhauser, Boston. [28] Bocher, M_ (1914). The smallest characteristic numbers in a certain exceptional case. Bull. Amer. Math. Soc., 21, 6-9. [29] Bratu, G. (1914). Sur les equations integrates non lineaires. Bull. Soc. Math. France, 42, 113-142. [30] Brezis, H. (1992). Analyse Fonctionnelle. Theorie, Methodes et Applications. Masson, Paris. [31] Brezis, H. and Browder, F. (1998). Partial differential equations in the 20th century. Adv. Math., 135, 76-144. [32] Brezis, H. and Kamin, S. (1992). Sublinear elliptic equations in III:N. Manus-

cripta Math., 74, 87-106. [33] Brezis, H. and Nirenberg, L. (1983). Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math., 36, 437-477. [34] Brezis, H. and Nirenberg, L. (1989). A minimization problem with critical

References

285

exponent and nonzero data. Symmetry in Nature, (a volume dedicated in honour of L. Radicati), vol. I, 129-140, Scuola Normale Superiore di Pisa. [35] Brezis, H. and Nirenberg, L. (1991). Remarks on finding critical points. Comm. Pure Appl. Math., 44, 939-964. [36] Brezis, H. and Nirenberg, L. (1993). H' versus C' local minimizers. C. R. Acad. Sci. Paris, Ser. I, 317, 465-472. [37] Brezis, H. and Oswald, L. (1986). Remarks on sublinear elliptic equations. Nonlinear Anal., T.M.A., 10, 55-64. [38] Callegari, A. and Nachman, A. (1980). A nonlinear singular boundary value

problem in the theory of pseudoplastic fluids. SIAM J. Appl. Math., 38, 275-281. [39] Chayes, J. T., Osher, S. J., and Ralston, J. V. (1993). On singular diffusion equations with applications to self-organized criticality. Comm. Pure Appl. Math., 46, 1363-1377. [40] Chen, H. (1997). On a singular nonlinear elliptic equation. Nonlinear Anal., T.M.A., 29, 337-345.

[41] Cheng, K. S. and Ni, W. M. (1992). On the structure of the conformal scalar curvature equation on 1RN. Indiana Univ. Math. J., 41, 261-278. [42] Choi, Y. S., Lazer, A. C., and McKenna, P. J. (1998). Some remarks on a singular elliptic boundary value problem. Nonlinear Anal., T.M.A., 3, 305-314.

[43] Choi, Y. S. and McKenna, P. J. (2000). A singular Gierer-Meinhardt system of elliptic equations. Ann. Inst. H. Poincare, Anal. Non Lineaire, 17, 503-522.

[44] Choi, Y. S. and McKenna, P. J. (2003). A singular Gierer-Meinhardt system of elliptic equations: The classical case. Nonlinear Anal., 55, 521-541. [45] Choquet-Bruhat, Y. and Leray, J. (1972). Sur le probleme de Dirichlet quasilineaire d'ordre 2. C. R. Acad. Sci. Paris, Ser. A, 274, 81-85. [46] Ciarlet, P. G.. (2002). The Finite Element Method for Elliptic Problems. SIAM Classics in Applied Mathematics, SIAM, Philadelphia. [47] Ciarlet, P. G. (2005). An Introduction to Differential Geometry with Applications to Elasticity. Springer-Verlag, Heidelberg. [48] Cirstea, F., Ghergu, M., and Radulescu, V. (2005). Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type. J. Math. Purer Appl., 84, 493-508. [49] Cirstea, F. and Radulescu, V. (2002). Blow-up boundary solutions of semilinear elliptic problems. Nonlinear Anal., T.M.A., 48, 521-534. [50] Cirstea, F. and RAdulescu, V. (2002). Existence and uniqueness of blowup solutions for a class of logistic equations. Commun. Contemp. Math., 4, 559-586. [51] Cirstea, F. and Radulescu, V. (2002). Uniqueness of the boundary blowup solution of logistic equation with absorbtion. C. R. Acad. Sci. Paris, Ser. 1, 335, 447-452.

References

286

(52] Cirstea, F. and Radulescu, V. (2002). Entire solutions blowing up at infinity for semilinear elliptic systems. J. Math. Pures Appl., 81, 827-846. [53] Cirstea, F. and FtMulescu, V. (2003). Solutions with boundary blow-up for a class of nonlinear elliptic problems. Houston J. Math., 29, 821-829.

[54] Clement, P. and Peletier, L. A. (1979). An anti-maximum principle for second order elliptic operators. J. Differential Equations, 34, 218-229. [55] Coclite, M. M. and Palmieri, G. (1989). On a singular nonlinear Dirichlet problem. Comm. Partial Differential Equations, 14, 1315-1327. (56] Cohen, D. S. and Keller, H. B. (1967). Some positive problems suggested by nonlinear heat generators. J. Math. Mech., 16, 1361-1376. [57] Crandall, M. G., Rabinowitz, P. H., and Tartar, L. (1977). On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations, 2, 193-222. [58] Dancer, E. N. (1996). Some remarks on classical problems and fine properties of Sobolev spaces, Differential Integral Equations, 9, 437-446. [59] Davila, J. and Montenegro, M. (2003). Positive versus free boundary solutions to a singular equation. J. Anal. Math., 90, 303-335.

[60] de Gennes, P. G. (1985). Wetting: Statics and dynamics. Rev. of Mod. Physics, 57, 827-863. [61] del Pino, M. (1992). A global estimate for the gradient in a singular elliptic boundary value problem. Proc. R. Soc. Edinburgh Sect. A, 122, 341-352. [62] del Pino;. M. (1994). Positive solutions of a semilinear elliptic equation on a compact manifold. Nonlinear Anal., T.M.A., 22, 1423-1430. [63] del Pino, M., Kowalczyk, M., and Chen, X. (2001). The Gierer-Meinhardt system: The breaking of homoclinics and multi-bump ground states. Commun. Conternp. Math., 3, 419-439. [64] del Pino, M., Kowalczyk, M., and Wei, J. (2003). Multi-bump ground states of the Gierer-Meinhardt system in R2. Ann. Inst. H. Poincare, Anal. Non Lineaire, 20, 53-85. [65] Diaz, G. and Letelier, R. (1993). Explosive solutions of quasilinear elliptic equations: Existence and uniqueness. Nonlinear Anal., T.M.A., 20, 97-125. [66] Diaz, J. I. (1986). Nonlinear Partial Differential Equations and Free Boundaries. Elliptic Equations. Pitman Adv. Publ., Boston. [67] Diaz, J. I., Morel, J. M., and Oswald, L. (1987). An elliptic equation with

singular nonlinearity. Comm. Partial Differential Equations, 12, 13331344.

[68] Donsker, M. and Varadhan, S. R. S. (1976). On the principal eigenvalue of second-order elliptic differential operators. Comm. Pure Appl. Math., 29, 595-621. [69] Doob, J. L. (2001). Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, Berlin. [70] Du, Y. and Huang, Q. (1999). Blow-up solutions for a class of semilinear elliptic and parabolic equations. SIAM J. Math. Anal., 31, 1-18.

[71] Dumont, S., Dupaigne, L., Goubet, O., and Radulescu, V. (2007). Back.

References

287

to the Keller-Osserman condition for boundary blow-up solutions. Adv. Nonlinear Studies, 7, 271-298. [72] Dupaigne, L., Ghergu, M., and Radulescu, V. (2007). Lane-Emden-Fowler equations with convection and singular potential. J. Math. Purrs Appl., 87, 563-581.

[73] Dynkin, E. B. (2002). Diffusions, Superdiffusions, and Partial Differential Equations. Colloquium Publications, vol. 50. American Mathematical Society, Providence, RI. [74] Edelson, A. (1989). Entire solutions of singular elliptic equations. J. Math. Anal. Appl., 139, 523-532. [75] Ekeland, I. (1974). On the variational principle. J. Math. Anal. Appl., 47; 324-353. [76] Emden, R. (1907). Die Gaskiigeln. Teubner, Leipzig. [77] Fowler, R. H. (1931). Further studies of Emden and similar differential equations. Quart. J. Math., 2, 259-288. [78] Fowler, A. C. (1997). Mathematical Models in the Applied Sciences. Cambridge University Press, Cambridge. [79] Fraile, J. M., Koch Medina, P., Ldpez-Gdmez, J., and Merino, S. (1996). Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation. J. Differential Equations, 127, 295-319. [80] Frank-Kamenetskii, D. A. (1955). Diffusion and Heat Exchange in Chemical Kinetics. Princeton University Press, Princeton. [81] Fulks, W. and Maybee, J. (1960). A singular nonlinear equation. Osaka

J. Math., 12, 1-19. [82] Gelfand, I. M. (1963). Some problems in the theory of quasilinear equations. Amer. Math. Soc. Transl., 29, 295-381. [83] Ghergu, M. and Radulescu, V. (2003). Sublinear singular elliptic problems with two parameters. J. Differential Equations, 195, 520-536. [84] Ghergu, M. and RAdulescu, V. (2003).. Bifurcation and asymptotics for the Lane-Emden-Fowler equation. C. R. Acad. Sci. Paris, Ser. I, 337, 259-264.

[85] Ghergu, M. and Radulescu, V. (2004). Bifurcation for a class of singular elliptic problems with quadratic convection term. C. R. Acad. Sci. Paris; Ser. I, 338, 831-836. [86] Ghergu, M. and Radulescu, V. (2005). Multiparameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with convection term. Proc. R. Soc. Edinburgh Sect. A, 135, 61-84. [87] Ghergu, M. and Radulescu, V. (2005). On a class of sublinear singular elliptic problems with convection term. J. Math. Anal. Appl., 311, 635646.

[88] Ghergu, M. and Radulescu, V. (2006). Singular elliptic problems with convection term in anisotropic media, In C. Niculescu and V. Radulescu (Eds.), Mathematical Analysis and Applications, 74-89, Amer. Inst. Phys., Melville, NY.

288

References

[89] Ghergu, M. and Radulescu, V. (2007). On the influence of a subquadratic convection term in singular elliptic problems. In O. Carja and I. Vrabie (Eds.), Applied Analysis and Differential Equations, 127-138, World Scientific, Singapore. [90] Chergu, M. and Radulescu, V. (2007). Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term. J. Math. Anal. Appl., 333, 265-273.

[91] Ghergu, M. and Radulescu, V. (2007). On a class of singular GiererMeinhardt systems arising in morphogenesis. C. R. Acad. Sci. Paris, Ser. I, 344, 163-168. [92] Ghergu, M. and Radulescu, V. (2008). A singular Gierer-Meinhardt system with different source terms. Proc. R. Soc. Edinburgh Sect. A, in press.

[93] Gidas, B., Ni, W. M., and Nirenberg, L. (1979). Symmetry and related properties via the maximum principle. Comm. Math. Phys., 68, 209-243. [94] Gierer, A. and Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik, 12, 30-39. [95] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin.

[96] Games, S. M. (1986). On a singular nonlinear elliptic problem. SIAM J. Math. Anal. 17, 1359-1369. [97] Gossez, J. P. and Omari, P. (1994). Non-ordered lower and upper solutions in semilinear elliptic problems. Comm. Partial Differential Equations, 19, 1163-1184. [98] Granas, A. and Dugundji, J. (2003). Fixed Point Theory. Springer-Verlag, New York.

[99] Gui, C. and Lin, F. H. (1993). Regularity of an elliptic problem with a singular nonlinearity. Proc. R. Soc. Edinburgh Sect. A, 123, 1021-1029. [100] Habets, P. and Omari, P. (1996). Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order. Topol. Methods Nonlinear Anal., 8, 25-56. [101] Haitao, Y. (2003). Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J. Differential Equations, 189, 487-512. [102] Hernindez, J. and Mancebo, F. J. (2006). Singular elliptic and parabolic problems. In M. Chipot and P. Quittner (Eds.), Handbook of Differential Equations: Stationary Partial Differential Equations, 317-400, Elsevier, North-Holland. [103] Hern indez, J., Mancebo, F. J., and Vega, J. M. (2005). Nonlinear singular elliptic problems: Recent results and open problems. In H. Brezis et al. (Eds.), Progress in Nonlinear Differential Equations and Their Applications, 227-242, Birkhi user, Basel. [104] Hernandez, J., Mancebo, F. J., and Vega, J. M. (2002). On the linearization of some singular elliptic problems. Ann. Inst. H. Poincare, Anal. Non Lineaire, 19, 777-813.

References

289

[105] Hess, P. (1981). An anti-maximum principle for linear elliptic equations with an indefinite weight function. J. Differential Equations, 41, 369-374. [106] Hess, P. (1991). Periodic-Parabolic Boundary Value Problems and Positiuity. Longman Scientific and Technical, Harlow, UK. (107] Hess, P. and Kato, T. (1980). On some linear and nonlinear eigenvalue problems with indefinite weight function. Comm. Partial Differential Equations, 5, 999-1030. [108] Hopf, E. (1927). Elementare Bemerkungen caber die Losungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitz. Ber. Preuf3. Akad. Wissensch. Berlin, 19, 147-152. [109] Huiqiang, J. (2006). Global existence of solutions of an activator-inhibitor system. Discrete Contin. Dyn. Syst., 14, 737-751.

[110] Iscoe, I. (1998). On the supports of measure-valued critical branching Brownian motions, Ann. Probab., 16, 200-221. [111] Jacobsen, J. and Schmitt, K. (2002). The Liouville-Bratu-Gelfand problem for radial operators. J. Differential Equations, 184, 283-298. [112] Joseph, D. D. and Lundgren, T. S. (1973). Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal., 49, 241-269.

[113] Karamata J.(1933). Sur un mode de croissance reguliere de fonctions. Theoremes fondamentaux. Bull. Soc. Math. France, 61, 55-62. [114] Kazdan, J. L. and Warner, F. W. (1975). Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math., 28, 567-597. [115] Kazdan, J. L. and Warner, F. W. (1975). Scalar curvature and conformal deformation of Riemannian structure. J. Differential Geometry, 10, 113134.

[116] Keener, J. P. (1978). Activators and inhibitors in pattern formation. Stud. Appl. Math., 59, 1-23. [117] Keller, J. B. (1956). Electrohydrodynamics. I. The equilibrium of a charged gas in a container. J. Rational Mech. Anal., 5, 715-724. [118] Keller, J. B. (1957). On solution of Au = f (u). Comm. Pure AppI. Math., 10, 503-510. (119] Kim, E. H. (2004). A class of singular Gierer-Meinhardt systems of elliptic boundary value problems. Nonlinear Anal., 59, 305-318. [120] Kim, E. H. (2005). Singular Gierer-Meinhardt systems of elliptic boundary value problems. J. Math. Anal. Appl., 308, 1-10. [121] Krantz, S. G. and Parks, H. P. (2002). The Implicit Function Theorem. History, Theory, and Applications. Birkbauser, Boston. [122] Kusano, T. and Swanson, C. A. (1985). Entire positive solutions of singular semilinear elliptic equations. Japan J. Math., 11, 145-155. [123] Ladyzhenskaya, O. A. and Ural'tseva, N. N. (1968). Linear and Quasilinear Elliptic Equations. Academic Press, New York. [124] Lair, A. V. and Shaker, A. W. (1996). Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl., 200, 498-505.

290

References

[125] Lair, A. V. and Shaker, A. W. (1997). Classical and weak solutions of a singular semilinear elliptic problem. J. Math. Anal. Appl., 211, 371-385. [126] Lair, A. V. and Shaker, A. W. (2000). Existence of entire large positive solutions of semilinear elliptic systems. J. Differential Equations, 164, 380394.

[127] Lazer, A. C. and McKenna, P. J. (1991). On a singular nonlinear elliptic boundary value problem. Proc. Amer. Math. Soc., 3, 721-730. [128] Lazer, A. C. and McKenna, P. J. (1993). On a problem of Bieberbach and Rademacher. Nonlinear Anal., T.M.A., 21, 327-335. [129] Lazer, A. C. and McKenna, P. J. (1994). Asymptotic behavior of solutions of boundary blow-up problems. Differential and Integral Equations, 7, 1001-1020.

[130] Le Gall, J. F. (1997). A probabilistic Poisson representation for positive

solutions Au = u2 in a planar domain. Comm. Pure Appl. Math., 50, 69-103. [131] Leray, J. and Schauder, J. P. (1934). Topologie et equations fonctionnelles. Ann. Sci. Ec. Norm. Sup., 51, 43-78. [132] Liouville, J. (1853). Sur 1'equation aux derivees partielles a2 log a/aua f 2a2A = 0. J. Math. Pares Appl., 18, 71-72. [133] Loewner, C. and Nirenberg, L. (1974). Partial differential equations invariant under conformal or projective transformations. In L.V. Ahlfors et al. (Eds.), Contributions to Analysis, 245-272, Academic Press, New York.

[134] Luning, C. D. and Perry, W. L. (1984). An interactive method for solution of a boundary value problem in non-Newtonian fluid flow. J. NonNewtonian Fluid Mech., 15, 145-154. [135] Marcus, M. (1992). On solutions with blow-up at the boundary for a class of semilinear elliptic equations. In G. Buttazzo et al. (Eds.), Developments in Partial Differential Equations and Applications to Mathematical Physics, 65-77, Plenum Press, New York. [136] Marcus, M. and Veron, L. (1997). Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations. Ann. Inst. H_ Poincare Anal. Non Lineaire, 14, 237-274. [137] McKenna, P. J., and Reichel, W. (2001). Sign-changing solutions to singular second-order boundary value problems. Adv. Differential Equations, 6, 441-460. [138] Meadows, A. (2004). Stable and singular solutions of the equation Au =

1/u. Indiana Univ. Math. J., 53, 1681-1703. [139] Meinhardt, H. (1976). Morphogenesis of lines and nets. Differentiation, 6, 117-123. [140] Meinhardt, H. and Gierer, A. (1980). Generation and regeneration of sequence of structures during morphogenesis. J. Theoret. Biol., 85, 429-450. [141] Minakshisundaran, S. and Pleijel, A. (1949). Some properties of the eigenfunctions of the Laplace operator on Riemann manifolds. Can. J. Math., 1, 242-256.

References

291

[142] Mironescu, P. and Radulescu, V. (1993). A bifurcation problem associated to a convex, asymptotically linear function. C. R. Acad. Sci. Paris, Ser. I, 316, 667-672. [143] Mironescu, P. and Radulescu, V. (1996). The study of a bifurcation problem associated to an asymptotically linear function. Nonlinear Anal., 26, 857-875. [144] Miyamoto, Y. (2006). An instability criterion for activator-inhibitor systems in a two-dimensional ball. J. Differential Equations, 229, 494-508. [145] Murray, J. D. (2003). Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, New York. [146] Nagumo, M. (1937). Ueber die Differentialgleichung y" = f (x, y, y'), Proc. Phys. Math. Soc. Japan, 19, 861-866. [147] Nash, J. (1956). The imbedding problem for Riemannian manifolds. Ann. Math. 63, 20-63. [148] Ni, W. M. (1998). Difussion, cross-difusion, and their spike-layer steady states. Notices Amer. Math. Soc., 45, 9-18. [149] Ni, W. M. (2004). Diffusion and cross-diffusion in pattern formation. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 15, 197-214. [150] Ni, W. M., Suzuki, K., and Takagi, I. (2006). The dynamics of a kynetics activato-inhibitor system. J. Differential Equations, 229, 426-465.

[151] Ni, W. M. and Wei, J. (2006). On positive solutions concentrating on spheres for the Gierer-Meinhardt system. J. Differential Equations, 221, 158-189. [152] O'Reagan, D. (1997). Existence Theory for Nonlinear Ordinary Differential Equations. Kluwer Academic Publishers, Dordrecht.

[153] O'Regan, D. and Precup, R. (2001). Theorems of Leray-Schauder Type and Applications. Gordon and Breach Science Publishers, Amsterdam. [154] Osserman, R. (1957). On the inequality Du > f (u). Pacific J. Math., 7, 1641-1647.

[155] Ouyang, T. (1992). On the positive solutions of semilinear equations Du+ Au - hup = 0 on the compact manifolds. Trans. Amer. Math. Soc., 331, 503-527.

[156] Ouyang, T., Shi J., and Yao, M. Exact multiplicity of positive solutions for a singular equation in unit ball, preprint. [157] Palais, R. (1966). Ljusternik-Schnirelmann theory on Banach manifolds. Topology, 5, 115-132.

[158] Palais, R. and Smale, S. (1964). A generalized Morse theory. Bull. Amer. Math. Soc., 70, 165-171. [159] Perron, O. (1923). Eine neue Behandlung der Randwertaufgabve fur Du = 0. Math. Z., 18, 42-54. [160] Perry, W. L. (1984). A monotone iterative technique for solution of pth order (p < 0) reaction-diffusion problems in permeable catalysis. J. Comput. Chemistry, 5, 353-357.

292

References

[161] Phelps, R. R. (1974). Support cones in Banach spaces and their applications. Adv. Math., 13, 1-19. (162] Pleijel, A. (1950). On the eigenvalues and eigenfunctions of elastic plates. Comm. Pure Appl. Math., 3, 1-10. (163] Pohozaev, S. (1965). On the eigenfunctions of the equation Au+Af (u) = 0. Dokl. Akad. Nauk.SSSR, 165, 36-39. [164] Poincare, H. (1890). Sur les equations aux derivees partielles de la physique mathematique. Amer. J. Math., 12, 211-294. [165] Poincare, H. (1894). Sur les equations de la physique mathematique. Rend. Circ. Mat. Palermo, 8, 57-155.

[166] Poincare, H. (1898). Les fonctions fuchsiennes et 1'equation Au = e". J. Math. Pures Appl., 4, 137-230. [167] Pucci, P. and Serrin, J. (1984). Extensions of the mountain pass theorem. J. Funct. Anal., 59, 185-210. [168] Pucci, P. and Serrin, J. (1986). A general variational identity. Indiana Univ. Math. J., 35, 681-703.

[169] Pucci, P. and Serrin, J. (2000). A note on the strong maximum principle for singular elliptic inequalities. J. Math. Pures Appl., 79, 57-71. (170] Pucci, P. and Serrin, J. (2004). The strong maximum principle revisited. J. Differential Equations, 196, 1-66. [171] Pucci, P. and Serrin, J. (2006). Dead cores and bursts for quasilinear singular elliptic equations. SIAM J. Math. Anal., 38, 259-278. [172] Pucci, P. and Serrin, J. (2007). Maximum principles for elliptic partial differential equations. In M. Chipot (Ed.), Handbook of Differential Equations: Stationary Partial Differential Equations, 355-482, Elsevier, NorthHolland. [173] Rademacher, H. (1943). Einige besondere problem partieller Differentialgleichungen. Die differential and Integralgleichungen der Mechanik and Physik I, 838-845, Rosenberg, New York. [174] Radulescu, V. (2005). Bifurcation and asymptotics for elliptic problems with singular nonlinearity. In C. Bandle et al. (Eds.), Studies in Nonlinear Partial Differential Equations: In Honor of Haim Brezis, Fifth European Conference on Elliptic and Parabolic Problems: A special tribute to the work of Haim Brezis, 349-362, Birkhauser, Basel. [175] Radulescu, V. (2007). Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities. In M. Chipot (Ed.), Handbook of Differential Equations: Stationary Partial Differential Equations, 483-591, Elsevier, North-Holland. [176] Rellich, F. (1940). Darstellung der Eigenwerte von Au + Au = 0 durch ein Randintegral. Math. Z., 46,.635-636.

[177] Riesz, F. (1926) Sur les fonctions subharmoniques et leur rapport a la theorie du potentiel. Acta Math., 48, 329-343. [178] Sattinger, D. H. (1972). Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J., 21, 979-1000.

References

293

[179] Scorza Dragoni, G. (1931). Il problema dei valori ai limiti studiato in grande per gli integrals di una equazione differenziale del secondo ordine. Giornale

di Mat. (Battaglini), 69, 77-112. [180] Seneta, E. (1976). Regularly Varying Functions. Springer-Verlag, Berlin. (181] Serrin, J. (1964). Local behavior of solutions of quasi-linear equations. Acta Math., 111, 247-302.

[182] Shi, J. and Yao, M. (1998). On a singular nonlinear semilinear elliptic problem. Proc. R. Soc. Edinburgh Sect. A, 128, 1389-1401. [183] Stampacchia, G. (1966). Equations elliptiques du second ordre a coefficients discontinus. Presses de 1'Universite de Montreal, Montreal. [184] Struwe, M. (1990). Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer-Verlag, Berlin.

[185] Sullivan, F. (1981). A characterization of complete metric spaces. Proc. Amer. Math. Soc., 83, 345-346. (186] Taliaferro, S. D. (1979). A nonlinear singular boundary value problem. Nonlinear Anal., T.M.A., 3, 897-904. [187] Tarantello, G. (1992). On nonhomogeneous elliptic equations involving crit-

ical Sobolev exponent. Ann. Inst. H. Poincare, Anal. Non Lineaire, 9, 281-304. [188] Trembley, A. (1744), Memoires pour servir a l'histoire dun genre de polype d'eau douce, a bras en forme de come. Verbeek, Leiden, Netherland. [189] Turing, A. M. (1952). The chemical basis of morphogenesis. Philos. Trans. R. Soc. London B, 237, 37-72. [190] Vazquez, J. L. (1984). A strong maximum principle for some quasilinear elliptic equations. Apps. Math. Optim., 12, 191-202. [191] Veron, L. (1992). Semilinear elliptic equations with uniform blow-up on the boundary. J. Anal. Math., 59, 231-250. [192] Wei, J. (1998). On the interior spike layer solutions for some singular perturbation problems. Proc. R. Soc. Edinburgh Sect. A, 128, 849-874. [193] Wei, J. and Winter, M. (2002). Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case. J. Differential Equations, 178, 478-518. [194] Wei, J. and Winter, M. (2004). Existence and stability analysis of asym-

metric patterns for the Gierer-Meinhardt system. J. Math. Pures Appl., 83, 433-476.

[195] Widman, K. O. (1967). Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand., 21, 17-37. [196] Willgoose, G., Bras, R. L., and Rodriguez-Iturbe, I. (1991). A coupled channel network growth and hillslope evolution model. Water Resourc. Res., 27, 1671-1684. [197] Wong, J. S. W. (1975). On the generalized Emden-Fowler equation. SIAM Rev., 17, 339-360.

References

294

[198] Yanagida, E. (2002). Mini-maximizers for reaction-diffusion systems with skew-gradient structure. J. Differential Equations, 179, 311-335. [199] Yijing, W. and Shujie, L. (2003). Structure of ground state solutions of singular semilinear elliptic equations. Nonlinear Anal., 55, 399-417. [200] Zhang, Z. (1996). A remark on the existence of entire solutions of a singular semilinear elliptic problem. J. Math. Anal. Appl., 215, 579-582. [201] Zhang, Z. (2006). The existence and asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem with a convection term. Proc. It. Soc. Edinburgh Sect. A, 136, 209-222. [202] Zhang, Z. and Cheng, J. (2004). Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems. Nonlinear Anal., 57, 473484.

INDEX a priori estimates, 249, 264 absorption, 72 activation energy, 142 activator, 243, 244, 262, 263 almost critical point, 275, 280 anisotropic media, 75 antimaximum principle, 33 asymptotic behavior, 63, 153 at the boundary, 228, 244, 247, 266 with respect to parameters, 134. 158, 184, 187

Bellman function, 241 best Sobolev constant, 174 bifurcation, 93, 107, 123, 126, 129, 130, 142, 147, 157, 190, 207 bifurcation point, 158, 215 bifurcation problem, 89, 107, 126, 127, 141, 143 bifurcation theory, 32, 123 bilinear form, 10, 11 blow-up rate, 25, 26 boundary condition Dirichlet, 243, 255 Neumann, 243, 255 boundary layer, 93 boundary operator, 52 Dirichlet, 27, 33, 47, 52, 72 Neumann, 52, 72 Robin, 52, 72 Broyden method, 260

catastrophe theory, 93 Cauchy problem, 257 cellular automata, 93 cellular differentiation, 243, 261 central value, 76 central value set, 75, 76, 89 coercive bilinear form, 11 collocation method, 134 compact perturbation of the identity, 196 comparison principle, 9, 17, 33, 53, 189 for singular nonlinearities, 17, 244 cone, 197, 201, 266 continuity method, 32 continuous dependence on data, 195 convection term, 207 convex conjugate, 220

cost functional, 241 critical exponent, ] 83 point, 149, 275, 277; 280, 282 value, 278, 280, 282 dead core, 16, 17, 122 dead core solution, 34 decouplization, 263 degenerate logistic equation. 73 differential operator, 9, 11 diffraction of X-rays by crystals, 72 diffusion, 72, 262 diffusion-driven instability, 262 discount factor, 241 distribution space, 153 dual variational method, 282 dynamic programming principle, 241 eigenfunction, 8, 28, 126, 127, 206. 244, 266 eigenvalue, 8, 27, 28, 33, 34, 47, 54, 62, 71, 93, 126, 127, 147, 197-199, 265, 266, 282

-Ekeland's variational principle, 158, 169, 170, 174, 179, 189, 273, 275. 278, 279 electrostatic potential, 72 embedding compact, 7. 106, 157, 164, 196, 197 continuous, 157, 174, 184 Emden-Fowler equation, 281 energy functional, 28, 149, 158, 161 entire solution, 37 blow up, 37 equation degenerate logistic, 73 Emden-Fowler, 281 Emden-Fowler(perturbed), 282 integral, 76, 81, 86 Lane-Emden-Fowler, 122 Liouville-Gelfand, 142, 155 logistic, 23, 24, 45, 52, 73 scalar curvature, 74 exterior cone condition. 47 extremal solution, 155 extremal value, 142 fixed point, 3, 4, 263

Index

296

fluid

heterogeneous catalyst, 93

non-Newtonian, 93 viscous, 93 Frechet differentiability, 201, 205 Frank-Kamenetskii constant, 142 Frank-Kamenetskii parameter, 142 free boundary, 16 Fu ik spectrum, 34 function

ideal gas, 72 identity Pucci-Serrin, 281 Rellich-Pohozaev, 281 identity operator, 201 impermeable plate, 93

concave, 95, 211, 213, 215, 229 convex, 50, 126-129, 131, 134, 140, 143, 153, 219

FLchsian,. 32

Holder continuous, 7, 94, 99, 104, 107, 113, 117, 207, 208, 210, 215, 231, 237, 238 harmonic, 33 Lebesgue integrable, 65 locally Lipschitz, 79 lower semicontinuous, 131 measurable, 64, 65 subharmonic, 9, 20, 39 superharmonic, 9, 12, 20, 33, 131 varying regular, 65 varying slowly, 65 weakly differentiable, 10, 11 with compact support, 123 functional coercive, 159, 273, 277 continuous, 273 continuous differentiable, 275 convex, 273 lower semicontinuous, 28, 274 weak lower semicontinuous, 159, 273

Gaussian curvature, 72 Gelfand transform, 221 geomorphology, 262 Gierer-Meinhardt system, 243, 255, 262 gradient term, 94, 207, 208, 215, 221, 238, 241

Green function, 102. 264 identity, 31, 132

operator, 195-198, 201 ground-state, 238 growth condition asymptotically linear monotone case, 126, 130, 134, 140,

implicit function theorem, 125, 128, 129, 144, 198, 269-271

inequality Gronwall, 80 Holder, 133, 141; 164, 183, 186; 198 Hardy-Sobolev, 197 Poincare, 265 Sobolev, 163, 164, 184 Sobolev-Morrey, 7 inhibitor, 243, 244, 262, 263 integral equation, 76, 81, 86 interior sphere condition, 10, 33 invariance at homotopy, 251 inverse function theorem, 269

Karamata class, 66 Karamata theory, 37, 64 Keller-Osserman condition around the origin, 15, 33, 123, 231 at infinity, 37, 38, 49, 51, 73, 81, 89 strong, 73 weak, 73 l'Hospital's rule, 50, 52, 66-68, 70, 119, 120

Lane-Emden-Fowler equation, 122 Laplace operator, 12, 102, 265-267 Lebesgue measure, 161, 168, 172 lemma Brezis-Lieb, 165 Fatou, 30, 97, 132, 147, 167

linear eigenvalue problem, 126, 196, 266 linear homeomorphism, 196 linearized operator, 34, 93. 128 linearized problem, 126, 192, 199, 205 Liouville-Gelfand problem, 142, 155 local minima, 13 local minimizer, 158, 166 local minimizer (in Hl topology), 158, 161, 166, 169 logistic equation, 23, 24, 45, 52, 73. 204

143

nonmonotone case, 143, 147, 148 sublinear, 89, 99, 142, 207, 241 superlinear, 157, 158, 169, 189 Holder inequality, 133, 141 Holder regularity, 7, 20, 25, 251, 266

maximum principle, 9, 12, 22, 26, 27, 89, 208, 221, 223, 241 for weakly differentiable functions, 10 of Pucci and Serrin, 15, 16 of Stampacchia, 11, 12, 33, 34, 128, 129 of Vazquez, 12, 14, 20, 33

Index

strong, 10, 11, 22, 49, 95, 101, 102, 109, 111, 112, 131, 134, 136, 145, 194, 195, 199, 200, 216, 234, 245. 246, 255 weak, 9, 39, 41, 42, 83, 84, 232, 244, 245, 250, 260

mean value theorem, 38, 104, 159, 163, 176, 186, 202, 256 min-max characterization. 147, 199, 200 minimizing sequence, 274, 277 morphogen, 243, 262 concentration, 243, 262 morphogenesis, 243, 261 Morse index of a solution, 126 Morse theory, 277 Moser iteration technique, 183 mountain pass theorem, 143, 147, 149 Nash-Moser theory, 269 Nobel Prize in physics, 72 non-Newtonian fluid, 93 noncooperative system, 244 nonlinear diffusion, 204 normalized sequence in L2, 135, 146 numerical methods Broyden method, 260 collocation method, 134 shooting method, 33, 260

operator p-Laplace, 15, 34 compact, 196-198, 266 continuous, 195, 265 differential, 15, 194, 266 elliptic, 9 generalized mean curvature, 15 Laplace, 12, 15, 16, 33, 102, 265-267 Laplace-Beltrami, 74 mean curvature, 15 nonhomogeneous differential, 15 positive, 266 self-adjoint, 47, 266 uniformly elliptic. 10, 33 optimal stochastic control, 241 orbit, 93

Palais-Smale condition. 150, 277, 282 parabolic system, 243 partial order relation, 266, 274 partition of unity, 179, 279 pattern formation, 261 perturbation sublinear, 93, 99, 116 perturbed Emden-Fowler equation, 282 point concentration phenomenon, 263 population density, 73

297

population dynamics, 204 positive cone, 197, 201 potential electrostatic, 72 negative, 107, 231 positive, 45, 99, 239 radially symmetric, 75, 81 sign-changing. 112 singular, 195, 210, 231, 241 vanishing, 40, 46, 47 variable, 47, 73, 75, 76, 208, 266 prescribed condition at infinity, 238 property of compact support, 123 pseudogradient lemma, 278

quadratic form, 47 quantum physics, 15 reaction-diffusion system, 262 regular variation theory, 64 regular varying function, 65 regularity of solution, 7, 20, 25, 42, 43. 49, 94, 96, 102, 122, 152 regularized system, 248 returning point, 154 Riemannian geometry, 73, 74 Riemannian manifold, 74 Riemannian metric, 71 Riesz theory on compact operators, 196, 266

Schauder regularity, 20, 25, 106, 251, 266 self-adjoint operator, 47 shadow system, 263 shooting method, 33, 260 singular elliptic problem, 93, 99, 113, 116, 121-123. 190. 207: 210, 221, 238, 241

elliptic system, 243, 255, 262 nonlinearity, 19, 94, 107, 122, 140, 142. 207; 231 solution, 24

weight, 207 slowly varying function, 65 Sobolev exponent critical, 183 subcritical, 281 supercritical, 281 solid fuel ignition model, 142 solution blow-up, 24, 37, 38, 40-43, 45-47, 49, 57, 58, 60, 62-64. 68, 69, 71-73, 89 classical, 5, 45, 94, 100, 106, 113, 116, 117, 121, 122, 158. 183, 188,

Index

298

207, 208, 210, 214, 215, 219, 221, 223, 224, 226-229, 231-233, 235, 236, 241, 244, 248, 249, 251, 254, 263 dead core, 16, 23, 122, 205

degenerate, 126 entire, 42, 75, 76, 86, 88, 113, 281 entire bounded, 75, 76, 79 explosive, 24, 37 extremal, 155 finite time blow-up, 263 global, 263 ground-state, 113, 123, 238, 263

large, 37 maximal, 3, 5, 56-58, 110, 122 minimal, 3, 5, 46, 56-58, 112, 127, 140, 158, 188, 189

neutrally stable, 126 nondegenerate, 126 oscilating, 263 periodic, 263 positive, 89, 114, 122, 221

radially symmetric, 24, 39, 41, 75, 76, 78, 79, 82, 84-86, 89, 116, 117, 123, 224, 264 sign-changing, 89, 122 singular, 24, 52 stable, 126-130, 134, 198, 204, 205

stationary, 204 steady-state, 142, 263 unstable, 126, 127, 144, 147-149, 152-154 weak, 155, 157, 190, 277

source terms common, 263 different, 244 spectral radius, 266 sphere condition exterior, 10, 252 interior, 10, 33, 194 spherical average, 114, 131 stability, 93, 126, 191, 192, 204, 205 stable system, 93 starshaped domain, 281 stationary solution, 204 steady-state system, 263 Sturm-Liouville problem, 4, 12, 32 subharmonic function, 9, 20, 39 sublinear perturbation, 93 subsolution, 5, 6, 8, 12, 19, 20, 23, 24, 32. 33, 46-49, 55, 69, 96, 100, 101, 108, 109, 111, 112, 209, 215,

218, 219, 221, 223-228, 233-236, 254

successive approximations method. 32 superharmonic function, 9, 12, 20, 33 supersolution. 5, 6, 8, 12, 19, 20, 23, 25, 32, 33, 46-49, 54, 55, 69, 95, 100, 101, 108, 110, 112, 113, 115, 185, 209, 210; 215-219, 221, 223, 224, 226, 227, 232, 233, 236, 237, 246

theorem Amann, 126, 127 Ambrosetti-Rabinowitz. 143, 147, 149, 152, 277 Arzela.-Ascoli, 146, 188, 214 Bishop-Phelps, 273 Brezis-Oswald, 26, 27 Crandall-Rabinowitz, 123 Fermat, 275 implicit function, 125. 128, 129, 198, 269-271 inverse function, 269

Karamata, 65 Krein-Rutman, 197, 266 Lebesgue, 133, 159, 160, 165, 171 Leray-Schauder, 3 mean value, 104, 159, 163, 176, 186, 202, 256 mountain pass, 143, 147, 149, 277 Pucci-Serrin, 15, 16, 282 Rellich-Kondrachov, 266 Rolle, 257, 258 Schauder, 263 topological degree, 251 topological dynamics, 93 topological space, 28

uniformly charged gas, 72 uniformly elliptic operator, 9, 10 value function, 241 Van der Waals interactions, 93 vector field, 93 viscous fluid, 93 weak

solution, 155, 159, 190, 277 subsolution, 158, 159 supersolution, 158, 159 weakly differentiable function, 10, 11