Topics in Critical Point Theory

CAMBRIDGE TRACTS IN MATHEMATICS General Editors

´ S, W. FULTON, A. KATOK, B . BOL L OB A F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO 198

Topics in Critical Point Theory

CAMBRIDGE TRACTS IN MATHEMATICS GENERAL EDITORS ´ W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. BOLLOBAS, B. SIMON, B. TOTARO A complete list of books in the series can be found at www.cambridge.org/mathematics. Recent titles include the following: 166. The L´evy Laplacian. By M. N. Feller 167. Poincar´e Duality Algebras, Macaulay’s Dual Systems, and Steenrod Operations. By D. Meyer and L. Smith 168. The Cube-A Window to Convex and Discrete Geometry. By C. Zong 169. Quantum Stochastic Processes and Noncommutative Geometry. By K. B. Sinha and D. Goswami ˘ 170. Polynomials and Vanishing Cycles. By M. Tibar 171. Orbifolds and Stringy Topology. By A. Adem, J. Leida, and Y. Ruan 172. Rigid Cohomology. By B. Le Stum 173. Enumeration of Finite Groups. By S. R. Blackburn, P. M. Neumann, and G. Venkataraman 174. Forcing Idealized. By J. Zapletal 175. The Large Sieve and its Applications. By E. Kowalski 176. The Monster Group and Majorana Involutions. By A. A. Ivanov 177. A Higher-Dimensional Sieve Method. By H. G. Diamond, H. Halberstam, and W. F. Galway 178. Analysis in Positive Characteristic. By A. N. Kochubei ´ Matheron 179. Dynamics of Linear Operators. By F. Bayart and E. 180. Synthetic Geometry of Manifolds. By A. Kock 181. Totally Positive Matrices. By A. Pinkus 182. Nonlinear Markov Processes and Kinetic Equations. By V. N. Kolokoltsov 183. Period Domains over Finite and p-adic Fields. By J.-F. Dat, S. Orlik, and M. Rapoport ´ ´ and E. M. Vitale 184. Algebraic Theories. By J. Adamek, J. Rosicky, 185. Rigidity in Higher Rank Abelian Group Actions I: Introduction and Cocycle Problem. By A. Katok and V. Nit¸ica˘ 186. Dimensions, Embeddings, and Attractors. By J. C. Robinson 187. Convexity: An Analytic Viewpoint. By B. Simon 188. Modern Approaches to the Invariant Subspace Problem. By I. Chalendar and J. R. Partington 189. Nonlinear Perron–Frobenius Theory. By B. Lemmens and R. Nussbaum 190. Jordan Structures in Geometry and Analysis. By C.-H. Chu 191. Malliavin Calculus for L´evy Processes and Infinite-Dimensional Brownian Motion. By H. Osswald 192. Normal Approximations with Malliavin Calculus. By I. Nourdin and G. Peccati 193. Distribution Modulo One and Diophantine Approximation. By Y. Bugeaud 194. Mathematics of Two-Dimensional Turbulence. By S. Kuksin and A. Shirikyan ¨ 195. A Universal Construction for R-free Groups. By I. Chiswell and T. M uller 196. The Theory of Hardy’s Z-Function. By A. Ivi c´ 197. Induced Representations of Locally Compact Groups. E. Kaniuth and K. F. Taylor 198. Topics in Critical Point Theory. By K. Perera and M. Schechter 199. Combinatorics of Minuscule Representations. By R. M. Green ´ 200. Singularities of the Minimal Model Program. By J. Kollar

Topics in Critical Point Theory KANISHKA PERERA Florida Institute of Technology

MARTIN SCHECHTER University of California, Irvine

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107029668  C Kanishka

Perera and Martin Schechter 2013

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Perera, Kanishka, 1969– Topics in critical point theory / Kanishka Perera, Florida Institute of Technology; Martin Schechter, University of California, Irvine. pages cm. – (Cambridge tracts in mathematics ; 198) Includes bibliographical references and index. ISBN 978-1-107-02966-8 1. Fixed point theory. I. Schechter, Martin. II. Title. QA329.9.P47 2013 5141 .74 – dc23 2012025065 ISBN 978-1-107-02966-8 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To my wife Champa. K.P. To my wife, Deborah, our children, our grandchildren (twenty five so far), our great grandchildren (fifteen so far), and our extended family. May they all enjoy many happy years. M.S.

Contents

Preface

page ix

1

Morse theory 1.1 Introduction 1.2 Compactness conditions 1.3 Deformation lemmas 1.4 Critical groups 1.5 Minimizers 1.6 Nontrivial critical points 1.7 Mountain pass points 1.8 Three critical points theorem 1.9 Generalized local linking 1.10 p-Laplacian

1 1 9 9 18 22 22 24 24 25 25

2

Linking 2.1 Introduction 2.2 Minimax principle 2.3 Homotopical linking 2.4 Homological linking 2.5 Schechter and Tintarev’s notion of linking 2.6 Pairs of critical points with nontrivial critical groups 2.7 Nonstandard geometries

30 30 30 31 33 36 40 42

3

Applications to semilinear problems 3.1 Introduction 3.2 Local nature of critical groups 3.3 Critical groups at zero

47 47 48 50

vii

viii

Contents

3.4 Asymptotically linear problems 3.5 Problems with concave nonlinearities

57 63

4

Fuˇc´ık spectrum 4.1 Introduction 4.2 Examples 4.3 Preliminaries on operators 4.4 Variational formulation 4.5 Some estimates 4.6 Convexity and concavity 4.7 Minimal and maximal curves 4.8 Null manifold 4.9 Type II regions 4.10 Simple eigenvalues 4.11 Critical groups

67 67 69 72 74 75 77 78 93 100 101 101

5

Jumping nonlinearities 5.1 Introduction 5.2 Compactness 5.3 Critical groups at infinity 5.4 Solvability 5.5 Critical groups at zero 5.6 Nonlinearities crossing the Fuˇc´ık spectrum

107 107 109 110 115 116 122

6

Sandwich pairs 6.1 Introduction 6.2 Flows 6.3 Cohomological index 6.4 Semilinear problems 6.5 p-Laplacian problems 6.6 Anisotropic systems

125 125 125 126 131 133 136

Appendix Sobolev spaces A.1 Sobolev inequality A.2 Sobolev spaces

143 143 144

Bibliography Index

147 156

Preface

Critical point theory has become a very powerful tool for solving many problems. The theory has enjoyed significant development over the past several years. The impetus for this development is the fact that many new problems could not be solved by the older theory. There have been several excellent books written on critical point theory from various points of view; see, e.g., Berger [19], Zeidler [161], Rabinowitz [129], Mawhin and Willem [91], Chang [29, 30], Ghoussoub [56], Ambrosetti and Prodi [8], Willem [158], Chabrowski [26], Dacorogna [36], and Struwe [153] (see also Schechter [143, 144, 147], Zou and Schechter [163], and Perera et al. [113]). In this book we present more recent developments in the subject that do not seem to be covered elsewhere, including some results of the authors dealing with nonstandard linking geometries and sandwich pairs. Chapter 1 is a brief review of Morse theory in Banach spaces. We prove the first and second deformation lemmas under the Cerami compactness condition. As the variational functionals associated with applications given later in the book will only be C 1 , we discuss critical groups of C 1 -functionals. We include discussions on minimizers, nontrivial critical points, mountain pass points, and the three critical points theorem. We also give a generalized notion of local linking that yields a nontrivial critical group, which will be applied to problems with jumping nonlinearities in Chapter 5. We close the chapter with a recent result of Perera [110] on nontrivial critical groups in p-Laplacian problems. Chapter 2 is on linking. We say that subsets A, B of a Banach space E link if every C 1 -functional G on E satisfying ´8 ă a :“ sup G ď inf G “: b ă `8, B

A

and a suitable compactness condition, has a critical point u with Gpuq ě b. There are three main notions of linking; homological, homotopical, and a more ix

x

Preface

recent one introduced by Schechter and Tintarev [148]. We discuss all three and show that homological linking

ùñ

homotopical linking

ùñ

Schechter–Tintarev linking.

We also discuss some results of Schechter and Tintarev [148] on pairs of critical points produced by linking subsets and some results on their critical groups due to Perera [104]. We close with some recent results of the authors [118] that give critical points with nontrivial critical groups under nonstandard geometrical assumptions that do not involve a finite-dimensional closed loop. Chapter 3 contains applications of the Morse theoretic and linking methods of the first two chapters to semilinear elliptic boundary value problems. We discuss the local nature of critical groups and critical groups at zero. We consider asymptotically linear problems and problems with concave nonlinearities, and obtain multiple nontrivial solutions using our nonstandard linking theorems. Chapter 4 considers the Fuˇc´ık spectrum in an abstract operator setting, that includes many concrete problems arising in applications as special cases. We construct the minimal and maximal curves of the spectrum locally near the points where it intersects the main diagonal of the plane. We give a sufficient condition for the region between them to be nonempty, and show that it is free of the spectrum in the case of a simple eigenvalue. Finally we compute the critical groups in various regions separated by these curves. We compute them precisely in Type I regions, and prove a shifting theorem that gives a finite-dimensional reduction to the null manifold for Type II regions. Chapter 5 is a continuation of the previous chapter that considers problems with jumping nonlinearities in the same abstract framework. We discuss compactness and critical groups at infinity and zero. We compute critical groups in both resonant and nonresonant problems. This allows us to establish solvability in Type I regions, and obtain nontrivial solutions for nonlinearities crossing a curve of the Fuˇc´ık spectrum constructed in Chapter 4. Chapter 6 is on sandwich pairs. We say that a pair of subsets A, B of a Banach space E is a sandwich pair if every C 1 -functional G on E satisfying ´8 ă b :“ inf G ď sup G “: a ă `8, B

A

and a suitable compactness condition, has a critical point u with b ď Gpuq ď a. We construct a very general class of sandwich pairs with wide applicability

Preface

xi

using a family of flows on E and the Fadell–Rabinowitz cohomological index. We use a special case of this where the sandwich pair is a certain pair of cones to solve p-Laplacian problems. We also solve anisotropic p-Laplacian systems using another special case with a curved sandwich pair made up of certain orbits of an associated group action on a product of Sobolev spaces.

1 Morse theory

1.1 Introduction The purpose of this chapter is to introduce the reader to Morse theoretic methods used in variational problems. General references are Milnor [93], Mawhin and Willem [91], Chang [29], and Benci [17]; see also Perera et al. [113]. We begin by briefly collecting some basic results of Morse theory. These include the Morse inequalities, Morse lemma and its generalization splitting lemma, the shifting theorem of Gromoll and Meyer, and the handle body theorem. Results that are needed later in the text will be proved in subsequent sections. Let G be a real-valued function defined on a real Banach space E. We say that G is Fr´echet differentiable at u P E if there is an element G1 puq of the dual E 1 , called the Fr´echet derivative of G at u, such that ` ˘ Gpu ` vq “ Gpuq ` G1 puq, v ` op}v}q as v Ñ 0 in E, where p¨, ¨q is the duality pairing. The functional G is continuously Fr´echet differentiable on E, or belongs to the class C 1 pE, Rq, if G1 is defined everywhere and the map E Ñ E 1 , u ÞÑ G1 puq is continuous. We assume that G P C 1 pE, Rq for the rest of the chapter. Replacing G with G ´ Gp0q if necessary, we may also assume that Gp0q “ 0. The functional G is called even if Gp´uq “ Gpuq

@u P E.

Then G1 is odd, i.e., G1 p´uq “ ´G1 puq

@u P E.

We say that u is a critical point of G if G1 puq “ 0. A value c of G is a critical value if there is a critical point u with Gpuq “ c, otherwise it is a regular value. 1

2

Morse theory

We use the standard notations  ( Ga “ u P E : Gpuq ě a , Gba “Ga X Gb ,  ( K“ u P E : G1 puq “ 0 , Kab “K X Gba ,

 ( Gb “ u P E : Gpuq ď b , p “ EzK, E

K c “ Kcc

for the various superlevel, sublevel, critical, and regular sets of G. It is usually necessary to assume some sort of a “compactness condition” when seeking critical points of a functional. The following condition was originally introduced by Palais and Smale [101]: G satisfies the Palais–Smale compactness condition at the level c, or pPSqc for short, if every sequence puj q Ă E such that Gpuj q Ñ c,

G1 puj q Ñ 0,

called a pPSqc sequence, has a convergent subsequence; G satisfies pPSq if it satisfies pPSqc for every c P R, or equivalently, if every sequence such that Gpuj q is bounded and G1 puj q Ñ 0, called a pPSq sequence, has a convergent subsequence. The following weaker version was introduced by Cerami [25]: G satisfies the Cerami condition at the level c, or pCqc for short, if every sequence such that ` ˘ Gpuj q Ñ c, 1 ` }uj } G1 puj q Ñ 0, called a pCqc sequence, has a convergent subsequence; G satisfies pCq if it satisfies pCqc `for every c, ˘ or equivalently, if every sequence such that Gpuj q is bounded and 1 ` }uj } G1 puj q Ñ 0, called a pCq sequence, has a convergent subsequence. This condition is weaker since a pCqc (resp. pCq) sequence is clearly a pPSqc (resp. pPSq) sequence also. The limit of a pPSqc (resp. pPSq) sequence is in K c (resp. K) since G and G1 are continuous. Since any sequence in K c is a pCqc sequence, it follows that K c is a compact set when pCqc holds. Some of the essential tools for locating critical points are the deformation lemmas, which allow to lower sublevel sets of a functional, away from its critical set. The main ingredient in their proofs is a suitable negative pseudogradient flow, a notion due to Palais [103]: a pseudo-gradient vector field for p is a locally Lipschitz continuous mapping V : E p Ñ E satisfying G on E › 1 › › ˘2 ` ˘ `› p }V puq} ď ›G puq› , 2 G1 puq, V puq ě ›G1 puq› @u P E. Such a vector field exists, and may be chosen to be odd when G is even. The first deformation lemma provides a local deformation near a (possibly critical) level set of a functional.

1.1 Introduction

3

Lemma 1.1.1 (first deformation lemma) If c P R, C is a bounded set containing K c , δ, k ą 0, and G satisfies pCqc , then there are an ε0 ą 0 and, for each ε P p0, ε0 q, a map η P CpE ˆ r0, 1s, Eq satisfying (i) (ii) (iii) (iv) (v) (vi)

ηp¨, 0q “ id E , ηp¨, tq is a homeomorphism of E for all t P r0, 1s, c`2ε ηp¨, tq is the identity ` outside ˘ A “ Gc´2ε zNδ{3 pCq for all t P r0, 1s, }ηpu, tq ´ u} ď 1 ` }u} δ{k @pu, tq P E ˆ r0, 1s, Gpηpu, ¨qq is nonincreasing for all u P E, ηpGc`ε zNδ pCq, 1q Ă Gc´ε .

When G is even and C is symmetric, η may be chosen so that ηp¨, tq is odd for all t P r0, 1s. The first deformation lemma under the pPSqc condition is due to Palais [102]; see also Rabinowitz [126]. The proof under the pCqc condition was given by Cerami [25] and Bartolo et al. [13]. The particular version given here will be proved in Section 1.3. The second deformation lemma implies that the homotopy type of sublevel sets can change only when crossing a critical level. Lemma 1.1.2 (second deformation lemma) If ´8 ă a ă b ď `8 and G has only a finite number of critical points at the level a, has no critical values in pa, bq, and satisfies pCqc for all c P ra, bs X R, then Ga is a deformation retract of Gb zK b , i.e. there is a map η P CppGb zK b q ˆ r0, 1s, Gb zK b q, called a deformation retraction of Gb zK b onto Ga , satisfying (i) ηp¨, 0q “ id Gb zK b , (ii) ηp¨, tq|Ga “ id Ga @t P r0, 1s, (iii) ηpGb zK b , 1q “ Ga . The second deformation lemma under the pPSqc condition is due to Rothe [135], Chang [28], and Wang [157]. The proof under the pCqc condition can be found in Bartsch and Li [14], Perera and Schechter [119], and in Section 1.3. In Morse theory the local behavior of G near an isolated critical point u is described by the sequence of critical groups Cq pG, uq “ Hq pGc X U, Gc X U z tuuq,

qě0

where c “ Gpuq is the corresponding critical value, U is a neighborhood of u, and H˚ denotes singular homology. They are independent of the choice of U by the excision property.

4

Morse theory

For example, if u is a local minimizer, Cq pG, uq “ δq0 G where δ is the Kronecker delta and G is the coefficient group. A critical point u with C1 pG, uq ‰ 0 is called a mountain pass point. Let ´8 ă a ă b ď `8 be regular values and assume that G has only isolated critical values c1 ă c2 ă ¨ ¨ ¨ in pa, bq, with a finite number of critical points at each level, and satisfies pPSqc for all c P ra, bs X R. Then the Morse type numbers of G with respect to the interval pa, bq are defined by ÿ Mq pa, bq “ rank Hq pGai`1 , Gai q, q ě 0 i

where a “ a1 ă c1 ă a2 ă c2 ă ¨ ¨ ¨ . They are independent of the ai by the second deformation lemma, and are related to the critical groups by ÿ Mq pa, bq “ rank Cq pG, uq. uPKab

Writing βj pa, bq “ rank Hj pGb , Ga q, we have the following. Theorem 1.1.3 (Morse inequalities) If there is only a finite number of critical points in Gba , then q ÿ

p´1q

q´j

j “0

Mj ě

q ÿ

p´1qq´j βj ,

q ě 0,

j “0

and 8 ÿ

p´1qj Mj “

j “0

8 ÿ

p´1qj βj

j “0

when the series converge. Critical groups are invariant under homotopies that preserve the isolatedness of the critical point; see Rothe [134], Chang and Ghoussoub [27], and Corvellec and Hantoute [32]. Theorem 1.1.4 If Gt , t P r0, 1s is a family of C 1 -functionals on E satisfying pPSq, u is a critical point of each Gt , and there is a closed neighborhood U such that (i) U contains no other critical points of Gt , (ii) the map r0, 1s Ñ C 1 pU, Rq, t ÞÑ Gt is continuous, then C˚ pGt , uq are independent of t.

1.1 Introduction

5

When the critical values are bounded from below and G satisfies pCq, the global behavior of G can be described by the critical groups at infinity introduced by Bartsch and Li [14] Cq pG, 8q “ Hq pE, Ga q,

qě0

where a is less than all critical values. They are independent of a by the second deformation lemma and the homotopy invariance of the homology groups. For example, if G is bounded from below, Cq pG, 8q “ δq0 G. If G is unbounded from below, Cq pG, 8q “ Hrq´1 pGa q where Hr denotes the reduced groups. Proposition 1.1.5 If Cq pG, 8q ‰ 0 and G has only a finite number of critical points and satisfies pCq, then G has a critical point u with Cq pG, uq ‰ 0. The second deformation lemma implies that Cq pG, 8q “ Cq pG, 0q if u “ 0 is the only critical point of G, so G has a nontrivial critical point if Cq pG, 0q ‰ Cq pG, 8q for some q. Now suppose that E is a Hilbert space pH, p¨, ¨qq and G P C 2 pH, Rq. Then the Hessian A “ G2 puq is a self-adjoint operator on H for each u. When u is a critical point the dimension of the negative space of A is called the Morse index of u and is denoted by mpuq, and m˚ puq “ mpuq ` dim ker A is called the large Morse index. We say that u is nondegenerate if A is invertible. The Morse lemma describes the local behavior of the functional near a nondegenerate critical point. Lemma 1.1.6 (Morse lemma) If u is a nondegenerate critical point of G, then there is a local diffeomorphism ξ from a neighborhood U of u into H with ξ puq “ 0 such that Gpξ ´1 pvqq “ Gpuq `

1 pAv, vq , 2

v P ξ pU q.

Morse lemma in Rn was proved by Morse [95]. Palais [102], Schwartz [149], and Nirenberg [98] extended it to Hilbert spaces when G is C 3 . Proof in the C 2 case is due to Kuiper [64] and Cambini [23]. A direct consequence of the Morse lemma is the following theorem. Theorem 1.1.7 If u is a nondegenerate critical point of G, then Cq pG, uq “ δqmpuq G. The handle body theorem describes the change in topology as the level sets pass through a critical level on which there are only nondegenerate critical points.

6

Morse theory

Theorem 1.1.8 (handle body theorem) If c is an isolated critical value of G for which there are only a finite number of nondegenerate critical points ui , i “ 1, . . . , k, with Morse indices mi “ mpui q, and G satisfies pPSq, then there are an ε ą 0 and homeomorphisms ϕi from the unit disk D mi in Rmi into H such that Gc´ε X ϕi pD mi q “ G´1 pc ´ εq X ϕi pD mi q “ ϕi pBD mi q Ť and Gc´ε Y ki“1 ϕi pD mi q is a deformation retract of Gc`ε . The references for Theorems 1.1.3, 1.1.7, and 1.1.8 are Morse [95], Pitcher [124], Milnor [93], Rothe [132, 133, 135], Palais [102], Palais and Smale [101], Smale [151], Marino and Prodi [89], Schwartz [149], Mawhin and Willem [91], and Chang [29]. The splitting lemma generalizes the Morse lemma to degenerate critical points. Assume that the origin is an isolated degenerate critical point of G and 0 is an isolated point of the spectrum of A “ G2 p0q. Let N “ ker A and write H “ N ‘ N K , u “ v ` w. Lemma 1.1.9 (splitting lemma) There are a ball B Ă H centered at the origin, a local homeomorphism ξ from B into H with ξ p0q “ 0, and a map η P C 1 pB X N, N K q such that 1 pAw, wq ` Gpv ` ηpvqq, u P B. 2 Splitting lemma when A is a compact perturbation of the identity was proved by Gromoll and Meyer [57] for G P C 3 and by Hofer [60] in the C 2 case. Mawhin and Willem [90, 91] extended it to the case where A is a Fredholm operator of index zero. The general version given here is due to Chang [29]. A consequence of the splitting lemma is the following. Gpξ puqq “

Theorem 1.1.10 (shifting theorem) We have Cq pG, 0q “ Cq´mp0q p G|N , 0q

@q

where N “ ξ pB X Nq is the degenerate submanifold of G at 0. The shifting theorem is due to Gromoll and Meyer [57]; see also Mawhin and Willem [91] and Chang [29]. Since dim N “ m˚ p0q ´ mp0q, the shifting theorem gives us the following Morse index estimates when there is a nontrivial critical group. Corollary 1.1.11 If Cq pG, 0q ‰ 0, then mp0q ď q ď m˚ p0q.

1.1 Introduction

7

It also enables us to compute the critical groups of a mountain pass point of nullity at most one. Corollary 1.1.12 If u is a mountain pass point of G and dim ker G2 puq ď 1, then Cq pG, uq “ δq1 G. This result is due to Ambrosetti [4, 5] in the nondegenerate case and to Hofer [60] in the general case. Shifting theorem also implies that all critical groups of a critical point with infinite Morse index are trivial, so the above theory is not suitable for studying strongly indefinite functionals. An infinite-dimensional Morse theory particularly well suited to deal with such functionals was developed by Szulkin [155]; see also Kryszewski and Szulkin [63]. The following important perturbation result is due to Marino and Prodi [88]; see also Solimini [152]. Theorem 1.1.13 If some critical value of G has only a finite number of critical points ui , i “ 1, . . . , k and G2 pui q are Fredholm operators, then for any sufficiently small ε ą 0 there is a C 2 -functional Gε on H such that (i) }Gε ´ G}C 2 pH q ď ε, Ť (ii) Gε “ G in H z ki“1 Bε puj q, (iii) Gε has only nondegenerate critical points in Bε puj q and their Morse indices are in rmpui q, m˚ pui qs, (iv) G satisfies pPSq ùñ Gε satisfies pPSq. Here  ( Br pu0 q “ u P H : }u ´ u0 } ď r is the closed ball of radius r centered at u0 . We will write Br for Br p0q in the sequel. Returning to the setting of a C 1 -functional on a Banach space E, in many applications G has the trivial critical point u “ 0 and we are interested in finding others. The notion of a local linking introduced by Li and Liu [70, 83] is useful for obtaining nontrivial critical points under various assumptions on the behavior of G at infinity; see also Brezis and Nirenberg [21] and Li and Willem [72]. Assume that the origin is a critical point of G with Gp0q “ 0. We say that G has a local linking near the origin if there is a direct sum decomposition E “ N ‘ M, u “ v ` w with N finite dimensional

8

such that

Morse theory

$ & Gpvq ď 0, %

Gpwq ą 0,

v P N, }v} ď r w P M, 0 ă }w} ď r

for sufficiently small r ą 0. Liu [82] showed that this yields a nontrivial critical group at the origin. Proposition 1.1.14 If G has a local linking near the origin with dim N “ d and the origin is an isolated critical point, then Cd pG, 0q ‰ 0. The following alternative obtained in Perera [106] gives a nontrivial critical point with a nontrivial critical group produced by a local linking. Theorem 1.1.15 If G has a local linking near the origin with dim N “ d, Hd pGb , Ga q “ 0 where ´8 ă a ă 0 ă b ď `8 are regular values, and G has only a finite number of critical points in Gba and satisfies pCqc for all c P ra, bs X R, then G has a critical point u ‰ 0 with either a ă Gpuq ă 0,

Cd´1 pG, uq ‰ 0

0 ă Gpuq ă b,

Cd`1 pG, uq ‰ 0.

or

When G is bounded from below, taking a ă inf GpEq and b “ `8 gives the following three critical points theorem; see also Krasnosel’skii [62], Chang [28], Liu and Li [83], and Liu [82]. Corollary 1.1.16 If G has a local linking near the origin with dim N “ d ě 2, is bounded from below, has only a finite number of critical points, and satisfies pCq, then G has a global minimizer u0 ‰ 0 with Gpu0 q ă 0,

Cq pG, u0 q “ δq0 G

and a critical point u ‰ 0, u0 with either Gpuq ă 0,

Cd´1 pG, uq ‰ 0

Gpuq ą 0,

Cd`1 pG, uq ‰ 0.

or

Proposition 1.1.14, Theorem 1.1.15, and Corollary 1.1.16 will be proved under a generalized notion of local linking in Section 1.9; see also Perera [107].

1.3 Deformation lemmas

9

1.2 Compactness conditions It is usually necessary to have some “compactness” when seeking critical points of a functional. The following condition was originally introduced by Palais and Smale [101]. Definition 1.2.1 G satisfies the Palais–Smale compactness condition at the level c, or pPSqc for short, if every sequence puj q Ă E such that Gpuj q Ñ c,

G1 puj q Ñ 0,

called a pPSqc sequence, has a convergent subsequence; G satisfies pPSq if it satisfies pPSqc for every c P R, or equivalently, if every sequence such that pGpuj qq is bounded,

G1 puj q Ñ 0,

called a pPSq sequence, has a convergent subsequence. The following weaker version was introduced by Cerami [25]. Definition 1.2.2 G satisfies the Cerami condition at the level c, or pCqc for short, if every sequence such that ` ˘ Gpuj q Ñ c, 1 ` }uj } G1 puj q Ñ 0, called a pCqc sequence, has a convergent subsequence; G satisfies pCq if it satisfies pCqc for every c, or equivalently, if every sequence such that ` ˘ pGpuj qq is bounded, 1 ` }uj } G1 puj q Ñ 0, called a pCq sequence, has a convergent subsequence. This condition is weaker since a pCqc (resp. pCq) sequence is clearly a pPSqc (resp. pPSq) sequence also. Note that the limit of a pPSqc (resp. pPSq) sequence is in K c (resp. K) since G is C 1 . Since any sequence in K c is a pCqc sequence, it follows that K c is compact when pCqc holds.

1.3 Deformation lemmas Deformation lemmas allow us to lower sublevel sets of a functional, away from its critical set, and are an essential tool for locating critical points. The main ingredient in their proofs is usually a suitable negative pseudo-gradient flow, a notion due to Palais [103].

10

Morse theory

p is a locally Definition 1.3.1 A pseudo-gradient vector field for G on E p Ñ E satisfying Lipschitz continuous mapping V : E › › ›2 ` ˘ › p }V puq} ď ›G1 puq› , 2 G1 puq, V puq ě ›G1 puq› @u P E. (1.1) p When G Lemma 1.3.2 There is a pseudo-gradient vector field V for G on E. is even, V may be chosen to be odd. p there is a wpuq P E satisfying Proof For each u P E, › › › ˘2 ` ˘ `› }wpuq} ă ›G1 puq› , 2 G1 puq, wpuq ą ›G1 puq› by the definition of the norm in E 1 . Since G1 is continuous, then › › › ˘2 ` ˘ `› }wpuq} ď ›G1 pvq› , 2 G1 pvq, wpuq ě ›G1 pvq› @v P Nu

(1.2)

p of u. for some open neighborhood Nu Ă E  ( p is a metric space and hence paracompact, the open covering Nu Since E p uPE  ( p such that has a locally finite refinement, i.e. an open covering Nλ λP of E p (i) each Nλ Ă Nuλ for some uλ P E, p (ii) each u P E has a neighborhood Uu that intersects Nλ only for λ in some finite subset u of  ( (see, e.g., Kelley  [61]). ( Let ϕλ λP be a Lipschitz continuous partition of unity subordinate to Nλ λP , i.e. ` ˘ p r0, 1s vanishes outside Nλ , (i) ϕλ P Lip E, p (ii) for each u P E, ÿ ϕλ puq “ 1, (1.3) λP

where the sum is actually over a subset of u , for example, p λq distpu, EzN ϕλ puq “ ÿ . p λq distpu, EzN λP

Now V puq “

ÿ

ϕλ puq wpuλ q

λP

is Lipschitz in each Uu and satisfies (1.1) by (1.2) and (1.3).

1.3 Deformation lemmas

11

When G is even, G1 is odd and hence ´V p´uq ` is also a pseudo-gradient, ˘ and therefore so is the odd convex combination 12 V puq ´ V p´uq .  For A Ă E, let

 ( Nδ pAq “ u P E : distpu, Aq ď δ

be the δ-neighborhood of A. The following deformation lemma improves that of Cerami [25]. Lemma 1.3.3 (first deformation lemma) If c P R, C is a bounded set containing K c , δ, k ą 0, and G satisfies pCqc , then there are an ε0 ą 0 and, for each ε P p0, ε0 q, a map η P CpE ˆ r0, 1s, Eq satisfying (i) (ii) (iii) (iv) (v) (vi)

ηp¨, 0q “ id E , ηp¨, tq is a homeomorphism of E for all t P r0, 1s, c`2ε ηp¨, tq is the identity ` outside ˘ A “ Gc´2ε zNδ{3 pCq for all t P r0, 1s, }ηpu, tq ´ u} ď 1 ` }u} δ{k @pu, tq P E ˆ r0, 1s, Gpηpu, ¨qq is nonincreasing for all u P E, ηpGc`ε zNδ pCq, 1q Ă Gc´ε .

When G is even and C is symmetric, η may be chosen so that ηp¨, tq is odd for all t P r0, 1s. First we prove a lemma. Lemma 1.3.4 If c P R, N is an open neighborhood of K c , k ą 0, and G satisfies pCqc , then there is an ε0 ą 0 such that › ` ˘› inf 1 ` }u} ›G1 puq› ě k ε @ε P p0, ε0 q. uPGc`ε c´ε zN

c`ε

Proof If not, there are sequences εj Œ 0 and uj P Gc´εjj zN such that › ` ˘› 1 ` }uj } ›G1 puj q› ă k εj . Then puj q Ă EzN is a pCqc sequence and hence has a subsequence converging to some u P K c zN “ H, a contradiction.  Proof of Lemma 1.3.3 Taking k larger if necessary, we may assume that ` ˘ 1 ` }u} {k ă 1{3 @u P Nδ pCq. (1.4) By Lemma 1.3.4, there is an ε0 ą 0 such that for each ε P p0, ε0 q, `

› ˘› 1 ` }u} ›G1 puq› ě

8ε logp1 ` δ{kq

@u P A.

(1.5)

12

Morse theory

Let V be a pseudo-gradient vector field for G, g P Liploc pE, r0, 1sq satisfy g “ 0 outside A and g “ 1 on B “ Gc`ε c´ε zN2δ{3 pCq, for example, gpuq “

distpu, EzAq , distpu, EzAq ` distpu, Bq

and ηpu, tq, 0 ď t ă T puq ď 8 the maximal solution of η9 “ ´4ε gpηq

V pηq }V pηq}2

For 0 ď s ă t ă T puq, }ηpu, tq ´ ηpu, sq} ď 4ε

żt s

,

t ą 0,

ηpu, 0q “ u P E.

gpηpu, τ qq dτ }V pηpu, τ qq}

żt

gpηpu, τ qq dτ 1 s }G pηpu, τ qq} żt ` ˘ ď logp1 ` δ{kq 1 ` }ηpu, τ q} dτ ď 8ε

(1.6)

by (1.1) by (1.5)

s

ď logp1 ` δ{kq

„ż t

}ηpu, τ q ´ ηpu, sq} dτ j ` ˘ ` 1 ` }ηpu, sq} pt ´ sq , s

and integrating gives

` ˘` ˘ }ηpu, tq ´ ηpu, sq} ď 1 ` }ηpu, sq} p1 ` δ{kqt´s ´ 1 .

(1.7)

Taking s “ 0 we see that }ηpu, ¨q} is bounded if T puq ă 8, so T puq “ 8 and (i)–(iv) follow. By (1.6) and (1.1), ˘ ` ˘ pG1 pηq, V pηqq d ` Gpηpu, tqq “ G1 pηq, η9 “ ´4ε gpηq dt }V pηq}2 ď ´2ε gpηq ď 0

(1.8)

and hence (v) holds. To see that (vi) holds, let u P Gc`ε zNδ pCq and suppose that ηpu, 1q R Gc´ε . Then ηpu, tq P Gc`ε c´ε for all t P r0, 1s, and we claim that ηpu, tq R N2δ{3 pCq. Otherwise there are 0 ă s ă t ď 1 such that distpηpu, sq, Cq “ δ,

2δ{3 ă distpηpu, τ q, Cq ă δ,

τ P ps, tq,

distpηpu, tq, Cq “ 2δ{3.

1.3 Deformation lemmas

13

But, then ` ˘ δ{3 ď }ηpu, tq ´ ηpu, sq} ď 1 ` }ηpu, sq} δ{k ă δ{3 by (1.7) and (1.4), a contradiction. Thus, ηpu, tq P B and hence gpηpu, tqq “ 1 for all t P r0, 1s, so (1.8) gives Gpηpu, 1qq ď Gpuq ´ 2ε ď c ´ ε, a contradiction. When G is even and C is symmetric, A and B are symmetric and hence g is even, so η may be chosen to be odd in u by choosing V to be odd.  If we assume pPSqc instead of pCqc in Lemma 1.3.3, then (iv) can be strengthened as follows. Lemma 1.3.5 If c P R, C is a set containing K c , δ ą 0, k ą 3, and G satisfies pPSqc , then there are an ε0 ą 0 and, for each ε P p0, ε0 q, a map η P CpE ˆ r0, 1s, Eq satisfying (i) (ii) (iii) (iv) (v) (vi)

ηp¨, 0q “ id E , ηp¨, tq is a homeomorphism of E for all t P r0, 1s, ηp¨, tq is the identity outside A “ Gc`2ε c´2ε zNδ{3 pCq for all t P r0, 1s, }ηpu, tq ´ u} ď δ{k @pu, tq P E ˆ r0, 1s, Gpηpu, ¨qq is nonincreasing for all u P E, ηpGc`ε zNδ pCq, 1q Ă Gc´ε .

When G is even and C is symmetric, η may be chosen so that ηp¨, tq is odd for all t P r0, 1s. First a lemma. Lemma 1.3.6 If c P R, N is an open neighborhood of K c , k ą 0, and G satisfies pPSqc , then there is an ε0 ą 0 such that › › inf ›G1 puq› ě k ε @ε P p0, ε0 q. c`ε

uPGc´ε zN

c`ε

Proof If not, there are sequences εj Œ 0 and uj P Gc´εjj zN such that › 1 › ›G puj q› ă k εj . Then puj q Ă EzN is a pPSqc sequence and hence has a subsequence converging to some u P K c zN “ H, a contradiction. 

14

Morse theory

Proof of Lemma 1.3.5 By Lemma 1.3.6, there is an ε0 ą 0 such that for each ε P p0, ε0 q, › 1 › ›G puq› ě 8k ε @u P A. (1.9) δ Let V be a pseudo-gradient vector field for G, g P Liploc pE, r0, 1sq satisfy g “ 0 outside A and g “ 1 on B “ Gc`ε c´ε zN2δ{3 pCq, for example, gpuq “

distpu, EzAq , distpu, EzAq ` distpu, Bq

and ηpu, tq, 0 ď t ă T puq ď 8 the maximal solution of η9 “ ´4ε gpηq

V pηq }V pηq}2

Since }ηpu, tq ´ u} ď 4ε ď 8ε ď

,

żt

t ą 0,

ηpu, 0q “ u P E.

0

gpηpu, τ qq dτ }V pηpu, τ qq}

0

gpηpu, τ qq dτ }G1 pηpu, τ qq}

żt

δt k

(1.10)

by (1.1) by (1.9),

}ηpu, ¨q} is bounded if T puq ă 8, so T puq “ 8 and (i)–(iv) follow. By (1.10) and (1.1), ˘ ` ˘ d ` pG1 pηq, V pηqq Gpηpu, tqq “ G1 pηq, η9 “ ´4ε gpηq dt }V pηq}2 ď ´2ε gpηq ď 0

(1.11)

and hence (v) holds. To see that (vi) holds, let u P Gc`ε zNδ pCq and suppose that c`ε ηpu, 1q R Gc´ε . Then ηpu, tq P Gc´ε for all t P r0, 1s, and ηpu, tq R N2δ{3 pCq by (iv) since k ą 3. Thus, ηpu, tq P B and hence gpηpu, tqq “ 1 for all t P r0, 1s, so (1.11) gives Gpηpu, 1qq ď Gpuq ´ 2ε ď c ´ ε, a contradiction. When G is even and C is symmetric, A and B are symmetric and hence g is even, so η may be chosen to be odd in u by choosing V to be odd.  First deformation lemma under the pPSqc condition is due to Palais [102]; see also Rabinowitz [126]. The proof under the pCqc condition was given by Cerami [25] and Bartolo et al. [13]; see also Perera et al. [113].

1.3 Deformation lemmas

15

Lemma 1.3.3 provides a local deformation near a (possibly critical) level set of a functional. The following lemma shows that the homotopy type of sublevel sets can change only when crossing a critical level. Lemma 1.3.7 (second deformation lemma) If ´8 ă a ă b ď `8 and G has only a finite number of critical points at the level a, has no critical values in pa, bq, and satisfies pCqc for all c P ra, bs X R, then Ga is a strong deformation retract of Gb zK b , i.e. there is a map η P CppGb zK b q ˆ r0, 1s, Gb zK b q, called a strong deformation retraction of Gb zK b onto Ga , such that (i) ηp¨, 0q “ id Gb zK b , (ii) ηp¨, tq|Ga “ id Ga @t P r0, 1s, (iii) ηpGb zK b , 1q “ Ga . Proof Let V be a pseudo-gradient vector field for G and ζ pu, tq the solution of V pζ q p (1.12) ζ9 “ ´ , t ą 0, ζ pu, 0q “ u P Gb zpGa Y K b q Ă E. }V pζ q}2 Then ˘ d ` pG1 pζ q, V pζ qq 1 Gpζ pu, tqq “ ´ ď´ 2 dt 2 }V pζ q} by (1.1) and hence 1 pt ´ sq, 0 ď s ă t. (1.13) 2 Taking s “ 0 and using Gpζ pu, 0qq “ Gpuq ď b gives a T puq P p0, 2pb ´ aqs such that Gpζ pu, tqq Œ a as t Õ T puq. We set T puq “ 0 and ζ pu, 0q “ u for u P Ga . For δ ą 0, let  ( Tδ “ t P r0, T puqq : distpζ pu, tq, K a q ě δ . Gpζ pu, tqq ď Gpζ pu, sqq ´

Then

› ` ˘› mδ :“ inf 1 ` }ζ pu, tq} ›G1 pζ pu, tqq› ą 0 tPTδ

by pCq, so }ζ pu, tq ´ ζ pu, sq} ď

żt

dτ }V pζ pu, τ qq}

s

ď2

żt s

2 ď mδ

dτ }G1 pζ pu, τ qq}

żt s

` ˘ 1 ` }ζ pu, τ q} dτ,

by (1.12) by (1.1) rs, ts Ă Tδ .

(1.14)

16

Morse theory

Case 1: ζ pu, ¨q is bounded away from K a . Then Tδ “ r0, T puqq for some δ ą 0 and }ζ pu, ¨q} is bounded as in the proof of Lemma 1.3.3, so }ζ pu, tq ´ ζ pu, sq} ď C pt ´ sq,

0 ď s ă t ă T puq

(1.15)

for some constant C ą 0. Let tj Õ T puq. Taking t “ tj and s “ tk in (1.15) shows that pζ pu, tj qq is a Cauchy sequence and hence converges to some v P G´1 paqzK a . Now taking s “ tj shows that ζ pu, tq Ñ v as t Õ T puq. We set ζ pu, T puqq “ v and note that T is continuous in u in this case. Case 2: ζ pu, ¨q is not bounded away from K a . We claim that then ζ pu, tq converges to some v P K a as t Õ T puq. Since K a is a finite set, otherwise there are a v0 P K a , δ ą 0 such that the ball B3δ pv0 q contains no other points of K a , and sequences sj , tj Õ T puq, sj ă tj such that }ζ pu, sj q ´ v0 } “ δ,

δ ă }ζ pu, τ q ´ v0 } ă 2δ,

τ P psj , tj q,

}ζ pu, tj q ´ v0 } “ 2δ. But, then δ ď }ζ pu, tj q ´ ζ pu, sj q} ď C ptj ´ sj q Ñ 0 by (1.14), a contradiction. We set ζ pu, T puqq “ v and claim that T is continuous in u in this case also. To see this, suppose that uj Ñ u. We will show that T :“ lim T puj q ě T puq ě lim T puj q “: T and hence T puj q Ñ T puq. If T ă T puq, then passing to a subsequence, ζ puj , T puj qq Ñ ζ pu, T q and hence a “ Gpζ puj , T puj qqq Ñ Gpζ pu, T qq ą a, a contradiction. If T ą T puq, then for a subsequence and any t ă T puq, a “ Gpζ puj , T puj qqq ď Gpζ puj , tqq ´

1 pT puj q ´ tq 2

by (1.13), and passing to the limit gives a ď Gpζ pu, tqq ´

1 1 pT ´ tq Ñ a ´ pT ´ T puqq ă a as t Õ T puq, 2 2

again a contradiction. We will show that ζ is continuous. Then ηpu, tq “ ζ pu, t T puqq will be a strong deformation retraction of Gb zK b onto Ga .

1.3 Deformation lemmas

17

Case 1: u P Gb zpGa Y K b q, 0 ď t ă T puq. Then ζ is continuous at pu, tq by standard ODE theory. Case 2: u P Gb zpGa Y K b q, t “ T puq. Suppose that uj P Gb zpGa Y K b q, 0 ď tj ď T puj q, puj , tj q Ñ pu, T puqq, but ζ puj , tj q Û ζ pu, T puqq “: v. Then there is a δ ą 0 such that pB3δ pvqz tvuq X K a “ H and }ζ puj , tj q ´ v} ě 2δ

(1.16)

for a subsequence. Since ζ pu, sq converges to v as s Õ T puq, }ζ pu, sq ´ v} ď δ{2

(1.17)

for all s ă T puq sufficiently close to T puq. For each such s, }ζ puj , sq ´ ζ pu, sq} ď δ{2

(1.18)

for all sufficiently large j by Case 1. Taking a sequence sj Õ T puq and combining (1.17) and (1.18) gives tj ą sj and }ζ puj , sj q ´ v} ď δ

(1.19)

for a further subsequence of puj , tj q. By (1.16) and (1.19), there are sequences sj1 , tj1 Õ T puq, sj ď sj1 ă tj1 ď tj such that › › ›ζ puj , s 1 q ´ v › “ δ, j

δ ă }ζ pτ, uj q ´ v} ă 2δ, τ P psj1 , tj1 q, › › ›ζ puj , t 1 q ´ v › “ 2δ. (1.20) j

Then › › δ ď ›ζ puj , tj1 q ´ ζ puj , sj1 q› ď C ptj1 ´ sj1 q Ñ 0

(1.21)

by (1.14), a contradiction. Case 3: u P G´1 paq, t “ 0. Suppose that uj P Gb zK b , 0 ď tj ď T puj q, puj , tj q Ñ pu, 0q, but ζ puj , tj q Û ζ pu, 0q “ u. Then there is a δ ą 0 such that pB3δ puqz tuuq X K a “ H and }ζ puj , tj q ´ u} ě 2δ

(1.22)

for a subsequence. Since ζ puj , 0q “ uj Ñ u, }ζ puj , 0q ´ u} ď δ

(1.23)

for sufficiently large j . By (1.22) and (1.23), uj P Gb zpGa Y K b q, tj ą 0, and there are sequences 0 ď sj1 ă tj1 ď tj for which (1.20), and hence also (1.21), holds. Case 4: u P Ga zG´1 paq, t “ 0. Then ζ pu, 0q “ u. 

18

Morse theory

Remark 1.3.8 Note that Gpηpu, tqq ď Gpuq for all t P r0, 1s by (1.13) and hence the restriction of η to G´1 p´8, bq ˆ r0, 1s is a strong deformation retraction of G´1 p´8, bq onto Ga . The second deformation lemma under the pPSqc condition is due to Rothe [135], Chang [28], and Wang [157]. The proof under the pCqc condition was given by Bartsch and Li [14] and Perera and Schechter [119]; see also Perera et al. [113].

1.4 Critical groups In Morse theory the local behavior of G near an isolated critical point u is described by the sequence of critical groups Cq pG, uq “ Hq pGc X U, Gc X U z tuuq,

qě0

(1.24)

where c “ Gpuq is the corresponding critical value, U is a neighborhood of u, and H˚ denotes singular homology with Z2 -coefficients. They are independent of U , and hence well-defined, by the excision property. Critical groups help distinguish between different types of critical points and are extremely useful for obtaining multiple critical points of a functional (see, e.g., Chang [29]). One of the consequences of the second deformation lemma is the following proposition relating the change in the topology of sublevel sets across a critical level to the critical groups of the critical points at that level. Proposition 1.4.1 If ´8 ă a ă b ď `8 and G has only a finite number of critical points at the level c P pa, bq, has no other critical values in ra, bs, and satisfies pCqc1 for all c1 P ra, bs X R, then à Hq pGb , Ga q « Cq pG, uq @q. uPK c

In particular, rank Hq pGb , Ga q “

ÿ

rank Cq pG, uq

@q.

uPK c

Proof We have Hq pGb , Ga q « Hq pGc , Ga q « Hq pGc , Gc zK c q

(1.25)

since Gc and Ga are strong deformation retracts of Gb and Gc zK c , respectively, by Lemma 1.3.7. Taking δ ą 0 so small that the balls Bδ puq, u P K c Ť are mutually disjoint and then excising Gc z uPK c Bδ puq, we see that the last

1.4 Critical groups

19

group in (1.25) is isomorphic to à à Hq pGc X Bδ puq, Gc X Bδ puqz tuuq “ Cq pG, uq. uPK c



uPK c

For the change in the topology across multiple critical levels, we have the following. Proposition 1.4.2 If ´8 ă a ă b ď `8 are regular values and G has only a finite number of critical points in Gba and satisfies pCqc for all c P ra, bs X R, then ÿ rank Hq pGb , Ga q ď rank Cq pG, uq @q. uPKab

In particular, G has a critical point u with a ă Gpuq ă b and Cq pG, uq ‰ 0 when Hq pGb , Ga q ‰ 0. First we prove a lemma of a purely topological nature. Lemma 1.4.3 If X1 Ă ¨ ¨ ¨ Ă Xk`1 are topological spaces, then rank Hq pXk`1 , X1 q ď

k ÿ

rank Hq pXi`1 , Xi q @q.

(1.26)

i“1

Proof In the exact sequence i˚

¨ ¨ ¨ ÝÝÝÝÑ Hq pXk , X1 q ÝÝÝÝÑ Hq pXk`1 , X1 q j˚

ÝÝÝÝÑ

Hq pXk`1 , Xk q ÝÝÝÝÑ ¨ ¨ ¨

of the triple pXk`1 , Xk , X1 q, im j˚ « Hq pXk`1 , X1 q{ ker j˚ “ Hq pXk`1 , X1 q{ im i˚ and hence rank Hq pXk`1 , X1 q “ rank i˚ ` rank j˚ ď rank Hq pXk , X1 q ` rank Hq pXk`1 , Xk q. Since equality holds in (1.26) when k “ 1, the conclusion now follows by induction on k.  Proof of Proposition 1.4.2 Let c1 ă ¨ ¨ ¨ ă ck be the critical values in pa, bq and a “ a1 ă c1 ă a2 ă c2 ă ¨ ¨ ¨ ă ck´1 ă ak ă ck ă ak`1 “ b.

20

Morse theory

Applying Lemma 1.4.3 with Xi “ Gai and using Proposition 1.4.1 gives rank Hq pGb , Ga q ď

k ÿ

rank Hq pGi`1 , Gi q

i“1

“

k ÿ ÿ

rank Cq pG, uq

i“1 uPK ci

“

ÿ

rank Cq pG, uq.



uPKab

We refer to Chang and Ghoussoub [27] or Corvellec and Hantoute [32] for the proof of the following theorem, which shows that critical groups are invariant under homotopies that preserve the isolatedness of the critical point. Theorem 1.4.4 If Gt , t P r0, 1s is a family of C 1 -functionals on E and u is a critical point of each Gt , with a closed neighborhood U such that (i) Gt satisfies pPSq over U , (ii) U contains no other critical points of Gt , (iii) the map r0, 1s Ñ C 1 pU, Rq, t ÞÑ Gt is continuous, then Cq pG0 , uq « Cq pG1 , uq

@q.

When the critical values are bounded from below and G satisfies pCq, the global behavior of G can be described by the critical groups at infinity Cq pG, 8q “ Hq pE, Ga q,

qě0

(1.27)

where a is less than all critical values. They are independent of a, and hence well-defined, by Lemma 1.3.7 and the homotopy invariance of the homology groups. Let $ &1, q “ d δqd “ %0, q ‰ d be the Kronecker delta and denote by Hr reduced homology. We have the following. Proposition 1.4.5 Assume that G satisfies pCq. (i) If G is bounded from below, then Cq pG, 8q « δq0 Z2 .

1.4 Critical groups

21

(ii) If G is unbounded from below, then Cq pG, 8q « Hrq´1 pGa q @q. In particular, C0 pG, 8q “ 0. Proof This follows from Lemma 1.4.6 below since E is contractible and Ga “ H if and only if G is bounded from below.  Lemma 1.4.6 Let pX, Aq be a pair with X contractible. (i) If A “ H, then Hq pX, Aq « δq0 Z2 . (ii) If A ‰ H, then Hq pX, Aq « Hrq´1 pAq

@q.

In particular, H0 pX, Aq “ 0. Proof (i) Since A “ H and X is contractible, Hq pX, Aq “ Hq pXq « δq0 Z2 . (ii) Since A ‰ H and the reduced groups are trivial in all dimensions for a contractible space, this follows from the exact sequence ¨ ¨ ¨ ÝÝÝÝÑ Hrq pXq ÝÝÝÝÑ Hq pX, Aq ÝÝÝÝÑ Hrq´1 pAq ÝÝÝÝÑ Hrq´1 pXq ÝÝÝÝÑ

¨¨¨ .



When studying the existence and multiplicity of critical points of a functional we may often assume without loss of generality that there are only finitely many critical points. The following proposition relating the critical groups of G at infinity to those of its (finite) critical points is then immediate from Proposition 1.4.2 with b “ `8. Proposition 1.4.7 If G satisfies pCq, then ÿ rank Cq pG, uq rank Cq pG, 8q ď

@q.

uPK

In particular, G has a critical point u with Cq pG, uq ‰ 0 when Cq pG, 8q ‰ 0.

22

Morse theory

1.5 Minimizers In this section we give sufficient conditions for G to have a global minimizer and compute the critical groups at an isolated local minimizer. Proposition 1.5.1 If G is bounded from below and satisfies pCqc for c “ inf G, then G has a global minimizer. Proof If not, there are an ε ą 0 and a map η P CpE ˆ r0, 1s, Eq satisfying ηpGc`ε , 1q Ă Gc´ε by the first deformation lemma. Taking u P E with Gpuq ď c ` ε, then we have Gpηpu, 1qq ď c ´ ε ă inf G, a contradiction.  Turning to critical groups, we have the next proposition. Proposition 1.5.2 If u is an isolated local minimizer of G, then Cq pG, uq « δq0 Z2 . Proof Let c “ Gpuq. Then for sufficiently small r ą 0, Gpvq ě c

@v P Br puq

and there are no other critical points of G in Br puq. Since any v P Br puq with Gpvq “ c is then a local minimizer, Gpvq ą c,

v P Br puqz tuu ,

and hence Cq pG, uq “ Hq pGc X Br puq, Gc X Br puqz tuuq “ Hq ptuu , Hq « δq0 Z2 .  Combining Propositions 1.4.5 (i), 1.5.1, and 1.5.2 gives the following corollary. Corollary 1.5.3 If G is bounded from below, satisfies pCq, and has only a finite number of critical points, then Cq pG, 8q « δq0 Z2 and G has a global minimizer u with Cq pG, uq « δq0 Z2 .

1.6 Nontrivial critical points In many applications G has the trivial critical point u “ 0 and we are interested in finding others. We assume that G has only a finite number of critical points. When u “ 0 is the only critical point of G, Cq pG, 0q « Cq pG, 8q for all q by Proposition 1.4.1 with ´8 ă a ă 0 ă b “ `8, so we have the following.

1.6 Nontrivial critical points

23

Proposition 1.6.1 If Cq pG, 0q ­« Cq pG, 8q for some q and G satisfies pCq, then G has a critical point u ‰ 0. The following proposition is useful for obtaining a nontrivial critical point with a nontrivial critical group. Proposition 1.6.2 Assume that G satisfies pCq. (i) If Cq pG, 0q “ 0 and Cq pG, 8q ‰ 0 for some q, then G has a critical point u ‰ 0 with Cq pG, uq ‰ 0. (ii) If Cq pG, 0q ‰ 0 and Cq pG, 8q “ 0 for some q, then G has a critical point u ‰ 0 with either Gpuq ă 0 and Cq´1 pG, uq ‰ 0, or Gpuq ą 0 and Cq`1 pG, uq ‰ 0. First a purely topological lemma. Lemma 1.6.3 If X1 Ă X2 Ă X3 Ă X4 are topological spaces, then rank Hq´1 pX2 , X1 q ` rank Hq`1 pX4 , X3 q ě rank Hq pX3 , X2 q ´ rank Hq pX4 , X1 q

@q.

Proof From the exact sequence B˚

j˚

¨ ¨ ¨ ÝÝÝÝÑ Hq pX3 , X1 q ÝÝÝÝÑ Hq pX3 , X2 q ÝÝÝÝÑ Hq´1 pX2 , X1 q ÝÝÝÝÑ ¨ ¨ ¨ of the triple pX3 , X2 , X1 q, we have rank Hq pX3 , X1 q ě rank j˚ “ nullity B˚ “ rank Hq pX3 , X2 q ´ rank B˚ ě rank Hq pX3 , X2 q ´ rank Hq´1 pX2 , X1 q, and from the exact sequence B˚

i˚

¨ ¨ ¨ ÝÝÝÝÑ Hq`1 pX4 , X3 q ÝÝÝÝÑ Hq pX3 , X1 q ÝÝÝÝÑ Hq pX4 , X1 q ÝÝÝÝÑ ¨ ¨ ¨ of the triple pX4 , X3 , X1 q, rank Hq pX3 , X1 q “ rank i˚ ` nullity i˚ ď rank Hq pX4 , X1 q ` rank B˚ ď rank Hq pX4 , X1 q ` rank Hq`1 pX4 , X3 q, so the conclusion follows.



Proof of Proposition 1.6.2 (i) By Proposition 1.4.7, G has a critical point u with Cq pG, uq ‰ 0 since Cq pG, 8q ‰ 0, and u ‰ 0 since Cq pG, 0q “ 0. (ii) Let ε ą 0 be so small that zero is the only critical value in r´ε, εs and a be less than ´ε and all critical values. Since rank Hq pGε , G´ε q ě rank Cq pG, 0q

24

Morse theory

by Proposition 1.4.1 and Hq pE, Ga q “ Cq pG, 8q, applying Lemma 1.6.3 to Ga Ă G´ε Ă Gε Ă E gives rank Hq´1 pG´ε , Ga q ` rank Hq`1 pE, Gε q ě rank Cq pG, 0q ´ rank Cq pG, 8q ą 0. Then either Hq´1 pG´ε , Ga q ‰ 0, or Hq`1 pE, Gε q ‰ 0, and the conclusion follows from Proposition 1.4.2.  Remark 1.6.4 The alternative in Proposition 1.6.2 (ii) and Lemma 1.6.3 were proved by Perera [106, 107].

1.7 Mountain pass points A critical point u of G with C1 pG, uq ‰ 0 is called a mountain pass point. Since homology groups, and hence also critical groups, are trivial in negative dimensions, the special case q “ 0 of Proposition 1.6.2 (ii) reduces to the following. Corollary 1.7.1 If C0 pG, 0q ‰ 0, C0 pG, 8q “ 0, and G satisfies pCq, then G has a mountain pass point u ‰ 0 with Gpuq ą 0. This implies the well-known mountain pass lemma of Ambrosetti and Rabinowitz [9]. Indeed, if the origin is a local minimizer and G is unbounded from below, then C0 pG, 0q « Z2 by Proposition 1.5.2 and C0 pG, 8q “ 0 by Proposition 1.4.5 (ii), so Corollary 1.7.1 gives a positive mountain pass level.

1.8 Three critical points theorem Another consequence of Proposition 1.6.2 (ii) is the following corollary. Corollary 1.8.1 If Cq pG, 0q ‰ 0 for some q ě 1 and G is bounded from below and satisfies pCq, then G has a critical point u1 ‰ 0. If q ě 2, then there is a second critical point u2 ‰ 0. Proof By Corollary 1.5.3, Cq pG, 8q “ 0 and G has a global minimizer u1 with Cq pG, u1 q “ 0. Since Cq pG, 0q ‰ 0, u1 ‰ 0 and there is a critical point u2 ‰ 0 with either Cq´1 pG, u2 q ‰ 0 or Cq`1 pG, u2 q ‰ 0. When q ě 2, u2 ‰ u1 since Cq´1 pG, u1 q “ Cq`1 pG, u1 q “ 0. 

1.10 p-Laplacian

25

1.9 Generalized local linking The notion of a generalized local linking is useful for obtaining nontrivial critical groups at zero and hence nontrivial critical points via Proposition 1.6.2 (ii). Definition 1.9.1 We say that G has a generalized local linking near zero in dimension q, 1 ď q ă 8 if there are (i) a direct sum decomposition E “ N ‘ M, u “ v ` w with dim N “ q, ( (ii) r ą 0 and a homeomorphism h from C “ u P E : }v} ď r, }w} ď r onto a neighborhood U of zero such that hp0q “ 0 and G|hpCXN q ď 0 ă G|hpCXMqzt0u .

(1.28)

Proposition 1.9.2 If G has a generalized local linking near zero in dimension q, then Cq pG, 0q ‰ 0. Proof We have Cq pG, 0q “ Hq pG0 X U, G0 X U z t0uq and the commutative diagram Hq pC X N, BC X Nq

h˚ «

/ Hq phpC X N q, hpBC X N qq  Hq pG0 X U, G0 X U z t0uq

i˚

 Hq pC, CzMq

h˚ «

 / Hq phpCq, hpCzMqq

where the vertical arrows are induced by inclusions. Since C X N is a q-dimensional ball and BC X N is its boundary, Hq pC X N, BC X N q « Z2 , and since they are strong deformation retracts of C and CzM, respectively, i˚ is an isomorphism.  Remark 1.9.3 Definition 1.9.1 is a special case of the more general notion of a homological local linking introduced by Perera [107], which also yields a nontrivial critical group at zero. The strict inequality in (1.28) can be relaxed to less than or equal to; see Degiovanni et al. [46].

1.10 p-Laplacian The p-Laplacian operator

` ˘ p u “ div |∇u|p´2 ∇u ,

p P p1, 8q

26

Morse theory

arises in non-Newtonian fluid flows, turbulent filtration in porous media, plasticity theory, rheology, glacelogy, and in many other application areas; see, e.g., Esteban and V´azquez [50] and Padial et al. [100]. Problems involving the p-Laplacian have been studied extensively in the literature during the Past 50 years. In this section we present a result on nontrivial critical groups associated with the p-Laplacian obtained in Perera [110]; see also Dancer and Perera [41]. Consider the nonlinear eigenvalue problem # ´ p u “ λ |u|p´2 u in (1.29) u“0 on B ` ˘ where is a bounded domain in Rn , n ě 1 and p u “ div |∇u|p´2 ∇u is the p-Laplacian of u, p P p1, 8q. The set σ p´ p q of eigenvalues is called the Dirichlet spectrum of ´ p on . It is known that the first eigenvalue λ1 is positive, simple, has an associated eigenfunction that is positive in , and is isolated in the spectrum; see Anane [10] and Lindqvist [80, 81]. So the second eigenvalue λ2 “ inf σ p´ p q X pλ1 , 8q is also defined; see Anane and Tsouli [11]. In the ODE case n “ 1, where is an interval, the spectrum consists of a sequence of simple eigenvalues λk Õ 8 and the eigenfunction associated with λk has exactly k ´ 1 interior zeroes; see Cuesta [34] or Dr´abek [48]. In the semilinear PDE case n ě 2, p “ 2 also σ p´ q consists of a sequence of eigenvalues λk Õ 8, but in the quasilinear PDE case n ě 2, p ‰ 2 a complete description of the spectrum is not available. Eigenvalues of (1.29) are the critical values of the C 1 -functional ż  ( 1, p I puq “ |∇u|p , u P S “ u P W “ W0 p q : }u}Lp p q “ 1 ,

which satisfies pPSq. Denote by A the class of closed symmetric subsets of S and by  ( γ ` pAq “ sup k ě 1 : D an odd continuous map S k´1 Ñ A ,  ( γ ´ pAq “ inf k ě 1 : D an odd continuous map A Ñ S k´1 the cogenus and the genus of A P A, respectively, where S k´1 is the unit sphere in Rk . Then λ˘ k “

inf

sup I puq,

APA uPA γ pAqěk

kě1

˘

are two increasing and unbounded sequences of eigenvalues (see Dr´abek and Robinson [49]), but, in general, ` it˘ is not known whether either sequence is a complete list. The sequence λ` was introduced by Dr´abek and Robinson k [49]; γ ´ is also called the Krasnosel’skii genus [62].

1.10 p-Laplacian

27

Solutions of (1.29) are the critical points of the functional ż 1, p Iλ puq “ |∇u|p ´ λ |u|p , u P W0 p q.

When λ R σ p´ p q, the origin is the only critical point of Iλ and hence the critical groups Cq pIλ , 0q are defined. Again we take the coefficient group to be Z2 . The following theorem is our main result on them. Theorem 1.10.1 ([110, Proposition 1.1]) The spectrum of ´ p contains a ` sequence of eigenvalues λk Õ 8 such that λ´ k ď λk ď λk and λ P pλk , λk`1 qzσ p´ p q ùñ Ck pIλ , 0q ‰ 0.

(1.30)

The eigenvalues λk are defined using the Yang index, whose definition and some properties we recall below. Various applications of this sequence of eigenvalues can be found in Perera [111, 112], Liu and Li [84], Perera and Szulkin [123], Cingolani and Degiovanni [31], Guo and Liu [58], Degiovanni and Lancelotti [44, 45], Tanaka [156], Fang and Liu [52], Medeiros and Perera [92], Motreanu and Perera [96], and Degiovanni et al. [46]. It is not known ` ˘ whether (1.30) holds for the sequences λ˘ . k Yang [160] considered compact Hausdorff spaces with fixed-point-free conq tinuous involutions and used the Cech homology theory, but for our purposes here it suffices to work with closed symmetric subsets of Banach spaces that do not contain the origin and singular homology groups. Following [160], we first construct a special homology theory defined on the category of all pairs of closed symmetric subsets of Banach spaces that do not contain the origin and all continuous odd maps of such pairs. Let pX, Aq, A Ă X be such a pair and CpX, Aq its singular chain complex with Z2 -coefficients, and denote by T# the chain map of CpX, Aq induced by the antipodal map T puq “ ´u. We say that a q-chain c is symmetric if T# pcq “ c, which holds if and only if c “ c1 ` T# pc1 q for some q-chain c1 . The symmetric q-chains form a subgroup Cq pX, A; T q of Cq pX, Aq, and the boundary operator Bq maps Cq pX, A; T q into Cq´1 pX, A; T q, so these subgroups form a subcomplex CpX, A; T q. We denote by  ( Zq pX, A; T q “ c P Cq pX, A; T q : Bq c “ 0 ,  ( Bq pX, A; T q “ Bq`1 c : c P Cq`1 pX, A; T q , Hq pX, A; T q “ Zq pX, A; T q{Bq pX, A; T q the corresponding cycles, boundaries, and homology groups. A continuous odd map f : pX, Aq Ñ pY, Bq of pairs as above induces a chain map f# : CpX, A; T q Ñ CpY, B; T q and hence homomorphisms f˚ : Hq pX, A; T q Ñ Hq pY, B; T q.

28

Morse theory

For example, Hq pS k ; T q “

$ &Z2 ,

0ďqďk

%0,

q ě k`1

(see [160, example 1.8]). Let X be as above, and define homomorphisms ν : Zq pX; T q Ñ Z2 inductively by $ &Inpcq, q “ 0 νpzq “ %νpBcq, q ą 0 ř if z “ c ` T# pcq, where the index of a 0-chain c “ i ni σi is defined by ř Inpcq “ i ni . As in [160], ν is well-defined and ν Bq pX; T q “ 0, so we can define the index homomorphism ν˚ : Hq pX; T q Ñ Z2 by ν˚ przsq “ νpzq. If F is a closed subset of X such that F Y T pF q “ X and A “ F X T pF q, then there is a homomorphism : Hq pX; T q Ñ Hq´1 pA; T q such that ν˚ p rzsq “ ν˚ przsq (see [160, proposition 2.8]). Taking F “ X we see that if ν˚ Hk pX; T q “ Z2 , then ν˚ Hq pX; T q “ Z2 for 0 ď q ď k. We define the Yang index of X by  ( ipXq “ inf k ě ´1 : ν˚ Hk`1 pX; T q “ 0 , taking inf H “ 8. Clearly, ν˚ H0 pX; T q “ Z2 if X ‰ H, so ipXq “ ´1 if and only if X “ H. For example, ipS k q “ k (see [160, Example 3.4]). Proposition 1.10.2 ([160, proposition 2.4]) If f : X Ñ Y is as above, then ν˚ pf˚ przsqq “ ν˚ przsq for rzs P Hq pX; T q, and hence ipXq ď ipY q. In particular, this inequality holds if X Ă Y . Thus, k ` ´ 1 ď ipXq ď k ´ ´ 1 if there are odd continuous maps S k ´ X Ñ S k ´1 , so γ ` pXq ď ipXq ` 1 ď γ ´ pXq.

`

´1

Ñ

(1.31)

Proposition 1.10.3 ([110, proposition 2.6]) If ipXq “ k ě 0, then Hrk pXq ‰ 0. Proof We have ν˚ Hq pX; T q “

$ &Z2 ,

0ďqďk

%0,

q ě k ` 1.

1.10 p-Laplacian

29

We show that if rzs P Hk pX; T q is such that ν˚ przsq ‰ 0, then rzs ‰ 0 in Hrk pXq. Arguing indirectly, assume that z P Bk pXq, say, z “ Bc. Since z P Bk pX; T q, T# pzq “ z. Let c1 “ c ` T# pcq. Then c1 P Zk`1 pX; T q since Bc1 “ z ` T# pzq “ 2z “ 0 mod 2, and ν˚ prc1 sq “ νpc1 q “ νpBcq “ νpzq ‰ 0, contradicting ν˚ Hk`1 pX; T q “ 0.  Lemma 1.10.4 We have Cq pIλ , 0q « Hrq´1 pI λ q @q.  ( Proof Taking U “ u P W : }u}Lp p q ď 1 in (1.24) gives Cq pIλ , 0q “ Hq pIλ0 X U, Iλ0 X U z t0uq. Since Iλ is positive homogeneous, Iλ0 X U radially contracts to the origin via pIλ0 X U q ˆ r0, 1s Ñ Iλ0 X U,

pu, tq ÞÑ p1 ´ tq u

and Iλ0 X U z t0u deformation retracts onto Iλ0 X S via pIλ0 X U z t0uq ˆ r0, 1s Ñ Iλ0 X U z t0u ,

pu, tq ÞÑ p1 ´ tq u ` t u{ }u}Lp p q ,

so it follows from the exact sequence of the pair pIλ0 X U, Iλ0 X U z t0uq that Hq pIλ0 X U, Iλ0 X U z t0uq « Hrq´1 pIλ0 X Sq. Since Iλ |S “ I ´ λ, Iλ0 X S “ I λ .



We are now ready to prove Theorem 1.10.1. Proof of Theorem 1.10.1 Set λk “

inf

sup I puq,

APA uPA ipAqěk´1

k ě 1.

Then pλk q is an increasing sequence of critical values of I , and hence eigenvalues of ´ p , by a standard deformation argument (see [110, proposition 3.1]). ` By (1.31), λ´ k ď λk ď λk , in particular, λk Ñ 8. Let λ P pλk , λk`1 qzσ p´ p q. By Lemma 1.10.4, Ck pIλ , 0q « Hrk´1 pI λ q, and λ I P A since I is even. Since λ ą λk , there is an A P A with ipAq ě k ´ 1 such that I ď λ on A. Then A Ă I λ and hence ipI λ q ě ipAq ě k ´ 1 by Proposition 1.10.2. On the other hand, ipI λ q ď k ´ 1 since I ď λ ă λk`1 on I λ . So ipI λ q “ k ´ 1 and hence Hrk´1 pI λ q ‰ 0 by Proposition 1.10.3. 

2 Linking

2.1 Introduction The purpose of this chapter is to introduce the reader to linking methods used in variational problems. General references are Rabinowitz [129], Struwe [153], Chang [29], Benci [17], and Schechter [143]; see also Perera et al. [113]. Our focus here will be the notion of homological linking, in particular, we will obtain pairs of critical points with nontrivial critical groups. Nonstandard geometries without a finite-dimensional closed loop will also be considered.

2.2 Minimax principle Minimax principle originated in the work of Ljusternik and Schnirelmann [85] and is a useful tool for finding critical points of a functional. By Lemma 1.3.3, if c is a regular value, G satisfies pCqc , and ε ą 0 is sufficiently small, there is a map η P CpE ˆ r0, 1s, Eq such that, writing ηpu, tq “ ηptq u, (i) ηp0q “ id E , (ii) ηptq is a homeomorphism of E for all t P r0, 1s, and the mapping η´1 : E ˆ r0, 1s Ñ E, pu, tq ÞÑ ηptq´1 u is continuous, (iii) ηptq is the identity on EzGc`2ε c´2ε for all t P r0, 1s, (iv) Gpηp¨q uq is nonincreasing for each u P E, (v) ηp1q Gc`ε Ă Gc´ε , and ` ˘ }ηptq u ´ u} ď 1 ` }u} {2 30

@pu, tq P E ˆ r0, 1s.

2.3 Homotopical linking

31

Then

› › ` ˘ }ηptq u} ď 1 ` 3 }u} {2, ›η´1 ptq u› ď 1 ` 2 }u}

@pu, tq P E ˆ r0, 1s,

so we have (vi) for each bounded subset A of E, ´ › ›¯ sup }ηptq u} ` ›η´1 ptq u› ă 8. pu,tqPAˆr0,1s

Denote by Dc, ε the set of all maps η P CpE ˆ r0, 1s, Eq satisfying (i)–(vi). We say that a family F of subsets of E is invariant under Dc, ε if M P F , η P Dc, ε ùñ ηp1q M P F . Proposition 2.2.1 (minimax principle) If F is a family of subsets of E, c :“ inf sup Gpuq MPF uPM

is finite, F is invariant under Dc, ε for all sufficiently small ε ą 0, and G satisfies pCqc , then c is a critical value of G. Proof If not, taking ε ą 0 sufficiently small, M P F with sup GpMq ď c ` ε, and η P Dc, ε , we have ηp1q M P F and sup Gpηp1q Mq ď c ´ ε by (v), contradicting the definition of c.  Some references for Proposition 2.2.1 are Palais [103], Nirenberg [99], Rabinowitz [129], Schechter and Tintarev [148], Ghoussoub [56], and Schechter [142, 143]. Minimax methods were introduced in Morse theory by Marino and Prodi [89]; see also Liu [82].

2.3 Homotopical linking The notion of homotopical linking is useful for obtaining critical points via the minimax principle. Definition 2.3.1 Let D be a subset of E homeomorphic to the unit disk in Rn for some n ě 1, A the relative boundary of D, B a nonempty subset of E disjoint from A, and  (  “ h P CpD, Eq : h|A “ id A . We say that A homotopically links B if hpDq X B ‰ H

@h P .

(2.1)

32

Linking

Some standard examples are the following. Proofs are postponed till the next section, where we will show that these sets link homologically and that homological linking implies homotopical linking. Example 2.3.2 If u0 P E, U is a bounded neighborhood of u0 , and u1 R U , then A “ tu0 , u1 u homotopically links B “ BU . Example 2.3.3 If E “ N ‘ M, u “ v ` w is adirect sum decomposition with ( N nontrivial and finite dimensional, then A “ v P N : }v} “ R homotopically links B “ M for any ( R ą 0. In particular, if E is finite dimensional, then A “ u P E : }u} “ R homotopically links B “ t0u. Example 2.3.4  If M and N (are as above and w0 P M with }w0 } “ 1, ( then A “ v P N : }v} ď R Y u “ v ` ( tw0 : v P N, t ě 0, }u} “ R homotopically links B “ w P M : }w} “ r for any 0 ă r ă R. Theorem 2.3.5 If A homotopically links the closed set B, c :“ inf max Gphpuqq, hP uPD

a :“ sup GpAq ď inf GpBq “: b, and G satisfies pCqc , then c ě b and is a critical value of G. If c “ b, then G has a critical point with critical value c on B. Proof By (2.1), c ě b. First suppose that c ą b, and let 2ε ă c ´ a. Then for any η P Dc, ε , ηptq is the identity on A for all t P r0,  1s by (iii) in the ( definition of Dc, ε , so ηp1q ˝ h P  whenever h P . So F “ hpDq : h P  is invariant under Dc, ε , and hence c is a critical value of G by Proposition 2.2.1. Now suppose that c “ b and K c X B “ H. Since A is compact and so is c K by pCqc , C “ A Y K c is compact. Since B is closed, then distpC, Bq ą 0. Applying Lemma 1.3.3 to ´G with this C and δ ă distpC, Bq gives an ε ą 0 and a homeomorphism ζ of E such that ζ is the identity outside Gc`2ε c´2ε zNδ{3 pCq and ζ pGc´ε zNδ pCqq Ă Gc`ε , in particular, ζ is the identity on A and G ě c ` ε on ζ pBq. Then taking a r :“ ζ ´1 ˝ h P  and hence h P  with G ă c ` ε on hpDq, we have h r hpDq X ζ pBq “ ζ phpDq X Bq ‰ H by (2.1), a contradiction.



Many authors have contributed to this result. The special cases that correspond to Examples 2.3.2, 2.3.3, and 2.3.4 are the well-known mountain pass

2.4 Homological linking

33

lemma of Ambrosetti and Rabinowitz [9] and the saddle point and linking theorems of Rabinowitz [127, 128], respectively. See also Ahmad et al. [2], Castro and Lazer [24], Benci and Rabinowitz [18], Ni [97], Chang [29], Qi [125], and Ghoussoub [55]. Morse index estimates for a critical point produced by a homotopical linking have been obtained by Lazer and Solimini [67], Solimini [152], Ghoussoub [55], Ramos and Sanchez [130], and others. However, the notion of homological linking introduced by Benci [15, 16] and Liu [82] is better suited for obtaining critical points with nontrivial critical groups.

2.4 Homological linking The notion of homological linking is useful for obtaining pairs of sublevel sets with nontrivial relative homology and hence critical points with nontrivial critical groups via Proposition 1.4.2. Definition 2.4.1 Let A and B be disjoint nonempty subsets of E. We say that A homologically links B in dimension q ă 8 if the homomorphism i˚ : Hrq pAq Ñ Hrq pEzBq, induced by the inclusion i : A Ă EzB, is nontrivial. Note that q ď dim E ´ 1 when E is finite dimensional. The following three propositions show that the sets in the examples of homotopical linking given in the previous section link homologically also. Proposition 2.4.2 If u0 P E, U is a bounded neighborhood of u0 , and u1 R U , then A “ tu0 , u1 u homologically links B “ BU in dimension q “ 0. Proof Follows since u0 and u1 are in different path components of EzB.



Proposition 2.4.3 If E “ N ‘ M, u “  v ` w is a direct ( sum decomposition with 1 ď d :“ dim N ă 8, then A “ v P N : }v} “ R homologically links B “ M in dimension q “ d ´ 1 (for any R ą 0. In particular, if d :“ dim E ă  8, then A “ u P E : }u} “ R homologically links B “ t0u in dimension q “ d ´ 1. Proof The map pEzBq ˆ r0, 1s Ñ EzB,

pu, tq ÞÑ p1 ´ tq u ` t

Rv }v}

34

Linking

is a strong deformation retraction of EzB onto A and hence i˚ is an isomorphism, so rank i˚ “ rank Hrd´1 pAq “ 1.



Proposition 2.4.4 If E “ N ‘ M, u “ v ` w is a direct sum decomposition  with 1 ď(d :“ dim N ă 8 and w P M with }w } “ 1, then A “ vPN : 0 0  ( }v} ď R Y u “ v ` tw ( 0 : v P N, t ě 0, }u} “ R homologically links B “ w P M : }w} “ r in dimension q “ d for any 0 ă r ă R. Proof Let

 (  ( A0 “ v P N : }v} “ R , A1 “ v P N : }v} ď R ,  ( A2 “ u “ v ` tw0 : v P N, t ě 0, }u} “ R ,  ( A1 “ u “ v ` tw0 : v P N, t ě 0, }u} ď R ,  (  ( B 1 “ w P M : }w} ď r , B 2 “ E z w P M : }w} ě r ,

and consider the commutative diagram Hd pA1 , A0 q § § k˚ đ

«

ÝÝÝÝÑ

Hrd´1 pA0 q § §j đ˚

ÝÝÝÝÑ Hrd´1 pA1 q § § đ

Hd pB 2 , EzMq ÝÝÝÝÑ Hrd´1 pEzMq ÝÝÝÝÑ Hrd´1 pB 2 q where the rows come from the exact sequences of the pairs pA1 , A0 q Ă pB 2 , EzMq. Since j˚ ‰ 0 by Proposition 2.4.3 and A1 is contractible, k˚ ‰ 0. Now consider the commutative diagram Hd pA1 , A0 q § § k˚ đ

ÝÝÝÝÑ

Hd pA, A2 q § §l đ˚

«

Hd pB 2 , EzMq ÝÝÝÝÑ Hd pEzB, EzB 1 q induced by inclusions. The bottom  ( arrow is an isomorphism by the excision property since w P M : }w} ą r is a closed subset of EzB contained in the open subset EzB 1 . Since k˚ ‰ 0, then l˚ ‰ 0. Next consider the commutative diagram «

Hd`1 pA1 , Aq ÝÝÝÝÑ § § m˚ đ

Hd pA, A2 q § §l đ˚

ÝÝÝÝÑ Hd pA1 , A2 q § § đ

Hd`1 pE, EzBq ÝÝÝÝÑ Hd pEzB, EzB 1 q ÝÝÝÝÑ Hd pE, EzB 1 q where the rows come from the exact sequences of the triples pA1 , A, A2 q Ă pE, EzB, EzB 1 q. Since l˚ ‰ 0 and A1 and A2 are contractible, m˚ ‰ 0.

2.4 Homological linking

35

Finally consider the commutative diagram Hd`1 pA1 , Aq ÝÝÝÝÑ § § m˚ đ

Hd pAq § §i đ˚

ÝÝÝÝÑ Hd pA1 q § § đ

«

Hd`1 pE, EzBq ÝÝÝÝÑ Hd pEzBq ÝÝÝÝÑ Hd pEq where the rows come from the exact sequences of the pairs pA1 , Aq Ă pE, EzBq. Since m˚ ‰ 0 and E is contractible, i˚ ‰ 0.  The following proposition shows that homological linking is invariant under homeomorphisms of the space. Proposition 2.4.5 If A homologically links B in dimension q and h is a homeomorphism of E, then hpAq homologically links hpBq in dimension q. Proof Follows from the commutative diagram Hrq pAq § § h˚ đ «

i˚

ÝÝÝÝÑ

Hrq pEzBq § § h˚ đ«

j˚ Hrq phpAqq ÝÝÝÝÑ Hrq phpEzBqq

where j : hpAq Ă hpEzBq.



The following proposition shows that homological linking implies homotopical linking. Proposition 2.4.6 Let A and B be as in the Definition 2.3.1. If A homologically links B in dimension n ´ 1, then A homotopically links B. Proof Since i˚ ‰ 0 and the homology class rid A s generates Hrn´1 pAq, i˚ rid A s ‰ 0 in Hrn´1 pEzBq. So there is no singular n-chain of EzB with boundary id A , in particular, there is no map h P CpD, EzBq with h|A “ id A .  Proposition 2.4.7 If A homologically links B in dimension q and G|A ď a ă G|B , then Hq`1 pE, Ga q ‰ 0. In particular, Cq`1 pG, 8q ‰ 0 when a is less than all critical values and G satisfies pCq. Proof Since i˚ ‰ 0 in the commutative diagram Hrq pAq

/ Hrq pGa q JJ JJ JJ JJ i˚ J$  Hrq pEzBq

36

Linking

induced by the inclusions A Ă Ga Ă EzB, Hrq pGa q ‰ 0. Since E is contractible, Hq`1 pE, Ga q « Hrq pGa q by Lemma 1.4.6 (ii).  Combining Propositions 2.4.7 and 1.4.2 gives the following theorem. Theorem 2.4.8 If A homologically links B in dimension q, G|A ď a ă G|B where a is a regular value, and G has only a finite number of critical points in Ga and satisfies pCqc for all c ě a, then G has a critical point u with Gpuq ą a,

Cq`1 pG, uq ‰ 0.

2.5 Schechter and Tintarev’s notion of linking Schechter and Tintarev [148] introduced a different notion of linking that yields pairs of critical points, and the following refined version of their definition was given in Schechter [142]. Denote by  the set of all maps  P CpE ˆ r0, 1s, Eq such that, writing pu, tq “ ptq u, (i) p0q “ id E , (ii) ptq is a homeomorphism of E for all t P r0, 1q, and the mapping  ´1 : E ˆ r0, 1q Ñ E, pu, tq ÞÑ ptq´1 u is continuous, (iii) p1q E is a single point in E, and ptq u Ñ p1q E as t Ñ 1, uniformly on bounded subsets of E, (iv) for each bounded subset A of E and t0 P r0, 1q, ´ › ›¯ sup }ptq u} ` › ´1 ptq u› ă 8. pu,tqPAˆr0,t0 s

Remark 2.5.1 Note that if  P  and A is a bounded subset of E, pA ˆ pt0 , 1sq is bounded for t0 ă 1 sufficiently close to 1 by (iii), and hence pA ˆ r0, 1sq is bounded by (iv). Definition 2.5.2 Let A and B be disjoint nonempty subsets of E. We say that A links B if pA ˆ p0, 1sq X B ‰ H

@ P .

(2.2)

Note that if  P  and the map η P CpE ˆ r0, 1s, Eq is such that, writing ηpu, tq “ ηptq u, (i) ηp0q “ id E , (ii) ηptq is a homeomorphism of E for all t P r0, 1s, and the mapping η´1 : E ˆ r0, 1s Ñ E, pu, tq ÞÑ ηptq´1 u is continuous,

2.5 Schechter and Tintarev’s notion of linking

37

(iii) for each bounded subset A of E, ´ › ›¯ sup }ηptq u} ` ›η´1 ptq u› ă 8, pu,tqPAˆr0,1s

then the map η ¨  : E ˆ r0, 1s Ñ E,

pu, tq ÞÑ

$ &ηp2tq u,

t P r0, 1{2s

%ηp1q p2t ´ 1q u,

t P p1{2, 1s

is in . In this sense,  is invariant under the family of maps Dc, ε defined in Section 2.2. Analogous to Theorem 2.3.5, we have the following. Theorem 2.5.3 If A is bounded and links the closed set B, distpA, Bq ą 0, c :“ inf

sup

P pu,tqPAˆr0,1s

Gpptq uq

is finite, a :“ sup GpAq ď inf GpBq “: b, and G satisfies pCqc , then c ě b and is a critical value of G. If c “ b, then G has a critical point with critical value c on B. Proof By (2.2), c ě b. First suppose that c ą b, and let 2ε ă c ´ a. Then for any η P Dc, ε , ηptq is the identity on A for all t P r0, 1s by (iii) in the definition of Dc, ε , so ηp1q pA ( ˆ r0, 1sq “ η ¨  pA ˆ r0, 1sq for all  P . So F “ pA ˆ r0, 1sq :  P  is invariant under Dc, ε , and hence c is a critical value of G by Proposition 2.2.1. Now suppose that c “ b and K c X B “ H. Since K c is compact by pCqc and B is closed, distpK c , Bq ą 0. Applying Lemma 1.3.3 to ´G with C “ A Y K c and δ ă distpC, Bq gives an ε ą 0 and a map ζ P CpE ˆ r0, 1s, Eq such that, writing ζ pu, tq “ ζ ptq u, (i) ζ p0q “ id E , (ii) ζ ptq is a homeomorphism of E for all t P r0, 1s, and the mapping ζ ´1 : E ˆ r0, 1s Ñ E, pu, tq ÞÑ ζ ptq´1 u is continuous, (iii) ζ ptq is the identity outside Gc`2ε c´2ε zNδ{3 pCq for all t P r0, 1s, (iv) Gpζ p¨q uq is nondecreasing for each u P E, (v) ζ p1q pGc´ε zNδ pCqq Ă Gc`ε , (vi) for each bounded subset Q of E, ´ › ›¯ sup }ζ ptq u} ` ›ζ ´1 ptq u› ă 8. pu,tqPQˆr0,1s

In particular, ζ ptq is the identity on A for all t P r0, 1s and G ě c ` ε on ζ p1q B. Then taking a  P  with G ă c ` ε on pA ˆ r0, 1sq, we have

38

Linking

r :“ ζ ´1 ¨  P  and pA r ˆ r0, 1sq “ ζ ´1 p1q pA ˆ r0, 1sq, so  r ˆ r0, 1sq X Bq ‰ H pA ˆ r0, 1sq X ζ p1q B “ ζ p1q ppA 

by (2.2), a contradiction.

The following proposition shows that homotopical linking implies linking in this sense. Proposition 2.5.4 Let A and B be as in the Definition 2.3.1. If A homotopically links B, then A links B. Proof If  P  and ϕ is a homeomorphism of D onto the unit disk in Rn , then the map $ &p1 ´ }ϕpuq}q ϕ ´1 pϕpuq{ }ϕpuq}q, ϕpuq ‰ 0 h : D Ñ E, u ÞÑ %p1q E, ϕpuq “ 0 is in  and hpDq “ pA ˆ r0, 1sq, so (2.1) implies (2.2).



Combining Propositions 2.4.6 and 2.5.4 gives homological linking

ùñ

homotopical linking

ùñ

Schechter–Tintarev linking.

In particular, A links B in Examples 2.3.2 – 2.3.4. Note that when E is infinite dimensional the unit sphere in E is contractible and therefore does not link the origin homologically or homotopically. However, it does so according to Schechter and Tintarev’s definition of linking, as the following proposition shows. Proposition 2.5.5 If U is a bounded neighborhood of u0 P E, then A “ BU links B “ tu0 u. Proof Suppose not, say, u0 R 0 pBU ˆ r0, 1sq

(2.3)

where 0 P . Then 0 p1q E ‰ u0 and hence distp0 pt0 q U , u0 q ą 0 for t0 ă 1 sufficiently close to 1 by (iii) in the definition of , so 0´1 pt0 q u0 R U . Thus, the path γ ptq “ 0´1 ptq u0 , t P r0, t0 s satisfies γ p0q P U and γ pt0 q R U . But then γ pt1 q P BU for some t1 P p0, t0 q, so u0 P 0 pt1 q BU , contradicting (2.3). 

2.5 Schechter and Tintarev’s notion of linking

39

The following proposition implies that B links A in Example 2.3.4 when dim M ě 2. Proposition 2.5.6 If A and B are disjoint closed bounded subsets of E such that EzA is path connected, then A links B ùñ B links A. Proof Suppose not, say, 0 pB ˆ r0, 1sq X A “ H where 0 P(. Let u0 “ 0 p1q E, and take R ą 0 so large that C “ u P E : }u} ď R Ą A. Since EzA is path connected, there is a path γ P Cpr0, 1s, EzAq joining u0 “ γ p0q to some point γ p1q P EzC. By (iii) in the definition of ,  ( diamp0 pt0 q B ´ u0 q ă min distpγ pr0, 1sq, Aq, distpγ p1q, Cq for t0 ă 1 sufficiently close to 1. Then γrptq “ γ ppt ´ t0 q{p1 ´ t0 qq, t P rt0 , 1s satisfies γrpt0 q “ u0 , p0 pt0 q B ´ u0 ` γrptqq X A “ H

@t P rt0 , 1s,

and p0 pt0 q B ´ u0 ` γrp1qq X C “ H. So 1 ptq u “

$ &0 ptq u,

t P r0, t0 s

% pt q u ´ u ` γrptq, t P pt , 1s 0 0 0 0

carries B outside C without intersecting A, and hence B X 1´1 ptq A “ H for all t P r0, 1s and B X 1´1 p1q C “ H. Let 2 P  be such that 2 pC ˆ r :“  ´1 ¨ 2 P  and B X pA r ˆ r0, 1sq “ H, contrar0, 1sq Ă C. Then  1 dicting the fact that A links B.  Example 2.5.7 If E “ N ‘ M, u “ v ` w is a direct sum decompositionwith 1 ď dim N ă(8 and dim Mě 2, and w0 P M (with}w0 } “ 1, then B “ w P M : }w} “ (r links A “ v P N : }v} ď R Y u “ v ` tw0 : v P N, t ě 0, }u} “ R for any 0 ă r ă R. Combining Theorem 2.5.3 and Proposition 2.5.6 now gives the following corollary. Corollary 2.5.8 Let A and B be closed bounded subsets of E such that A links B, EzA is path connected, and distpA, Bq ą 0. If c :“ inf

sup

P pu,tqPAˆr0,1s

Gpptq uq,

d :“ sup

inf

P pu,tqPBˆr0,1s

Gpptq uq

40

Linking

are finite, a :“ sup GpAq ď inf GpBq “: b, and G satisfies pCqc and pCqd , then c ě b and d ď a are critical values of G.

2.6 Pairs of critical points with nontrivial critical groups The following analog of Corollary 2.5.8 for homologically linking sets was obtained by Perera [104], where it was shown that the second critical point also has a nontrivial critical group. We assume that G has only a finite number of critical points throughout this section. Theorem 2.6.1 If A homologically links B in dimension q ď dim E ´ 2 and B is bounded, G|A ď a ă G|B where a is a regular value, and G is bounded from below on bounded sets and satisfies pCq, then G has two critical points u1 and u2 with Gpu1 q ą a ą Gpu2 q,

Cq`1 pG, u1 q ‰ 0, Cq pG, u2 q ‰ 0.

This is a special case of Theorem 2.6.3 below. When E is infinite dimensional the assumption that q ď dim E ´ 2 is, of course, satisfied. When 2 ď d :“ dim E ă 8 it is related to the assumption in Corollary 2.5.8 that EzA is path connected. Indeed, suppose A homologically links B in dimension q and A is a compact neighborhood retract such that EzA is path connected. Since our coefficient group Z2 is a field, the vector space HompHd´1 pAq, Z2 q of linear maps from Hd´1 pAq to Z2 is isomorphic to the singular cohomology group H d´1 pAq « Hq d´1 pAq

since A is a neighborhood retract

« H1 pE, EzAq

by the Poincar´e duality theorem

« Hr0 pEzAq

as in Lemma 1.4.6 (ii)

“0

since EzA is path connected,

q where Hq ˚ denotes Cech cohomology. So Hd´1 pAq “ 0 and hence q ď d ´ 2. Combining Theorem 2.6.1 and Proposition 2.4.4 gives the following. Corollary 2.6.2 Let E “ N ‘ M, u  “ v ` w be a direct ( sum decomposition with d :“ dim N ă (8. If G ď a on v P N : }v} ď R Y u “ v ` tw0 : v P N, t ě 0, }u} “ R for ( some R ą 0 and w0 P M with }w0 } “ 1 and G ą a on w P M : }w} “ r for some 0 ă r ă R, where a is a regular value, and G is bounded from below on bounded sets and satisfies pCq, then G has two critical points u1 and u2 with Gpu1 q ą a ą Gpu2 q,

Cd`1 pG, u1 q ‰ 0, Cd pG, u2 q ‰ 0.

2.6 Pairs of critical points with nontrivial critical groups

41

It was also shown by Perera [104] that the assumptions that B is bounded and G is bounded from below on bounded sets can be relaxed as follows; see also Schechter [139]. Theorem 2.6.3 If A homologically links B in dimension q, G|A ď a ă G|B where a is a regular value, and G is bounded from below on a set C Ą B such that the inclusion-induced homomorphism Hrq pEzCq Ñ Hrq pEzBq is trivial and satisfies pCq, then G has two critical points u1 and u2 with Gpu1 q ą a ą Gpu2 q,

Cq`1 pG, u1 q ‰ 0, Cq pG, u2 q ‰ 0.

Proof Theorem 2.4.8 gives the critical point u1 . Let c ă min ta, inf GpCqu be a regular value. Then Gc Ă Ga Ă EzB and Gc Ă EzC Ă EzB, so we have the commutative diagram i˚ j˚ Hrq pGc q ÝÝÝÝÑ Hrq pGa q ÝÝÝÝÑ Hq pGa , Gc q § § § §k đ đ˚ l˚ Hrq pEzCq ÝÝÝÝÑ Hrq pEzBq

where the top row is a part of the exact sequence of the pair pGa , Gc q. We have l˚ “ 0, and k˚ ‰ 0 as in the proof of Proposition 2.4.7, so i˚ is not onto. Then j˚ ‰ 0 by exactness and hence Hq pGa , Gc q ‰ 0, so G has a second critical point u2 with c ă Gpu2 q ă a and Cq pG, u2 q ‰ 0 by Proposition 1.4.2.   Proof of Theorem 2.6.1 Apply Theorem 2.6.3 with C “ u P E : }u} ď ( R for sufficiently large R ą 0, noting that Hrq pEzCq “ 0 since q ď dim E ´ 2.  Corollary 2.6.4 Let E “ N ‘ M, u “ v` w be a direct sum ( decomposition with 1 ď d :“ dim N ă 8. If G ď a on v P N : }v} “ R for some R ą 0 and  G ą a on M, where a(is a regular value, and G is bounded from below on tv0 ` w : t ě 0, w P M for some v0 P N z t0u and satisfies pCq, then G has two critical points u1 and u2 with Gpu1 q ą a ą Gpu2 q,

Cd pG, u1 q ‰ 0, Cd´1 pG, u2 q ‰ 0.

Proof Apply Theorem 2.6.3  ( with A and B as in Proposition 2.4.3 and C “ tv0 ` w : t ě 0, w P M , noting that the map pEzCq ˆ r0, 1s Ñ EzC,

pu, tq ÞÑ p1 ´ tq u ´ tv0

is a contraction of EzC to ´v0 and hence Hrd´1 pEzCq “ 0.



42

Linking

2.7 Nonstandard geometries Note that when N is infinite dimensional in Example 2.3.3 the set A is contractible and therefore does not link B homologically or homotopically. However, it does so according to Schechter and Tintarev’s definition of linking if M is finite dimensional. This is a consequence of the following. Proposition 2.7.1 Let A and B be disjoint subsets of E with A bounded. If there is a sequence pBj q of subsets of E such that A links each Bj , and Bj “ Bj1 Y Bj2 where Bj1 Ă B and distpBj2 , 0q Ñ 8, then A links B. In particular, A links Ť 1 j Bj . Proof Let  P . For each j , pA ˆ p0, 1sq X Bj ‰ H. But for sufficiently large j , sup pu,tqPAˆr0,1s

}ptq u} ă distpBj2 , 0q

by Remark 2.5.1, so pA ˆ p0, 1sq X Bj1 ‰ H. Hence pA ˆ p0, 1sq X B ‰ H.  Corollary 2.7.2 If E “ N ‘ M, u “ v ` w is a direct sum  decomposition ( with N nontrivial and M finite dimensional, then A “ v P N : }v} “ R links B “ M for any R ą 0.  ( Proof Apply Proposition 2.7.1 with B(j1 “ w P M : }w} ď j and Bj2 “  u “ tv0 ` w : t ě 0, w P M, }u} “ j for some v0 P N with }v0 } “ 1, noting that A links Bj “ Bj1 Y Bj2 for j ą R by Example 2.5.7.  Ť Applying Theorem 2.5.3 with B “ j Bj1 in the setting of Proposition 2.7.1 now gives the following. Theorem 2.7.3 Let A be a bounded subset of E and pBj q a sequence of subsets of E such that A links each Bj “ Bj1 Y Bj2 where inf distpA, Bj1 q ą 0,

distpBj2 , 0q Ñ 8.

j

If c :“ inf

sup

P pu,tqPAˆr0,1s

Gpptq uq

is finite, a :“ sup Gpuq ď inf inf1 Gpuq “: b, uPA

j

uPBj

and G satisfies pCqc , then c ě b is a critical value of G.

2.7 Nonstandard geometries

43

Corollary 2.7.4 Let E “ N ‘ M, u “ v ` w be a direct sum decomposition with N nontrivial and M finite dimensional. If c :“ inf

sup

P pu,tqPAˆr0,1s

Gpptq uq

 ( is finite where A “ v P N : }v} “ R for some R ą 0, a :“ sup GpAq ď inf GpMq “: b, and G satisfies pCqc , then c ě b is a critical value of G. Proposition 2.7.1, Theorem 2.7.3, and Corollaries 2.7.2 and 2.7.4 are due to Schechter [142]; see also Ribarska et al. [131]. An analog of Theorem 2.7.3 for homologically linking sets does not seem to be known, but we have the following related result, which gives critical points with nontrivial critical groups under nonstandard geometrical assumptions that do not involve a finitedimensional closed loop. We assume that G has only a finite number of critical points for the rest of this section. Theorem 2.7.5 Let pAj q be a sequence of subsets of E and B a subset of E such that each Aj “ A1j Y A2j homologically links B in dimension q, where distpA2j , Bq Ñ 8. If G|A1j ď a ă G|B where a is a regular value, b :“ sup sup Gpuq ă 8, j

uPA2 j

(2.4)

and G satisfies pPSq, then G has a critical point u1 with Gpu1 q ą a,

Cq`1 pG, u1 q ‰ 0.

If, in addition, q ď dim E ´ 2, B is bounded, and G is bounded from below on bounded sets, then there is a second critical point u2 with Gpu2 q ă a,

Cq pG, u2 q ‰ 0.

First we prove a deformation lemma. Lemma 2.7.6 If ´8 ă a ă b ă `8, C is a set containing Kab , δ ą 0, and G satisfies pPSqc for all c P ra, bs, then there are an ε0 ą 0 and, for each ε P p0, ε0 q, a map η P CpE ˆ r0, 1s, Eq such that, writing ηpu, tq “ ηptq u, (i) (ii) (iii) (iv) (v) (vi)

ηp0q “ id E , ηptq is a homeomorphism of E for all t P r0, 1s, ηptq is the identity outside A “ Gb`ε a´ε zNδ{2 pCq for all t P r0, 1s, }ηptq u ´ u} ď 1{ε @pu, tq P E ˆ r0, 1s, Gpηp¨q uq is nonincreasing for each u P E, ηp1q pGb zNδ`1{ε pCqq Ă Ga .

First, we have another lemma.

44

Linking

Lemma 2.7.7 If ´8 ă a ă b ă `8, N is an open neighborhood of Kab , k ą 0, and G satisfies pPSqc for all c P ra, bs, then there is an ε0 ą 0 such that › › inf ›G1 puq› ě k ε @ε P p0, ε0 q. uPGb`ε a´ε zN

b`ε

Proof If not, there are sequences εj Œ 0 and uj P Ga´εjj zN such that › 1 › ›G puj q› ă k εj . Then puj q has a subsequence that is a pPSqc sequence for some c P ra, bs, which in turn has a subsequence converging to some u P K c zN “ H, a contradiction.  Proof of Lemma 2.7.6 By Lemma 2.7.7, there is an ε0 ą 0 such that for each ε P p0, ε0 q, › 1 › ›G puq› ě 4 pb ´ aq ε @u P A. (2.5) Let V be a pseudo-gradient vector field for G, g P Liploc pE, r0, 1sq satisfy g “ 0 outside A and g “ 1 on B “ Gba zNδ pCq, and ηptq u, 0 ď t ă T puq ď 8 the maximal solution of η9 “ ´2 pb ´ aq gpηq

V pηq }V pηq}2

Since }ηptq u ´ u} ď 2 pb ´ aq

żt 0

ď 4 pb ´ aq

żt 0

ď t{ε

,

t ą 0,

ηp0q u “ u P E.

(2.6)

gpηpτ q uq dτ }V pηpτ q uq} gpηpτ q uq dτ }G1 pηpτ q uq}

by (1.1) by (2.5),

}ηp¨q u} is bounded if T puq ă 8, so T puq “ 8 and (i)–(iv) follow. By (2.6) and (1.1), ˘ ` ˘ d ` pG1 pηq, V pηqq Gpηptq uq “ G1 pηq, η9 “ ´2 pb ´ aq gpηq dt }V pηq}2 ď ´pb ´ aq gpηq ď 0 (2.7) and hence (v) holds. To see that (vi) holds, let u P Gb zNδ`1{ε pCq and suppose that ηp1q u R Ga . Then ηptq u P Gba for all t P r0, 1s, and ηptq u R Nδ pCq by

2.7 Nonstandard geometries

45

(iv). Thus, ηptq u P B and hence gpηptq uq “ 1 for all t P r0, 1s, so (2.7) gives Gpηp1q uq ď Gpuq ´ pb ´ aq ď a, 

a contradiction.

Proof of Theorem 2.7.5 If a ě b, this follows from Theorems 2.4.8 and 2.6.1 with A “ any Aj , so we assume that a ă b. Applying Lemma 2.7.6 with C “ Kab Y B and δ ą 0, let ε ą 0 be sufficiently small and η the corresponding map. We claim that distpA2j , Cq Ñ 8. Since distpA2j , Bq Ñ 8 by assumption, it suffices to show that distpA2j , Kab q Ñ 8 to prove the claim. By pPSq, Kab is compact and hence bounded, so it is, in fact, enough to show that distpA2j , 0q Ñ 8. But this follows since distpA2j , 0q ě distpA2j , Bq ´ distpB, 0q by the triangle inequality. Fix j so large that distpA2j , Cq ą δ ` 1{ε. Then A2j Ă Gb zNδ`1{ε pCq by (2.4), so ηp1q A2j Ă Ga by (vi). Since A1j Ă Ga , ηp1q A1j Ă Ga by (v), and ηp1q B “ B by (iii), so G|ηp1q Aj ď a ă G|ηp1q B . By (ii) and Proposition 2.4.5, ηp1q Aj homologically links ηp1q B in dimension q, so the conclusions follow from Theorems 2.4.8 and 2.6.1.  The following corollaries of Theorem 2.7.5 were obtained by Perera and Schechter [118]; see also Perera and Schechter [114] and Lancelotti [65]. Corollary 2.7.8 Let E “ N ‘ M, u “ v ` w be a direct sum decomposition with d :“ dim M ă 8. If G ą a on N , where a is a regular value, and G is bounded from above on M and satisfies pPSq, then G has a critical point u with Gpuq ą a,

Cd pG, uq ‰ 0.

 ( Proof Apply Theorem 2.7.5 with A1j “ H and A2j “ w P M : }w} “ j , noting that each Aj “ A1j Y A2j homologically links B “ N in dimension q “ d ´ 1 and b ď sup GpMq ă 8.  Corollary 2.7.9 Let E “ N ‘ M, u “ v ` w be a direct sum decomposition with d :“ dim M ă 8. If G ă a on N, where a is a regular value, and G is bounded from below on M and satisfies pPSq, then G has a critical point u with Gpuq ă a, Proof Apply Corollary 2.7.8 to ´G.

Cd p´G, uq ‰ 0. 

46

Linking

Corollary 2.7.10 Let E “ N ‘ M, u “ v ` w be a direct ( sum decomposition with d :“ dim M ă 8. If G ă a on v P N : }v} “ R for some R ą 0 and Gě  a on M, where a is a(regular value, and G is bounded from below on tv0 ` w : t ě 0, w P M for some v0 P N with }v0 } “ 1, bounded from above on bounded sets, and satisfies pPSq, then G has two critical points u1 and u2 with Gpu1 q ă a ă Gpu2 q,

Cd`1 p´G, u1 q ‰ 0, Cd p´G, u2 q ‰ 0.  ( Proof Apply Theorem 2.7.5 to ´G with(A1j “ w P M : }w} ď j and A2j “  u “ tv0 ` w : t ě 0, w P M, }u} “(j , noting that Aj “ A1j Y A2j homologically links B “ v P N : }v} “ R in dimension q “ d for j ą R.  We believe that under the hypotheses of Corollary 2.7.4, G has a critical point u with Cd p´G, uq ‰ 0 where d “ dim M. More specifically, we conjecture that if E “ N ‘ M, u “ v ` w isa direct sum decomposition with dim N ě 1 and ( d :“ dim M ă 8, G ď a on v P N : }v} “ R for some R ą 0 and G ą a on M, where a is a regular value, and G satisfies pPSq, then G has a critical point u with Gpuq ą a,

Cd p´G, uq ‰ 0.

Note that when E is a Hilbert space and G is C 2 this implies mp´G, uq ď d ď m˚ p´G, uq via Corollary 1.1.11. It was shown in Perera and Schechter [114] that G has critical points u and u, not necessarily distinct, such that Gpuq ď Gpuq,

mp´G, uq ď d ď m˚ p´G, uq

when, in addition, ∇G is a Fredholm nonlinear map of index zero near the critical points of G.

3 Applications to semilinear problems

3.1 Introduction Now we give some applications of the previously discussed Morse theoretic and linking methods to the semilinear elliptic boundary value problem # ´ u “ f px, uq in (3.1) u“0 on B where is a bounded domain in Rn , n ě 1 and f is a Carath´eodory function on ˆ R satisfying the growth condition ` ˘ |f px, tq| ď C |t|r´1 ` 1 @px, tq P ˆ R (3.2) for some r P r2, 2˚ q and a constant C ą 0. Here # 2n{pn ´ 2q, n ą 2 ˚ 2 “ 8, nď2 is the critical Sobolev exponent. Weak solutions of (3.1) coincide with critical points of the C 1 -functional ż Gpuq “ |∇u|2 ´ 2F px, uq, u P E “ H01 p q

where F px, tq “

żt

f px, sq ds

0

is the primitive of f and H01 p q is the usual Sobolev space with the norm ˆż ˙1{2 2 }u} “ |∇u| .

47

48

Applications to semilinear problems

When verifying the pPSq or the pCq condition for G it suffices to check the boundedness of the sequence by Lemma 3.1.1 Every bounded sequence puj q Ă E such that G1 puj q Ñ 0 has a convergent subsequence. Proof Fix 2˚ {p2˚ ´ r ` 1q ă q ă 2˚ , so that, denoting by q 1 “ q{pq ´ 1q the H¨older conjugate of q, pr ´ 1q q 1 ă 2˚ . Since puj q is bounded, a renamed subsequence converges to some u weakly in E and strongly in Lq p q by the compactness of the Sobolev embedding E ãÑ Lq p q. We claim that uj Ñ u in E. Since E is a Hilbert space, it suffices to show that puj ´ u, uj q Ñ 0. We have ż ˘ 1` 1 puj ´ u, uj q “ G puj q, uj ´ u ` f px, uj q puj ´ uq. 2

Since G1 puj q Ñ 0 and puj ´ uq is bounded, pG1 puj q, uj ´ uq Ñ 0. By (3.2), ˇż ˇ ż ˇ ˇ ` ˘ ˇ f px, uj q puj ´ uqˇ ď C |uj |r´1 ` 1 |uj ´ u| ˇ ˇ

´ ¯ 1{q 1 ď C }uj }r´1 ` | | }uj ´ u}Lq p q 1 pr´1q q L p q 1

where | | denotes the volume of , and puj q is bounded in Lpr´1q q p q by the 1 Sobolev embedding E ãÑ Lpr´1q q p q. 

3.2 Local nature of critical groups Let u0 be an isolated critical point of G. Clearly, the critical groups of G at u0 depend only on the values of G near u0 . However, when n ě 2, E is not embedded in L8 p q and it would seem that C˚ pG, u0 q depend on the values of f everywhere. In this section we show that this is not the case. Set r Gpuq “ Gpu ` u0 q ´ Gpu0 q ż ` ˘ ` ˘ “ |∇u|2 ´ 2 F px, u ` u0 q ´ F px, u0 q ´ f px, u0 q u ` G1 pu0 q, u ż “ |∇u|2 ´ 2Frpx, uq,

where Fr is the primitive of frpx, tq “ f px, t ` u0 pxqq ´ f px, u0 pxqq.

3.2 Local nature of critical groups

49

Then fr also satisfies the growth condition (3.2) since u0 P L8 p q by a standard r regularity argument, 0 is an isolated critical point of G, r 0q “ C˚ pG, u0 q, C˚ pG, and the values of frpx, tq near t “ 0 depend only on the values of f px, tq near t “ u0 pxq. Thus, we may assume without loss of generality that u0 “ 0. Lemma 3.2.1 Let δ ą 0, ϑ : R Ñ r´δ, δs be a smooth nondecreasing function such that ϑptq “ ´δ for t ď ´δ, ϑptq “ t for ´δ{2 ď t ď δ{2, and ϑptq “ δ for t ě δ, and set ż G1 puq “ |∇u|2 ´ 2F px, ϑpuqq, u P E.

If 0 is an isolated critical point of G, then it is also an isolated critical point of G1 and Cq pG, 0q « Cq pG1 , 0q

@q.

Proof We apply Theorem 1.4.4 to the family ż Gs puq “ |∇u|2 ´ 2F px, p1 ´ sq u ` s ϑpuqq,

u P E, s P r0, 1s

in a small ball Bε , noting that G0 “ G. Lemma 3.1.1 implies that each Gs satisfies pPSq over Bε , and it is easy to see that the map r0, 1s Ñ C 1 pBε , Rq, s ÞÑ Gs is continuous, so it only remains to show that for sufficiently small ε ą 0, Bε contains no critical point of Gs other than 0 for all s P r0, 1s. Suppose uj Ñ 0, G1sj puj q “ 0, sj P r0, 1s, and uj ‰ 0. Then # ´ uj “ fj px, uj q in uj “ 0

on B

where fj px, tq “ p1 ´ sj ` sj ϑ 1 ptqq f px, p1 ´ sj q t ` sj ϑptqq also satisfies the growth condition (3.2) for some generic positive constant C independent of j . Passing to a subsequence, we may assume that uj Ñ 0 strongly in L2 p q and a.e. in , sj converges to some s P r0, 1s, and hence fj px, uj q Ñ f px, 0q “ 0 a.e. Thus, if uj Ñ 0 in Lq p q, then fj px, uj q Ñ 0 in Lq{pr´1q p q. Since }uj }Lnq{pnpr´1q´2qq p q ď C }uj }H 2,q{pr´1q p q ď C }fj px, uj q}Lq{pr´1q p q by the Sobolev and Calder´on–Zygmund inequalities, then uj Ñ 0 also in Lnq{pnpr´1q´2qq p q. Starting with q “ 2, iterating until q ą npr ´ 1q{2, and

50

Applications to semilinear problems

using the Sobolev embedding H 2,q{pr´1q p q ãÑ Cp q now gives uj Ñ 0 in Cp q. So for sufficiently large j , |uj | ď δ{2 and hence G1 puj q “ 0, contradicting our assumption that 0 is an isolated critical point of G.  The following theorem is now immediate from Lemma 3.2.1 and the remarks at the beginning of the section. Theorem 3.2.2 Let fi , i “ 1, 2 be Carath´eodory functions on ˆ R satisfying ` ˘ |fi px, tq| ď C |t|r´1 ` 1 @px, tq P ˆ R żt for some r P r2, 2˚ q and C ą 0, Fi px, tq “ fi px, sq ds, and set 0

ż Gi puq “

|∇u|2 ´ 2Fi px, uq,

u P E.

If u0 is an isolated critical point of G1 and f1 px, u0 pxq ` tq “ f2 px, u0 pxq ` tq @x P , |t| ď δ for some δ ą 0, then u0 is also an isolated critical point of G2 and Cq pG1 , u0 q « Cq pG2 , u0 q @q. Remark 3.2.3 The homotopy argument used in the proof of Lemma 3.2.1 was adapted from Degiovanni et al. [46].

3.3 Critical groups at zero In this section we compute the critical groups of G at zero under various assumptions on the behavior of the nonlinearity f px, tq near t “ 0. First we consider the linear case f px, tq “ λ t, λ P R, i.e. the eigenvalue problem # ´ u “ λu in (3.3) u“0 on B . The spectrum σ p´ q of the negative Laplacian consists of isolated eigenvalues λl , l ě 1 of finite multiplicities satisfying 0 ă λ1 ă λ2 ă ¨ ¨ ¨ ă λl ă ¨ ¨ ¨ , and }u}2 ě λ1 }u}2L2 p q

@u P E.

(3.4)

3.3 Critical groups at zero

51

Let El be the eigenspace of λl , Nl “

l à

Ml “ NlK .

Ej ,

j “1

Then E “ Nl ‘ Ml ,

u“v`w

(3.5)

is an orthogonal decomposition with respect to both the inner product in E and the L2 p q-inner product, and }v}2 ď λl }v}2L2 p q

@v P Nl ,

}w}2 ě λl`1 }w}2L2 p q

@w P Ml .

(3.6) (3.7)

When λ R σ p´ q, the origin is the only critical point of ż Gpuq “ |∇u|2 ´ λ u2 , u P E,

so the critical groups Cq pG, 0q are defined. Let dl “ dim Nl . We have the following theorem. Theorem 3.3.1 Let λ P Rzσ p´ q. (i) If λ ă λ1 , then Cq pG, 0q « δq0 Z2 . (ii) If λl ă λ ă λl`1 , then Cq pG, 0q « δqdl Z2 . Proof (i) By (3.4), Gpuq ě

ˆ ˙ λ 1´ }u}2 λ1

@u P E,

so 0 is the unique global minimizer of G and hence Proposition 1.5.2 applies. (ii) Taking U “ E in (1.24) gives Cq pG, 0q “ Hq pG0 , G0 z t0uq. By (3.6),

ˆ Gpvq ď ´

˙ λ ´ 1 }v}2 ď 0 λl

@v P Nl ,

52

Applications to semilinear problems

so Nl Ă G0 . On the other hand, by (3.7), ˆ ˙ λ Gpwq ě 1 ´ }w}2 ą 0 λl`1

@w P Ml z t0u ,

(3.8)

so G0 z t0u Ă EzMl . Referring to the decomposition (3.5), let ηpu, tq “ v ` p1 ´ tq w,

u P G0 , t P r0, 1s

and note that Gpηpu, tqq “ Gpvq ` p1 ´ tq2 Gpwq ď Gpvq ` Gpwq “ Gpuq ď 0 by orthogonality and (3.8). So η is a strong deformation retraction of G0 onto Nl and its restriction to G0 z t0u is a strong deformation retraction onto Nl z t0u. Thus, Cq pG, 0q « Hq pNl , Nl z t0uq “ δqdl Z2 .



In the general case we have the next theorem. Theorem 3.3.2 Assume that (3.2) holds and 0 is an isolated critical point of G. (i) If there is a δ ą 0 such that f px, tq ď λ1 t

@x P , 0 ă |t| ď δ,

(3.9)

then Cq pG, 0q « δq0 Z2 . (ii) If there is a δ ą 0 such that λl ď

f px, tq ď λl`1 t

@x P , 0 ă |t| ď δ,

(3.10)

then Cq pG, 0q « δqdl Z2 . Proof (i) In view of Theorem 3.2.2, we may assume that (3.9) holds for all px, tq P ˆ pRz t0uq, so 2F px, tq ď λ1 t 2 and hence Gpuq ě }u}2 ´ λ1 }u}2L2 p q ě 0 @u P E by (3.4). So 0 is a global minimizer of G, which is isolated by assumption, and hence Proposition 1.5.2 applies. (ii) Referring to the decomposition (3.5), let ż G1 puq “ |∇w|2 ´ |∇v|2 , u P E

3.3 Critical groups at zero

53

and note that Cq pG1 , 0q « δqdl Z2 by an argument similar to that in the proof of Theorem 3.3.1 (ii). We apply Theorem 1.4.4 to the family Gs puq “ p1 ´ sq Gpuq ` s G1 puq,

u P E, s P r0, 1s

in a small ball Bε . Lemma 3.1.1 implies that each Gs satisfies pPSq over Bε , and it is easy to see that the map r0, 1s Ñ C 1 pBε , Rq, s ÞÑ Gs is continuous. By assumption, 0 is an isolated critical point of G0 “ G, and we will show that it is the only critical point of Gs for s ą 0. Let u “ ´v ` w. Then ż ˘ ˘ 1` 1 1` 1 G puq, u “ }w}2 ´ }v}2 ´ f px, uq u, G1 puq, u “ }u}2 2 2

by orthogonality. In view of Theorem 3.2.2, we may assume that (3.10) holds for all px, tq P ˆ pRz t0uq, so when upxq ‰ 0, f px, uq u “ ď

f px, uq uu u # λl`1 pw2 ´ v 2 q, uu ě 0 ´λl pv 2 ´ w 2 q,

uu ă 0

ď λl`1 w ´ λl v . 2

2

When upxq “ 0, f px, upxqq “ 0 and vpxq “ ´wpxq, so this inequality still holds. So ˘ 1` 1 G puq, u ě }w}2 ´ }v}2 ´ λl`1 }w}2L2 p q ` λl }v}2L2 p q ě 0 2 by (3.6) and (3.7). Thus, ` 1 ˘ ` ˘ ` ˘ Gs puq, u “ p1 ´ sq G1 puq, u ` s G11 puq, u ě 2s }u}2 , and hence u “ 0 if G1s puq “ 0 and s ą 0.



Since λ1 ą 0, Theorem 3.3.2 (i) gives the following corollary. Corollary 3.3.3 If (3.2) holds, 0 is an isolated critical point of G, and tf px, tq ď 0 @x P , |t| ď δ for some δ ą 0, then Cq pG, 0q « δq0 Z2 . An important special case of Theorem 3.3.2 is the asymptotically linear case f px, tq “ λ t ` optq as t Ñ 0, uniformly a.e.

(3.11)

for some λ P R. Then we say that (3.1) is resonant at zero if λ P σ p´ q, otherwise it is nonresonant. We have a corollary, as follows.

54

Applications to semilinear problems

Corollary 3.3.4 Assume (3.2) and (3.11) with λ P Rzσ p´ q. (i) If λ ă λ1 , then Cq pG, 0q « δq0 Z2 . (ii) If λl ă λ ă λl`1 , then Cq pG, 0q « δqdl Z2 . Proof It only remains to show that 0 is an isolated critical point of G. If not, there is a sequence puj q Ă Ez t0u , ρj :“ }uj } Ñ 0 such that G1 puj q “ 0. rj :“ uj {ρj we have }r Then setting u uj } “ 1, so a renamed subsequence of r weakly in E, strongly in L2 p q, and a.e. in . By pr uj q converges to some u (3.11), ż pG1 puj q, vq rj ¨ ∇v ´ λ u rj v ` op1q }v} , v P E, 0“ “ ∇u 2ρj

and passing to the limit gives ż r ¨ ∇v ´ λ u r v “ 0 @v P E, ∇u

r solves (3.3). Taking v “ u rj and passing to the limit gives λ }r so u u}2L2 p q “ 1, r ‰ 0. So λ P σ p´ q, a contradiction. so u  Corollary 3.3.5 Assume that (3.2) and (3.11) hold and 0 is an isolated critical point of G. (i) If λ “ λ1 and there is a δ ą 0 such that f px, tq ď λ1 @x P , 0 ă |t| ď δ, t then Cq pG, 0q « δq0 Z2 . (ii) If λ “ λl and there is a δ ą 0 such that f px, tq ě λl @x P , 0 ă |t| ď δ, t or if λ “ λl`1 and there is a δ ą 0 such that f px, tq ď λl`1 t then Cq pG, 0q « δqdl Z2 .

@x P , 0 ă |t| ď δ,

Remark 3.3.6 Theorem 3.3.2 (ii) was proved by Li et al. [74]. Finally we consider the case with a concave nonlinearity f px, tq “ μ |t|σ ´2 t ` op|t|σ ´1 q as t Ñ 0, uniformly a.e. for some μ ‰ 0 and σ P p1, 2q. We have the following theorem.

(3.12)

3.3 Critical groups at zero

55

Theorem 3.3.7 Assume that (3.2) and (3.12) hold and 0 is an isolated critical point of G. (i) If μ ă 0, then Cq pG, 0q « δq0 Z2 . (ii) If μ ą 0, then Cq pG, 0q “ 0 @q. Proof (i) This is immediate from Corollary 3.3.3. (ii) This follows from Proposition 3.3.8 below since (3.12) and (3.2) imply ˆ ˙ μ 2 F px, tq “ |t|σ ` op|t|σ q, H px, tq “ ´ 1 μ |t|σ ` op|t|σ q σ σ and σ ă 2.



Let H px, tq “ 2F px, tq ´ tf px, tq. Proposition 3.3.8 If (3.2) holds, F px, tq ě c |t|σ

@x P , |t| ď δ

(3.13)

H px, tq ą 0 @x P , 0 ă |t| ď δ

(3.14)

and

for some σ P p1, 2q and c, δ ą 0, and 0 is an isolated critical point of G, then Cq pG, 0q “ 0 @q. Proof By (3.13) and (3.2), F px, tq ě c |t|σ ´ C |t|r

@px, tq P ˆ R

(3.15)

for some constant C ą 0. Set $ t ’ ’ ´f px, ´δq , t ă ´δ ’ ’ δ ’ ’ żt & |t| ď δ frpx, tq “ f px, tq, Frpx, tq “ frpx, sq ds ’ 0 ’ ’ ’ ’ t ’ %f px, δq , t ą δ, δ and apply Theorem 3.2.2 to G and ż r Gpuq “ |∇u|2 ´ 2Frpx, uq

56

Applications to semilinear problems

r 0q. Since Fr “ F on ˆ r´δ, δs, (3.13) holds with to get C˚ pG, 0q « C˚ pG, Fr in place of F also. A simple calculation shows that $ ’ ’ &H px, ´δq, t ă ´δ Hr px, tq “ 2Frpx, tq ´ t frpx, tq “ H px, tq, |t| ď δ ’ ’ %H px, δq, t ą δ, so (3.14) implies Hr ą 0 on ˆ pRz t0uq. Thus, we may assume without loss of generality that H ą 0 on ˆ pRz t0uq. We have Cq pG, 0q “ Hq pG0 X B, G0 X Bz t0uq, where

 ( B “ u P E : }u} ď 1

is the unit ball in E. We will show that G0 X B is contractible to 0 and G0 X Bz t0u is a strong deformation retract of Bz t0u » BB “: S. Since E is infinite dimensional and hence S is contractible, the conclusion will then follow. For u P S and 0 ă t ď 1, ż ´ ¯ 2 Gptuq “ t ´ 2F px, tuq ď t σ t 2´σ ´ 2c }u}σLσ p q ` C t r´σ

for some constant C ą 0 by (3.15) and the Sobolev embedding E ãÑ Lr p q. Since c ą 0 and σ ă 2 ď r, Gptuq ă 0 for all sufficiently small t (depending on u). Since ˆ ˙ ż ˘ d ` Gptuq “ 2 t ´ f px, tuq u dt

ˆ ˙ ż 2 “ Gptuq ` H px, tuq t

2 ą Gptuq, (3.16) t ˘ d ` Gptuq ě 0 ùñ Gptuq ą 0. Thus, there is a unique 0 ă T puq ď 1 such dt that Gptuq ă 0 for 0 ă t ă T puq, GpT puq uq ď 0, and Gptuq ą 0 for T puq ă t ď 1. We claim that the map T : S Ñ p0, 1s is continuous. ( By (3.16) and the 1 implicit function theorem, T is C on u P S : T puq ă 1 , so it suffices to show that if uj Ñ u and T puq “ 1, then T puj q Ñ 1. But for any t ă 1, Gptuj q Ñ Gptuq ă 0, so T puj q ą t for j sufficiently large.

3.4 Asymptotically linear problems

57

Thus,  ( G0 X B “ tu : u P S, 0 ď t ď T puq and is radially contractible to 0, and pBz t0uq ˆ r0, 1s Ñ Bz t0u , # p1 ´ tq u ` t T pπ puqq π puq, u P BzG0 pu, tq ÞÑ u, u P G0 X Bz t0u , where π is the radial projection onto S, is a strong deformation retraction of Bz t0u onto G0 X Bz t0u.  Remark 3.3.9 Theorem 3.3.7 was proved by Perera [104, 105].

3.4 Asymptotically linear problems In this section we consider the solvability and the existence of nontrivial solutions of the problem (3.1) in the asymptotically linear case f px, tq “ λ t ´ ppx, tq

(3.17)

for some λ P R and a Carath´eodory function p on ˆ R satisfying ` ˘ |ppx, tq| ď C |t|τ ´1 ` 1 @px, tq P ˆ R

(3.18)

for some τ P r1, 2q and a constant C ą 0. We say that (3.1) is resonant at infinity if λ P σ p´ q, otherwise it is nonresonant. We have ż Gpuq “ |∇u|2 ´ λ u2 ` 2P px, uq, u P E (3.19) where P px, tq “

żt

ppx, sq ds satisfies 0

|P px, tq| ď C p|t|τ ` 1q

@px, tq P ˆ R

for some constant C ą 0, and ż ˘ 1` 1 G puq, v “ ∇u ¨ ∇v ´ λ uv ` ppx, uq v, 2

u, v P E.

By (3.18), (3.20), and the Sobolev embedding E ãÑ Lτ p q, ˇ ˇż ´ ¯ ˇ ˇ ˇ ppx, uq v ˇ ď C }u}τ ´1 ` 1 }v} @u, v P E ˇ ˇ

(3.20)

(3.21)

(3.22)

58

Applications to semilinear problems

and

ˇż ˇ ˇ ˇ ˇ P px, uqˇ ď C p}u}τ ` 1q ˇ ˇ

@u P E

(3.23)

for some constant C ą 0. We assume that G has only a finite number of critical points throughout this section. The following lemma is useful for verifying the boundedness of pPSq sequences. Lemma 3.4.1 If G1 puj q Ñ 0 and ρj :“ }uj } Ñ 8, then a subsequence of rj :“ uj {ρj converges to a nontrivial solution of (3.3), in particular, λ P u σ p´ q. r weakly in Proof Since }r uj } “ 1, a renamed subsequence converges to some u 2 E, strongly in L p q, and a.e. in . By (3.21), (3.22), and the assumption that τ ă 2, ż pG1 puj q, vq rj v ` op1q }v} , v P E, “ ∇r uj ¨ ∇v ´ λ u (3.24) 2ρj

and passing to the limit gives ż r v “ 0 @v P E, ∇r u ¨ ∇v ´ λ u

(3.25)

r solves (3.3). Taking v “ u r in (3.25) gives }r so u u}2 “ λ }r u}2L2 p q , and taking rj in (3.24) and passing to the limit gives λ }r v“u u}2L2 p q “ 1, so }r u} “ 1. rj á u r and }r rj Ñ u r. Since u uj } Ñ }r u}, u  First we consider the nonresonant case. Lemma 3.4.2 If λ R σ p´ q, then every sequence puj q Ă E such that G1 puj q Ñ 0 has a convergent subsequence, in particular, G satisfies pPSq. Proof Since λ R σ p´ q, puj q is bounded by Lemma 3.4.1, so the conclusion follows from Lemma 3.1.1.  Theorem 3.4.3 Assume (3.17) and (3.18). (i) If λ ă λ1 , then (3.1) has a solution u with Cq pG, uq « δq0 Z2 . (ii) If λl ă λ ă λl`1 , then (3.1) has a solution u with Cdl pG, uq ‰ 0.

3.4 Asymptotically linear problems

59

Proof G satisfies pPSq by Lemma 3.4.2. (i) By (3.20), 2P px, tq ě ´pλ1 ´ λq t 2 ´ C

@px, tq P ˆ R

for some constant C ą 0, and hence ż Gpuq ě |∇u|2 ´ λ1 u2 ´ C ě ´C | | @u P E

by (3.19) and (3.4). So Propositions 1.5.1 and 1.5.2 apply. (ii) By (3.20), ´pλl`1 ´ λq t 2 ´ C ď 2P px, tq ď pλ ´ λl q t 2 ` C

@px, tq P ˆ R

for some constant C ą 0, and hence ż Gpvq ď |∇v|2 ´ λl v 2 ` C ď C | | @v P Nl

by (3.19) and (3.6), and ż Gpwq ě |∇w|2 ´ λl`1 w2 ´ C ě ´C | | @w P Ml

by (3.19) and (3.7). So Corollary 2.7.8 applies.



In the resonant case we assume that either ppl´ q λ “ λl and α˘ pxq :“ lim inf tÑ˘8

ż

ż

tppx, tq satisfy |t|τ

α` pxq |y| `

α´ pxq |y|τ ą 0 @y P El z t0u ,

τ

yą0

yă0

or ppl` q λ “ λl and α ˘ pxq :“ lim sup tÑ˘8

ż

`

ż

tppx, tq satisfy |t|τ α ´ pxq |y|τ ă 0 @y P El z t0u .

α pxq |y| ` yą0

τ

yă0

Note that α˘ , α ˘ P L8 p q by (3.18) and α˘ pxq ď lim inf tÑ˘8

τ P px, tq τ P px, tq ď lim sup ď α ˘ pxq. τ |t| |t|τ tÑ˘8

(3.26)

Lemma 3.4.4 If ppl´ q or ppl` q holds, then every sequence puj q Ă E such that G1 puj q Ñ 0 has a convergent subsequence, in particular, G satisfies pPSq.

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Applications to semilinear problems

Proof It suffices to show that puj q is bounded by Lemma 3.1.1, so suppose rj :“ uj {ρj converges to ρj :“ }uj } Ñ 8. Then a renamed subsequence of u some y P El z t0u strongly in E and a.e. in by Lemma 3.4.1. Writing uj “ yj ` zj P El ‘ ElK , then r zj :“ zj {ρj Ñ 0. By (3.21), ż ˘ 1` 1 G puj q, v “ ∇zj ¨ ∇v ´ λl zj v ` ppx, uj q v, v P E, 2

and taking v “ yj {ρjτ gives ˇż ˇ ˇ y ppx, u q ˇ }G1 pu q} j ˇ j j ˇ Ñ0 ˇ ˇď τ τ ´1 ˇ ˇ ρj 2ρj since τ ě 1. By (3.22), ˇż ˇ ˜ ¸ ˇ z ppx, u q ˇ 1 ˇ j j ˇ zj } Ñ 0. ˇ ˇ ď C 1 ` τ ´1 }r ˇ ˇ ρjτ ρj Thus, ż

uj ppx, uj q |r uj |τ “ |uj |τ

ż

uj ppx, uj q Ñ 0. ρjτ

rj Ñ ˘8 a.e. on ty ż 0u, then Since uj “ ρj u ż ż ż ż α` pxq |y|τ ` α´ pxq |y|τ ď 0 ď α ` pxq |y|τ ` yą0

yă0

yą0

by Fatou’s lemma, contradicting ppl´ q and ppl` q.

α ´ pxq |y|τ yă0



Theorem 3.4.5 Assume (3.17) and (3.18). (i) If pp0´ q holds, then (3.1) has a solution u with Cq pG, uq « δq0 Z2 . ´ (ii) If ppl` q or ppl`1 q holds, then (3.1) has a solution u with

Cdl pG, uq ‰ 0. Proof G satisfies pPSq by Lemma 3.4.4. (i) We claim that every sequence puj q Ă E, ρj :“ }uj } Ñ 8 has a renamed subsequence such that Gpuj q Ñ `8, so that G is bounded from below and rj :“ uj {ρj “ yj {ρj ` wj {ρj “: Propositions 1.5.1 and 1.5.2 apply. Writing u

3.4 Asymptotically linear problems

61

r j P E1 ‘ M1 , we have yrj ` w ż Gpuj q “ |∇wj |2 ´ λ1 wj2 ` 2P px, uj q

ˆ ˙ ż λ1 2 2 r j } ` 2P px, uj q ě ρj 1 ´ }w λ2

(3.27)

by (3.19) and (3.7). Since ˇż ˇ ˇ ˇ ˇ P px, uj qˇ ď C pρ τ ` 1q j ˇ ˇ

r j } ą 0, then lim Gpuj q{ρj2 ą 0 and the claim by (3.23) and τ ă 2, if lim }w r j Ñ 0 for a renamed subsequence. Since }r follows, so suppose w yj } ď }r uj } “ 1 and E1 is finite dimensional, then a further subsequence of pr uj q converges to rj Ñ ˘8 a.e. on some y P E1 z t0u strongly in E and a.e. in . Then uj “ ρj u ty ż 0u, so ż Gpuj q 2P px, uj q lim ě lim |r uj |τ by (3.27) τ ρj |uj |τ

ˆż ˙ ż 2 τ τ ě α` pxq |y| ` α´ pxq |y| by Fatou’s lemma τ yą0 yă0 and (3.26) by pp0´ q.

ą0

(ii) We will show that G is bounded from above on Nl and from below on Ml , so that Corollary 2.7.8 applies. If ppl` q holds, 2P px, tq ě ´pλl`1 ´ λl q t 2 ´ C

@px, tq P ˆ R

for some constant C ą 0 by (3.20), and hence ż Gpwq ě |∇w|2 ´ λl`1 w2 ´ C ě ´C | | @w P Ml

by (3.19) and (3.7). We claim that every sequence puj q Ă Nl , ρj :“ }uj } Ñ 8 has a renamed subsequence such that Gpuj q Ñ ´8, so that G is bounded from rj :“ uj {ρj “ vj {ρj ` yj {ρj “: vrj ` yrj P Nl´1 ‘ El , above on Nl . Writing u we have ż |∇vj |2 ´ λl vj2 ` 2P px, uj q Gpuj q “

ˆ ˙ ż λl 2 2 ď ´ρj ´ 1 }r vj } ` 2P px, uj q (3.28) λl´1

62

Applications to semilinear problems

by (3.6). Since

ˇż ˇ ˇ ˇ ˇ P px, uj qˇ ď C pρ τ ` 1q j ˇ ˇ

by (3.23) and τ ă 2, if lim }r vj } ą 0, then lim Gpuj q{ρj2 ă 0 and the claim follows, so suppose vrj Ñ 0 for a renamed subsequence. Since }r yj } ď }r uj } “ 1 and El is finite dimensional, then a further subsequence of pr uj q converges to rj Ñ ˘8 a.e. on some y P El z t0u strongly in E and a.e. in . Then uj “ ρj u ty ż 0u, so ż Gpuj q 2P px, uj q lim ď lim |r uj |τ by (3.28) τ ρj |uj |τ

ˆż ˙ ż 2 ` τ ´ τ ď α pxq |y| ` α pxq |y| by Fatou’s lemma τ yą0 yă0 and (3.26) by ppl` q.

ă0 ´ If ppl`1 q holds,

2P px, tq ď pλl`1 ´ λl q t 2 ` C

@px, tq P ˆ R

for some constant C ą 0 by (3.20), and hence ż Gpvq ď |∇v|2 ´ λl v 2 ` C ď C | | @v P Nl

by (3.19) and (3.6). We claim that every sequence puj q Ă Ml , ρj :“ }uj } Ñ 8 has a renamed subsequence such that Gpuj q Ñ `8, so that G is bounded from rj :“ uj {ρj “ yj {ρj ` wj {ρj “: yrj ` w rj P El`1 ‘ below on Ml . Writing u Ml`1 , we have ż Gpuj q “ |∇wj |2 ´ λl`1 wj2 ` 2P px, uj q

ˆ ˙ ż λl`1 r j }2 ` 2P px, uj q ě ρj2 1 ´ }w (3.29) λl`2

by (3.7). Since

ˇż ˇ ˇ ˇ ˇ P px, uj qˇ ď C pρ τ ` 1q j ˇ ˇ

r j } ą 0, then lim Gpuj q{ρj2 ą 0 and the claim by (3.23) and τ ă 2, if lim }w rj Ñ 0 for a renamed subsequence. Since }r follows, so suppose w yj } ď }r uj } “ 1 and El`1 is finite dimensional, then a further subsequence of pr uj q converges

3.5 Problems with concave nonlinearities

63

rj Ñ ˘8 to some y P El`1 z t0u strongly in E and a.e. in . Then uj “ ρj u a.e. on ty ż 0u, so ż Gpuj q 2P px, uj q lim ě lim |r uj |τ by (3.29) τ ρjτ |u | j

ˆż ˙ ż 2 τ τ ě α` pxq |y| ` α´ pxq |y| by Fatou’s lemma τ yą0 yă0 and (3.26) ą0

´ by ppl`1 q.



Set d0 “ 0. By Theorems 3.4.3 and 3.4.5, (3.1) has a solution u with Cdl pG, uq ‰ 0 under the condition $ ´ ’ l“0 &λ ă λ1 , or pp0 q holds, l pf q ’ %λ ă λ ă λ , or pp ` q or pp´ q holds, l ě 1. l l`1 l l`1 By Theorem 3.3.2, Cq pG, 0q “ 0 for q ‰ dl0 under the condition pfl0 q there is a δ ą 0 such that $ f px, tq ’ & ď λ1 @x P , 0 ă |t| ď δ, t ’ %λl ď f px, tq ď λl `1 @x P , 0 ă |t| ď δ, 0 0 t

l0 “ 0 l0 ě 1.

Thus, we have the following theorem. Theorem 3.4.6 If (3.17), (3.18), pf l q, and pfl0 q hold with l0 ‰ l, then (3.1) has a nontrivial solution. Remark 3.4.7 The nonresonant case of Theorem 3.4.6 is due to Amann and Zehnder [3]. Related results can be found in Bartsch and Li [14], Costa and Silva [33], Hirano and Nishimura [59], Lazer and Solimini [67], Li and Liu [71, 73], Li and Willem [76], Li and Zhang [78], Li and Zou [79], Perera [108], Schechter [141], Silva [150], Su and Tang [154], and Zou and Liu [162].

3.5 Problems with concave nonlinearities In this section we consider the multiplicity of nontrivial solutions of (3.1) when f satisfies (3.17), (3.18), and f px, tq “ μ |t|σ ´2 t ` λ0 t ` optq as t Ñ 0, uniformly a.e.

(3.30)

64

Applications to semilinear problems

for some μ ‰ 0, σ P p1, 2q, and λ0 P R. Set qpx, tq “ f px, tq ´ μ |t|σ ´2 t,

Qpx, tq “

żt qpx, sq ds, 0

so that

ż Gpuq “

|∇u|2 ´

2μ σ |u| ´ 2Qpx, uq, σ

u P E.

(3.31)

Again we assume that G has only a finite number of critical points. For simplicity we only consider the nonresonant case. Let dl “ dim Nl . Theorem 3.5.1 Assume (3.17), (3.18), and (3.30) with λ R σ p´ q. (i) If λ0 ą λl ą λ and 2Qpx, tq ď λl`1 t 2

@px, tq P ˆ R

(3.32)

for some l, then there is a μ˚ ă 0 such that (3.1) has two nontrivial solutions u1 and u2 with Gpu1 q ě 0 ą Gpu2 q,

Cdl pG, u1 q ‰ 0, Cdl ´1 pG, u2 q ‰ 0

(3.33)

for all μ P pμ˚ , 0q. (ii) If λ0 ă λl`1 ă λ and 2Qpx, tq ě λl t 2

@px, tq P ˆ R

(3.34)

for some l, then there is a μ˚ ą 0 such that (3.1) has two nontrivial solutions u1 and u2 with Gpu1 q ą 0 ě Gpu2 q,

Cdl `1 pG, u1 q ‰ 0, Cdl pG, u2 q ‰ 0

(3.35)

for all μ P p0, μ˚ q. Proof G satisfies pPSq by Lemma 3.4.2. (i) We apply Corollary 2.6.4  to G with N “ Nl , M(“ Ml , and v0 P El z t0u, noting that then d “ dl and tv0 ` w : t ě 0, w P M Ă Ml´1 . For all μ ă 0, ż Gpwq ě |∇w|2 ´ λl`1 w2 ě 0 @w P Ml

by (3.31), (3.32), and (3.7). Since λl ą λ, 2P px, tq ě ´pλl ´ λq t 2 ´ C

@px, tq P ˆ R

3.5 Problems with concave nonlinearities

65

for some constant C ą 0 by (3.20), and hence ż Gpwq ě |∇w|2 ´ λl w2 ´ C ě ´C | | @w P Ml´1

by (3.19). Fixing 0 ă ε ă λ0 {λl ´ 1, 2F px, tq ě

2μ σ |t| ` p1 ` εq λl t 2 ´ C |t|r σ

@px, tq P ˆ R

for some r P p2, 2˚ q and a constant C ą 0 by (3.30), (3.17), and (3.18), and hence ` ˘ Gpvq ď ´ε }v}2 ` C |μ| }v}σ ` }v}r @v P Nl by (3.6) and the Sobolev embedding theorem, so  ( there are R ą 0, a ă 0, and μ˚ ă 0 such that G ď a on v P Nl : }v} “ R for all μ P pμ˚ , 0q. Taking a larger if necessary, we may assume that G has no critical values in ra, 0q, so Corollary 2.6.4 gives two critical points u1 and u2 satisfying (3.33); u1 ‰ 0 since Cq pG, 0q « δq0 Z2 by Theorem 3.3.7 (i), and u2 ‰ 0 since Gp0q “ 0. (ii) We apply Corollary 2.7.10 to ´G with  N “ Ml , M “ Nl , and (v0 P El`1 with }v0 } “ 1, noting that then d “ dl and tv0 ` w : t ě 0, w P M Ă Nl`1 . For all μ ą 0, ż ´Gpvq ě ´|∇v|2 ` λl v 2 ě 0 @v P Nl

by (3.31), (3.34), and (3.6). Since λ ą λl`1 , 2P px, tq ď pλ ´ λl`1 q t 2 ` C

@px, tq P ˆ R

for some constant C ą 0 by (3.20), and hence ż ´Gpvq ě ´|∇v|2 ` λl`1 v 2 ´ C ě ´C | | @v P Nl`1

by (3.19). Fixing 0 ă ε ă 1 ´ λ0 {λl`1 , 2F px, tq ď

2μ σ |t| ` p1 ´ εq λl`1 t 2 ` C |t|r σ

@px, tq P ˆ R

for some r P p2, 2˚ q and a constant C ą 0 by (3.30), (3.17), and (3.18), and hence ` ˘ ´Gpwq ď ´ε }w}2 ` C μ }w}σ ` }w}r @w P Ml by (3.7) and the Sobolev embedding theorem, so there are R ą 0, a ă 0, and  ( ˚ μ ą 0 such that ´G ă a on w P Ml : }w} “ R for all μ P p0, μ˚ q. Taking a larger if necessary, we may assume that ´G has no critical values in ra, 0q, so Corollary 2.7.10 gives two critical points u1 and u2 satisfying (3.35); u1 ‰ 0

66

Applications to semilinear problems

since Gp0q “ 0, and u2 ‰ 0 since Cq pG, 0q “ 0 for all q by Theorem 3.3.7 (ii).  Remark 3.5.2 Theorem 3.5.1 is a special case of a result due to Perera and Schechter [118]. Related results can be found in Ambrosetti et al. [6], Ambrosetti et al. [7], de Paiva and Massa [43], Li and Wang [75], Li et al. [77], Moroz [94], Perera [104, 105], and Wu and Yang [159].

4 Fuˇc´ık spectrum

4.1 Introduction Let us recall some terminology concerning mappings between Hilbert spaces. Definition 4.1.1 Let H, H 1 be Hilbert spaces. (i) f : H Ñ H 1 is bounded if it maps bounded sets into bounded sets. (ii) f : H Ñ H 1 is positive homogeneous of degree α ą 0 if f psuq “ s α f puq

@u P H, s ě 0.

Taking u “ 0 and s “ 0 gives f p0q “ 0. When α “ 1 we will simply say that f is positive homogeneous. (iii) f : H Ñ H is monotone if pf puq ´ f pvq, u ´ vq ě 0

@u, v P H.

(iv) f P CpH, H q is a potential operator if there is a F P C 1 pH, Rq, called a potential for f , such that F 1 puq “ f puq @u P H. Replacing F with F ´ F p0q gives a potential F with F p0q “ 0. Let H be a Hilbert space with the inner product p¨, ¨q and the associated norm }¨}. We assume that there are positive homogeneous monotone potential operators p, n P CpH, H q such that ppuq ` npuq “ u, pppuq, npuqq “ 0 67

@u P H.

(4.1)

Fuˇc´ık spectrum

68

We use the suggestive notation u` “ ppuq,

u´ “ ´npuq,

so that (4.1) becomes u “ u` ´ u´ ,

pu` , u´ q “ 0.

(4.2)

This implies }u}2 “ }u` }2 ` }u´ }2 ,

(4.3)

}u˘ } ď }u} .

(4.4)

in particular,

Let A be a self-adjoint operator on H with the spectrum σ pAq Ă p0, 8q and A´1 compact. Then σ pAq consists of isolated eigenvalues λl , l ě 1 of finite multiplicities satisfying 0 ă λ1 ă λ2 ă ¨ ¨ ¨ ă λl ă ¨ ¨ ¨ . Moreover, D “ DpA1{2 q is a Hilbert space with the inner product pu, vqD “ pA1{2 u, A1{2 vq “ pAu, vq and the associated norm }u}D “ }A1{2 u} “ pAu, uq1{2 . We have }u}2D “ pAu, uq ě λ1 pu, uq “ λ1 }u}2

@u P H,

so D ãÑ H , and the embedding is compact since A´1 is a compact operator. Let El be the eigenspace of λl , Nl “

l à j “1

Ej ,

Ml “ NlK X D.

Then D “ Nl ‘ M l is an orthogonal decomposition with respect to both p¨, ¨q and p¨, ¨qD . Moreover, }v}2D “ pAv, vq ď λl pv, vq “ λl }v}2

@v P Nl ,

}w}2D “ pAw, wq ě λl`1 pw, wq “ λl`1 }w}2

@w P Ml .

(4.5) (4.6)

4.2 Examples

69

We assume that w P M1 z t0u ùñ w ˘ ‰ 0.

(4.7)

Now let f P CpD, H q and consider the equation Au “ f puq,

u P D.

(4.8)

We say that (4.8) has a jumping nonlinearity at zero (resp. infinity) if f puq “ bu` ´ au´ ` op}u}D q as }u}D Ñ 0 (resp. 8)

(4.9)

for some pa, bq P R2 . The asymptotic equation associated with (4.8) when (4.9) holds is Au “ bu` ´ au´ ,

u P D.

(4.10)

The Fuˇc´ık spectrum pAq of A is the set of points pa, bq P R2 such that (4.10) has a nontrivial solution. Note that if a “ b “ λ, then (4.10) reduces to the eigenvalue problem Au “ λu,

u P D,

which has a nontrivial solution if and only if λ is one of the eigenvalues λl , so the points pλl , λl q are in pAq.

4.2 Examples Here are some concrete examples of the equation (4.10). Example 4.2.1 Perhaps the best-known example is the semilinear elliptic boundary value problem $ & ´ u “ bu` ´ au´ in (4.11) % u“0 on B where is a bounded domain in Rn , n ě 1 and u˘ “ max t˘u, 0u are the positive and negative parts of u, respectively. Here H “ L2 p q, D “ H01 p q is the usual Sobolev space, and A is the inverse of the solution operator S : H Ñ D, f ÞÑ u “ p´ q´1 f of the problem $ & ´ u “ f pxq in %

u“0

on B .

Since the embedding D ãÑ H is compact, A´1 “ S is compact on H .

Fuˇc´ık spectrum

70

The Fuˇc´ık spectrum p´ q of ´ was introduced by Dancer [37, 38] and Fuˇc´ık [53], who recognized its significance for the solvability of the problem $ & ´ u “ f px, uq in (4.12) % u“0 on B when f is a Carath´eodory function on ˆ R satisfying |f px, tq| ď C p|t| ` 1q

@px, tq P ˆ R

(4.13)

for some constant C ą 0 and f px, tq “ bt ` ´ at ´ ` optq as |t| Ñ 8, uniformly a.e. In the ODE case n “ 1, Fuˇc´ık showed that p´d 2 {dx 2 q consists of a sequence of hyperbolic-like curves passing through the points pλl , λl q, with one or two curves going through each point. In the PDE case n ě 2 also, p´ q consists, at least locally, of curves emanating from the points pλl , λl q; see Gallou¨et and Kavian [54], Ruf [136], Lazer and McKenna [66], Lazer [68], C´ac [22], Magalh˜aes [86], Cuesta and Gossez [35], de Figueiredo and Gossez [42], Schechter [140], and Margulies and Margulies [87]. In particular, it was shown in Schechter [140] that in the square pλl´1 , λl`1 q2 , p´ q contains two strictly decreasing curves, which may coincide, such that the points in the square that are either below the lower curve or above the upper curve are not in p´ q, while the points between them may or may not belong to p´ q when they do not coincide. Example 4.2.2 Let n ě 3 and consider the weighted problem $ & ´ u “ V pxq pbu` ´ au´ q in %

u“0

on B

with the weight V P Ln p q positive a.e. Here H “ L2 p q, D “ H01 p q, and A is the inverse of the solution operator S : H Ñ D, f ÞÑ u “ p´ q´1 pVf q of $ & ´ u “ V pxq f pxq in %

u“0

on B .

This includes singular weights such as V pxq “ |x|´q , 0 ă q ă 1.

4.2 Examples

71

Example 4.2.3 Consider the problem $ ’ & ´ u ` u “ 0

in

Bu “ bu` ´ au´ on B Bν where the parameters a, b appear in the boundary condition, B is now assumed to be C 1 , and B{Bν is the exterior normal derivative on B . Here H “ L2 pB q, D “ H 1 p q, and A is the inverse of the solution operator S : H Ñ D, f ÞÑ u of $ in ’ & ´ u ` u “ 0 ’ %

’ %

Bu “ f pxq Bν

on B .

Since the trace embedding D ãÑ H is compact, A´1 “ S is compact on H . Example 4.2.4 Consider the dynamic boundary value problem $ & ´pu ptqq “ b puσ ptqq` ´ a puσ ptqq´ , t P pa, bq X T % upaq “ upbq “ 0

(4.14)

where T, called a time scale, is a nonempty closed subset of ra, bs such that min T “ a, max T “ b,  ( σ ptq “ inf s P T : s ą t is the forward jump operator, u ptq “ lim

sÑt s‰σ ptq

upσ ptqq ´ upsq σ ptq ´ s

is the -derivative of u, and uσ ptq “ upσ ptqq. In particular, the top equation in (4.14) is an ordinary differential equation when T is continuous and a difference equation when T is discrete. Here H “ L2 ppa, bq X Tq, D “ H0,1 ppa, bq X Tq, and A is the inverse of the solution operator S : H Ñ D, f ÞÑ u of the problem $ & ´pu ptqq “ f ptq, t P pa, bq X T % upaq “ upbq “ 0. Since the embedding D ãÑ H is compact, A´1 “ S is compact on H . We refer to Agarwal et al. [1] for the details.

Fuˇc´ık spectrum

72

Note that in all the above examples p´uq˘ “ u¯

@u P H.

(4.15)

Then u solves (4.10) if and only if Ap´uq “ ap´uq` ´ bp´uq´ and hence pa, bq P pAq if and only if pb, aq P pAq, so pAq is symmetric about the line a “ b. We do not have this symmetry in the general theory we are developing here since we do not assume (4.15). Here is an example without symmetry. Example 4.2.5 Let ˘ be disjoint subdomains of such that ` Y ´ “ and consider the problem $ & ´ u “ b χ ` pxq u ` a χ ´ pxq u in %

u“0

on B

where χ ˘ are the characteristic functions of ˘ , respectively. Here ppuq “ χ ` pxq u,

npuq “ χ ´ pxq u

and H , D, and A are as in Example 4.2.1.

4.3 Preliminaries on operators Let us prove some basic properties of positive homogeneous and potential operators. Let H, H 1 be Hilbert spaces. Our first proposition gives a growth condition for a continuous positive homogeneous operator. Proposition 4.3.1 If f P CpH, H 1 q is positive homogeneous, then there is a constant C ą 0 such that }f puq} ď C }u}

@u P H,

in particular, f is bounded. Proof If not, there is a sequence puj q Ă H z t0u such that }f puj q} ą j 2 }uj } pj “ uj {pj }uj }q. Then Let u }p uj } “

1 Ñ 0, j

@j.

(4.16)

4.3 Preliminaries on operators

73

but }f pp uj q} “

}f puj q} ąj Ñ8 j }uj } 

by (4.16). This contradicts the continuity of f at zero. Our next proposition gives a formula for the potential of an operator.

Proposition 4.3.2 Let f P CpH, H q be a potential operator and F P C 1 pH, Rq its potential with F p0q “ 0. Then ż1 F puq “ pf psuq, uq ds, u P H, 0

in particular, if f is bounded, then so is F . If f is positive homogeneous, then 1 pf puq, uq , u P H, 2 in particular, F is positive homogeneous of degree 2, bounded, and there is a constant C ą 0 such that F puq “

|F puq| ď C }u}2

@u P H.

Proof We have ż1 ż1 ż1 ˘ ` 1 ˘ d ` F puq “ F psuq ds “ F psuq, u ds “ pf psuq, uq ds. 0 ds 0 0 The last integral equals ż1

s pf puq, uq ds “

0

1 pf puq, uq 2 

if f is positive homogeneous. Finally we have a proposition, as follows.

Proposition 4.3.3 If f P CpH, H q is a positive homogeneous potential operator and F P C 1 pH, Rq is its potential, then there is a constant C ą 0 such that |F puq ´ F pvq| ď C p}u} ` }v}q }u ´ v} Proof We have F puq ´ F pvq “

ż1 0

“

ż1 0

@u, v P H.

˘ d ` F psu ` p1 ´ sqvq ds ds pf psu ` p1 ´ sqvq, u ´ vq ds,

(4.17)

Fuˇc´ık spectrum

74

so |F puq ´ F pvq| ď

ż1

}f psu ` p1 ´ sqvq} }u ´ v} ds

0

ď

ˆż 1

˙ C }su ` p1 ´ sqv} ds }u ´ v} ,

0

where C is as in Proposition 4.3.1. Since }su ` p1 ´ sqv} ď s }u} ` p1 ´ sq }v} ď }u} ` }v} , 

(4.17) follows.

4.4 Variational formulation It is easy to see that A is a potential operator with the potential 1 1 pAu, uq “ }u}2D . 2 2 By Proposition 4.3.2 and (4.2), the potentials of p, n are 1 1 1 pppuq, uq “ pu` , u` ´ u´ q “ }u` }2 , 2 2 2 1 1 1 pnpuq, uq “ p´u´ , u` ´ u´ q “ }u´ }2 , 2 2 2 respectively. Let I pu, a, bq “ }u}2D ´ a }u´ }2 ´ b }u` }2 ,

u P D.

Then I p¨, a, bq P C pD, Rq with 1

I 1 puq “ 2pAu ` au´ ´ bu` q, so the critical points of I coincide with the solutions of (4.10). Thus, pa, bq P pAq if and only if I p¨, a, bq has a nontrivial critical point. Lemma 4.4.1 If I 1 puj q Ñ 0 and }uj }D “ 1, then a subsequence of puj q converges to a nontrivial critical point of I , in particular, pa, bq P pAq. Proof Since ´ 1 ´1 1 uj “ A´1 pbu` j ´ auj ` I puj q{2q “ A puj q,

where pu1j q is bounded and A´1 is compact, uj converges to some u P D for a renamed subsequence. Then I 1 puq “ 0 by the continuity of I 1 , and u ‰ 0 since }u}D “ 1. 

4.5 Some estimates

75

Proposition 4.4.2 If pa, bq R pAq, then every sequence puj q Ă D such that I 1 puj q Ñ 0 converges to zero, in particular, I satisfies pPSq. Proof If uj Û 0, then setting ρj :“ }uj }D we have inf ρj ą 0 for a renamed rj :“ uj {ρj . Then subsequence. Let u I 1 pr uj q “

I 1 puj q Ñ0 ρj

and }r uj }D “ 1, so pa, bq P pAq by Lemma 4.4.1, contrary to assumption.  Proposition 4.4.3 pAq is closed. Proof Let paj , bj q P pAq converge to pa, bq and let uj ‰ 0 satisfy ´ 1 Auj “ bj u` j ´ aj uj “: uj .

(4.18)

Replacing uj with uj { }uj }D if necessary, we may assume that }uj }D “ 1 and hence pu1j q is bounded. Since A´1 is compact, then uj “ A´1 pu1j q converges to some u P D with }u}D “ 1 for a renamed subsequence, and passing to the limit in (4.18) shows that u solves (4.10), so pa, bq P pAq. 

4.5 Some estimates In this section we derive some estimates for I and I 1 . Let λ “ min ta, bu ,

λ “ max ta, bu .

Then }u}2D ´ λ }u}2 ď I puq ď }u}2D ´ λ }u}2

@u P D

(4.19)

by (4.3). Lemma 4.5.1 There is a constant C ą 0 such that for all pa1 , b1 q, pa2 , b2 q P R2 , u1 , u2 P D, |I pu1 , a1 , b1 q ´ I pu2 , a2 , b2 q| ď p}u1 }D ` }u2 }D q }u1 ´ u2 }D ` C λ1 p}u1 } ` }u2 }q }u1 ´ u2 } ` p|a1 ´ a2 | ` |b1 ´ b2 |q }u2 }2 , where λ1 “ max ta1 , b1 u.

(4.20)

Fuˇc´ık spectrum

76

Proof We have I pu1 , a1 , b1 q ´ I pu2 , a2 , b2 q ´ 2 ` 2 ` 2 2 “ }u1 }2D ´ }u2 }2D ´ pa1 }u´ 1 } ´ a2 }u2 } q ´ pb1 }u1 } ´ b2 }u2 } q ´ 2 2 “ p}u1 }D ` }u2 }D qp}u1 }D ´ }u2 }D q ´ a1 p}u´ 1 } ´ }u2 } q ` 2 ´ 2 ` 2 2 ´ b1 p}u` 1 } ´ }u2 } q ´ pa1 ´ a2 q }u2 } ´ pb1 ´ b2 q }u2 } .

(4.21)

Since }u˘ }2 are the potentials of the positive homogeneous operators 2p, 2n, respectively, there is a constant C ą 0 such that ˇ ˘ 2 ˇ ˇ}u } ´ }u˘ }2 ˇ ď C p}u1 } ` }u2 }q }u1 ´ u2 } 1 2 by Proposition 4.3.3. So (4.20) follows from (4.21), the triangle inequality, and (4.4).  The following lemma is where we use the monotonicity of the operators p, n. Lemma 4.5.2 For all pa1 , b1 q, pa2 , b2 q P R2 , u1 , u2 P D, λ1 }u1 ´ u2 }2 ´ p|a1 ´ a2 | ` |b1 ´ b2 |q }u2 } }u1 ´ u2 } ď }u1 ´ u2 }2D ´

˘ 1` 1 I pu1 , a1 , b1 q ´ I 1 pu2 , a2 , b2 q, u1 ´ u2 2

ď λ1 }u1 ´ u2 }2 ` p|a1 ´ a2 | ` |b1 ´ b2 |q }u2 } }u1 ´ u2 } where λ1 “ min ta1 , b1 u , λ1 “ max ta1 , b1 u. Proof We have }u1 ´ u2 }2D ´

˘ 1` 1 I pu1 , a1 , b1 q ´ I 1 pu2 , a2 , b2 q, u1 ´ u2 2

´ ` ´ “ pb1 u` 1 ´ a1 u1 ´ b2 u2 ` a2 u2 , u1 ´ u2 q ` ´ ´ “ b1 pu` 1 ´ u2 , u1 ´ u2 q ´ a1 pu1 ´ u2 , u1 ´ u2 q ´ ` ppb1 ´ b2 q u` 2 ´ pa1 ´ a2 q u2 , u1 ´ u2 q.

Since p, n are monotone operators, ´ ` ` pu´ 1 ´ u2 , u1 ´ u2 q ď 0 ď pu1 ´ u2 , u1 ´ u2 q,

and 2 ` ´ ´ pu` 1 ´ u2 , u1 ´ u2 q ´ pu1 ´ u2 , u1 ´ u2 q “ }u1 ´ u2 } ,

4.6 Convexity and concavity

77

so 2 ` ´ ´ λ1 }u1 ´u2 }2 ď b1 pu` 1 ´u2 , u1 ´u2 q ´a1 pu1 ´u2 , u1 ´u2 q ď λ1 }u1 ´u2 } .

By the Schwarz inequality and (4.4), ˇ ˇ ˇppb1 ´b2 q u` ´pa1 ´a2 q u´ , u1 ´u2 q ˇ ď p|a1 ´a2 |`|b1 ´b2 |q }u2 } }u1 ´u2 } . 2 2  In the special case pa1 , b1 q “ pa2 , b2 q “ pa, bq, Lemmas 4.5.1 and 4.5.2 reduce, respectively, to the following. Lemma 4.5.3 There is a constant C ą 0 such that for all u1 , u2 P D, |I pu1 q´I pu2 q| ď p}u1 }D ` }u2 }D q }u1 ´ u2 }D ` C λ p}u1 } ` }u2 }q }u1 ´ u2 } . Lemma 4.5.4 For all u1 , u2 P D, λ }u1 ´ u2 }2 ď }u1 ´ u2 }2D ´

˘ 1` 1 I pu1 q ´ I 1 pu2 q, u1 ´ u2 ď λ }u1 ´ u2 }2 . 2

4.6 Convexity and concavity In this section we show that I pv ` y ` w, a, bq, v ` y ` w P Nl´1 ‘ El ‘ Ml is strictly concave in v and strictly convex in w when pa, bq is in the square Ql “ pλl´1 , λl`1 q2 ,

l ě 2.

Proposition 4.6.1 Let pa, bq P Ql . (i) If v1 ‰ v2 P Nl´1 , w P Ml´1 , then I pp1 ´ tqv1 ` t v2 ` wq ą p1 ´ tq I pv1 ` wq ` t I pv2 ` wq @t P p0, 1q. (ii) If v P Nl , w1 ‰ w2 P Ml , then I pv`p1 ´ tqw1 ` t w2 q ă p1´tq I pv ` w1 q ` t I pv ` w2 q

@t P p0, 1q.

Proof (i) We have I pp1 ´ tq v1 ` t v2 ` wq ´ p1 ´ tq I pv1 ` wq ´ t I pv2 ` wq “ p1 ´ tq rI pp1 ´ tq v1 ` t v2 ` wq ´ I pv1 ` wqs ´ t rI pv2 ` wq ´ I pp1 ´ tq v1 ` t v2 ` wqs ` ˘ “ p1 ´ tq t I 1 pu1 q ´ I 1 pu2 q, v2 ´ v1 ,

Fuˇc´ık spectrum

78

where u1 “ p1 ´ t1 q v1 ` t1 v2 ` w, u2 “ p1 ´ t2 q v1 ` t2 v2 ` w for some 0 ă t1 ă t ă t2 ă 1 by the mean-value theorem. Applying the first inequality in Lemma 4.5.4 gives ˘ 1` 1 I pu1 q ´ I 1 pu2 q, v2 ´ v1 2 ě pt2 ´ t1 qpλ }v1 ´ v2 }2 ´ }v1 ´ v2 }2D q ě pt2 ´ t1 qpλ ´ λl´1 q }v1 ´ v2 }2

by (4.5)

ą0 since t2 ą t1 , λ ą λl´1 , and v1 ‰ v2 . (ii) We have I pv ` p1 ´ tq w1 ` t w2 q ´ p1 ´ tq I pv ` w1 q ´ t I pv ` w2 q “ p1 ´ tq rI pv ` p1 ´ tq w1 ` t w2 q ´ I pv ` w1 qs ´ t rI pv ` w2 q ´ I pv ` p1 ´ tq w1 ` t w2 qs ` ˘ “ p1 ´ tq t I 1 pu1 q ´ I 1 pu2 q, w2 ´ w1 , where u1 “ v ` p1 ´ t1 q w1 ` t1 w2 , u2 “ v ` p1 ´ t2 q w1 ` t2 w2 for some 0 ă t1 ă t ă t2 ă 1 by the mean-value theorem. Applying the second inequality in Lemma 4.5.4 gives ˘ 1` 1 I pu1 q ´ I 1 pu2 q, w2 ´ w1 2 ď ´pt2 ´ t1 qp}w1 ´ w2 }2D ´ λ }w1 ´ w2 }2 q ď ´pt2 ´ t1 qpλl`1 ´ λq }w1 ´ w2 }2

by (4.6)

ă0 since t2 ą t1 , λl`1 ą λ, and w1 ‰ w2 .



4.7 Minimal and maximal curves In this section we construct the minimal and maximal curves of pAq in Ql . The development here closely follows Schechter [140]. Proposition 4.7.1 Let pa, bq P Ql . (i) There is a positive homogeneous map θ p¨, a, bq P CpMl´1 , Nl´1 q such that v “ θ pwq is the unique solution of I pv ` wq “ sup I pv 1 ` wq, v 1 PNl´1

w P Ml´1 .

(4.22)

4.7 Minimal and maximal curves

79

Moreover, I 1 pv ` wq K Nl´1 ðñ v “ θ pwq.

(4.23)

(ii) There is a positive homogeneous map τ p¨, a, bq P CpNl , Ml q such that w “ τ pvq is the unique solution of I pv ` wq “ inf I pv ` w1 q, 1 w PMl

v P Nl .

(4.24)

Moreover, I 1 pv ` wq K Ml ðñ w “ τ pvq.

(4.25)

Proof (i) For v P Nl´1 , w P Ml´1 , I pv ` wq ď }v ` w}2D ´ λ }v ` w}2

by (4.19)

“ }v}2D ` }w}2D ´ λ p}v}2 ` }w}2 q ď ´pλ ´ λl´1 q }v}2 ` }w}2D ´ λ }w}2

by (4.5).

Since λ ą λl´1 , this implies that I p¨ ` wq is bounded from above and anticoercive on the finite-dimensional space Nl´1 , so (4.22) has a solution v “ θ pwq satisfying I 1 pθ pwq ` wq K Nl´1 . It is unique by Proposition 4.6.1 (i). If I 1 pv ` wq K Nl´1 , then applying the first inequality in Lemma 4.5.4 with u1 “ v ` w, u2 “ θ pwq ` w and noting that u1 ´ u2 “ v ´ θ pwq K I 1 pu1 q ´ I 1 pu2 q gives λ }v ´ θpwq}2 ď }v ´ θ pwq}2D ď λl´1 }v ´ θ pwq}2 , so v “ θpwq. Next we show that θ is bounded. Applying the first inequality in Lemma 4.5.4 with u1 “ θ pwq ` w, u2 “ w and noting that u1 ´ u2 “ θ pwq K I 1 pu1 q gives λ }θ pwq}2 ď }θ pwq}2D `

˘ 1` 1 I pwq, θpwq 2

ď λl´1 }θ pwq}2 ` paw´ ´ bw ` , θpwqq since pAw, θpwqq “ pw, θ pwqqD “ 0.

Fuˇc´ık spectrum

80

Since

ˇ ˇ ˇpaw ´ ´ bw ` , θpwqq ˇ ď pa ` bq }w} }θ pwq} ,

it follows that }θ pwq} ď

2λ }w} . λ ´ λl´1

(4.26)

Now to see that θ is continuous, let w0 , w P Ml´1 . Applying the first inequality in Lemma 4.5.4 with u1 “ θpwq ` w, u2 “ θ pw0 q ` w0 and noting that θpwq ´ θ pw0 q K I 1 pu1 q ´ I 1 pu2 q gives λ p}θ pwq ´ θpw0 q}2 ` }w ´ w0 }2 q ď }θ pwq ´ θpw0 q}2D ` }w ´ w0 }2D ˘ 1` 1 ´ I pθ pwq ` wq ´ I 1 pθ pw0 q ` w0 q, w ´ w0 2 ď λl´1 }θ pwq ´ θ pw0 q}2 ´ pa pθ pwq ` wq´ ´ b pθpwq ` wq` ´ a pθ pw0 q ` w0 q´ ` b pθ pw0 q ` w0 q` , w ´ w0 q

(4.27)

since pApθ pwq ´ θ pw0 qq, w ´ w0 q “ pθpwq ´ θ pw0 q, w ´ w0 qD “ 0 and pApw ´ w0 q, w ´ w0 q “ }w ´ w0 }2D . Together with (4.26), (4.27) implies }θ pwq ´ θ pw0 q}2 ď C p}w} ` }w0 }q }w ´ w0 } for some constant C ą 0, so θ pwq Ñ θ pw0 q as w Ñ w0 . For s ě 0, by (4.23), I 1 ps θpwq ` swq “ s I 1 pθpwq ` wq K Nl´1 and hence θ pswq “ s θpwq. (ii) For v P Nl , w P Ml , I pv ` wq ě }v ` w}2D ´ λ }v ` w}2 “

}v}2D

`

}w}2D

2

by (4.19) 2

´ λ p}v} ` }w} q

ě p1 ´ λ{λl`1 q }w}2D ` }v}2D ´ λ }v}2

by (4.6).

(4.28)

4.7 Minimal and maximal curves

81

Since λ ă λl`1 , this implies that I pv ` ¨q is bounded from below and coercive on Ml . It is also weakly lower semicontinuous since the embedding D ãÑ H is compact. So (4.24) has a solution w “ τ pvq satisfying I 1 pv ` τ pvqq K Ml . It is unique by Proposition 4.6.1 (ii). If I 1 pv ` wq K Ml , then applying the second inequality in Lemma 4.5.4 with u1 “ v ` w, u2 “ v ` τ pvq and noting that u1 ´ u2 “ w ´ τ pvq K I 1 pu1 q ´ I 1 pu2 q gives λl`1 }w ´ τ pvq}2 ď }w ´ τ pvq}2D ď λ }w ´ τ pvq}2 , so w “ τ pvq. Next we show that τ is bounded. Applying the second inequality in Lemma 4.5.4 with u1 “ v ` τ pvq, u2 “ v and noting that u1 ´ u2 “ τ pvq K I 1 pu1 q gives ˘ 1` 1 }τ pvq}2D ď λ }τ pvq}2 ´ I pvq, τ pvq 2 ď λ }τ pvq}2D {λl`1 ´ pav ´ ´ bv ` , τ pvqq since pAv, τ pvqq “ pv, τ pvqqD “ 0. Since

ˇ ´ ˇ ˇpav ´ bv ` , τ pvqq ˇ ď pa ` bq }v} }τ pvq} {λ1 , D

it follows that }τ pvq}D ď

2λ }v} . λ1 p1 ´ λ{λl`1 q

(4.29)

Now to see that τ is continuous, let v0 , v P Nl . Applying the second inequality in Lemma 4.5.4 with u1 “ τ pvq ` v, u2 “ τ pv0 q ` v0 and noting that τ pvq ´ τ pv0 q K I 1 pu1 q ´ I 1 pu2 q gives }τ pvq ´ τ pv0 q}2D ď λ p}v ´ v0 }2 ` }τ pvq ´ τ pv0 q}2 q ´ }v ´ v0 }2D ˘ 1` 1 ` I pτ pvq ` vq ´ I 1 pτ pv0 q ` v0 q, v ´ v0 2 ď λ }v ´ v0 }2 ` λ }τ pvq ´ τ pv0 q}2D {λl`1 ` pa pτ pvq ` vq´ ´ b pτ pvq ` vq` ´ a pτ pv0 q ` v0 q´ ` b pτ pv0 q ` v0 q` , v ´ v0 q

(4.30)

Fuˇc´ık spectrum

82

since pApτ pvq ´ τ pv0 qq, v ´ v0 q “ pτ pvq ´ τ pv0 q, v ´ v0 qD “ 0 and pApv ´ v0 q, v ´ v0 q “ }v ´ v0 }2D . Together with (4.29), (4.30) implies }τ pvq ´ τ pv0 q}2D ď C p}v} ` }v0 }q }v ´ v0 } for some constant C ą 0, so τ pvq Ñ τ pv0 q as v Ñ v0 . For s ě 0, by (4.25), I 1 psv ` s τ pvqq “ s I 1 pv ` τ pvqq K Ml and hence τ psvq “ s τ pvq.



Next we show that θ, τ are locally bounded in and depend continuously on pa, bq. Lemma 4.7.2 Given pa0 , b0 q P Ql and a neighborhood N ĂĂ Ql of pa0 , b0 q, there is a constant C ą 0 such that (i) for all w P Ml´1 and pa, bq P N , }θ pw, a, bq} ď C }w}D , }θ pw, a, bq ´ θ pw, a0 , b0 q} ď C p|a ´ a0 | ` |b ´ b0 |q }w}D ,

(4.31) (4.32)

(ii) for all v P Nl and pa, bq P N , }τ pv, a, bq}D ď C }v} , }τ pv, a, bq ´ τ pv, a0 , b0 q}D ď C p|a ´ a0 | ` |b ´ b0 |q }v} .

(4.33) (4.34)

Proof (i) The estimate (4.26) gives (4.31). Applying the first inequality in Lemma 4.5.2 with pa1 , b1 q “ pa, bq, pa2 , b2 q “ pa0 , b0 q, u1 “ θ pw, a, bq` w, u2 “ θ pw, a0 , b0 q ` w and noting that u1 ´ u2 “ θ pw, a, bq ´ θpw, a0 , b0 q K I 1 pu1 , a1 , b1 q ´ I 1 pu2 , a2 , b2 q gives λ }θ pw, a, bq ´ θ pw, a0 , b0 q}2 ´ p|a1 ´ a2 | ` |b1 ´ b2 |q }θ pw, a0 , b0 q ` w} }θ pw, a, bq ´ θ pw, a0 , b0 q} ď }θ pw, a, bq ´ θpw, a0 , b0 q}2D ď λl´1 }θ pw, a, bq ´ θ pw, a0 , b0 q}2

(4.35)

4.7 Minimal and maximal curves

83

where λ “ min ta, bu. Since λ ´ λl´1 is uniformly positive in N , (4.32) follows from (4.31) and (4.35). (ii) The estimate (4.29) gives (4.33). Applying the second inequality in Lemma 4.5.2 with pa1 , b1 q “ pa, bq, pa2 , b2 q “ pa0 , b0 q, u1 “ v ` τ pv, a, bq, u2 “ v ` τ pv, a0 , b0 q and noting that u1 ´ u2 “ τ pv, a, bq ´ τ pv, a0 , b0 q K I 1 pu1 , a1 , b1 q ´ I 1 pu2 , a2 , b2 q gives }τ pv, a, bq ´ τ pv, a0 , b0 q}2D ´ p|a1 ´ a2 | ` |b1 ´ b2 |q }v ` τ pv, a0 , b0 q} }τ pv, a, bq ´ τ pv, a0 , b0 q} ď λ }τ pv, a, bq ´ τ pv, a0 , b0 q}2 ď λ }τ pv, a, bq ´ τ pv, a0 , b0 q}2D {λl`1

(4.36)

where λ “ max ta, bu. Since 1 ´ λ{λl`1 is uniformly positive in N , (4.34) follows from (4.33) and (4.36).  Corollary 4.7.3 We have (i) θ is continuous on Ml´1 ˆ Ql , (ii) τ is continuous on Nl ˆ Ql . Proof (i) If pwj , aj , bj q Ñ pw, a, bq in Ml´1 ˆ Ql , then }θpwj , aj , bj q ´ θ pw, a, bq} ď }θ pwj , aj , bj q ´ θ pwj , a, bq} ` }θ pwj , a, bq ´ θ pw, a, bq} and the last term goes to zero by the continuity of θ p¨, a, bq. By Lemma 4.7.2 (i) there is a constant C ą 0 such that for sufficiently large j , }θpwj , aj , bj q ´ θ pwj , a, bq} ď C p|aj ´ a| ` |bj ´ b|q }wj }D Ñ 0. (ii) If pvj , aj , bj q Ñ pv, a, bq in Nl ˆ Ql , then }τ pvj , aj , bj q ´ τ pv, a, bq}D ď }τ pvj , aj , bj q ´ τ pvj , a, bq}D ` }τ pvj , a, bq ´ τ pv, a, bq}D and the last term goes to zero by the continuity of τ p¨, a, bq. By Lemma 4.7.2 (ii) there is a constant C ą 0 such that for sufficiently large j , }τ pvj , aj , bj q ´ τ pvj , a, bq}D ď C p|aj ´ a| ` |bj ´ b|q }vj } Ñ 0. When a “ b “ λl , we have the following proposition.



Fuˇc´ık spectrum

84

Proposition 4.7.4 We have (i) θpw, λl , λl q “ 0 @w P Ml´1 , (ii) τ pv, λl , λl q “ 0 @v P Nl . Proof (i) Follows from (4.23) since I 1 pw, λl , λl q “ 2pA ´ λl q w K Nl´1 . (ii) Follows from (4.25) since I 1 pv, λl , λl q “ 2pA ´ λl q v K Ml .



For pa, bq P Ql , let σ pw, a, bq “ θ pw, a, bq ` w,

w P Ml´1 ,

Sl pa, bq “ σ pMl´1 , a, bq, ζ pv, a, bq “ v ` τ pv, a, bq,

v P Nl ,

S l pa, bq “ ζ pNl , a, bq. Then Sl , S l are topological manifolds modeled on Ml´1 , Nl , respectively. Thus, Sl is infinite dimensional, while S l is dl -dimensional, where dl “ dim Nl . For B Ă D, set r “ B X S, B where

(4.37)

 ( S “ u P D : }u}D “ 1

is the unit sphere in D. We say that B is a radial set if  ( r sě0 . B “ su : u P B, Since θ, τ are positive homogeneous, so are σ, ζ , and hence Sl , S l are radial manifolds. Let  ( Kpa, bq “ u P D : I 1 pu, a, bq “ 0 be the set of critical points of I p¨, a, bq. Since I 1 is positive homogeneous, K is a radial set. Moreover, ˘ 1` 1 I puq “ I puq, u (4.38) 2

4.7 Minimal and maximal curves

85

by Proposition 4.3.2 and hence I puq “ 0 @u P K.

(4.39)

Since D “ Nl´1 ‘ El ‘ Ml , Proposition 4.7.1 implies

 ( K “ u P Sl X S l : I 1 puq K El .

Together with (4.39), it also implies  ( K Ă u P Sl X S l : I puq “ 0 .

(4.40)

(4.41)

Set nl´1 pa, bq “

inf

sup I pv ` w, a, bq,

r l´1 vPNl´1 wPM

ml pa, bq “ sup inf I pv ` w, a, bq. rl vPN

wPMl

Since I pu, a, bq is nonincreasing in a for fixed u, b and in b for fixed u, a, nl´1 pa, bq, ml pa, bq are nonincreasing in a for fixed b and in b for fixed a. By Proposition 4.7.1, nl´1 pa, bq “ inf I pσ pw, a, bq, a, bq,

(4.42)

ml pa, bq “ sup I pζ pv, a, bq, a, bq.

(4.43)

r l´1 wPM

rl vPN

Proposition 4.7.5 Let pa, bq, pa 1 , b1 q P Ql . (i) Assume that nl´1 pa, bq “ 0. Then I pu, a, bq ě 0 @u P Sl pa, bq,  ( Kpa, bq “ u P Sl pa, bq : I pu, a, bq “ 0 ,

(4.44) (4.45)

and pa, bq P pAq. (a) If a 1 ď a, b1 ď b, and pa 1 , b1 q ‰ pa, bq, then nl´1 pa 1 , b1 q ą 0, I pu, a 1 , b1 q ą 0

@u P Sl pa 1 , b1 qz t0u ,

(4.46)

and pa 1 , b1 q R pAq. (b) If a 1 ě a, b1 ě b, and pa 1 , b1 q ‰ pa, bq, then nl´1 pa 1 , b1 q ă 0 and there is a u P Sl pa 1 , b1 qz t0u such that I pu, a 1 , b1 q ă 0.

Fuˇc´ık spectrum

86

(ii) Assume that ml pa, bq “ 0. Then I pu, a, bq ď 0 @u P S l pa, bq,  ( Kpa, bq “ u P S l pa, bq : I pu, a, bq “ 0 ,

(4.47) (4.48)

and pa, bq P pAq. (a) If a 1 ě a, b1 ě b, and pa 1 , b1 q ‰ pa, bq, then ml pa 1 , b1 q ă 0, I pu, a 1 , b1 q ă 0 @u P S l pa 1 , b1 qz t0u ,

(4.49)

and pa 1 , b1 q R pAq. (b) If a 1 ď a, b1 ď b, and pa 1 , b1 q ‰ pa, bq, then ml pa 1 , b1 q ą 0 and there is a u P S l pa 1 , b1 qz t0u such that I pu, a 1 , b1 q ą 0. Proof (i) By (4.42), I pσ pwqq ě }w}2D nl´1 “ 0 @w P Ml´1 ,

(4.50)

so (4.44) holds. By (4.41), K is contained in the set B on the right-hand side of (4.45). Noting that  ( B “ σ pwq : w P Ml´1 , I pσ pwqq “ 0 , suppose σ pwq P B, so I pσ pwqq “ 0.

(4.51)

To show that I 1 pσ pwqq “ 0 and hence σ pwq P K, it suffices to check that ` 1 ˘ I pσ pwqq, z “ 0 @z P Ml´1 (4.52) since I 1 pσ pwqq K Nl´1 by Proposition 4.7.1 (i). For t P R, I pθpw ` tzq ` w ` tzq´I pθpw ` tzq ` wq ě I pσ pw ` tzqq ´ I pσ pwqq ě 0 by Proposition 4.7.1 (i), (4.50), and (4.51). So ż1 ˘ d ` I pθpw ` tzq ` w ` stzq ds ě 0, ds 0 or ż1 ` 1 ˘ t I pθ pw ` tzq ` w ` stzq, z ds ě 0. 0

Dividing by t ą 0 and letting t Œ 0 gives pI 1 pσ pwqq, zq ě 0, and dividing by t ă 0 and letting t Õ 0 gives pI 1 pσ pwqq, zq ď 0, so (4.52) holds.

4.7 Minimal and maximal curves

87

To show that pa, bq P pAq, it now suffices to produce a nonzero element of r l´1 be a minimizing sequence for nl´1 in (4.42), so B. Let pwj q Ă M I pσ pwj qq Ñ 0. A renamed subsequence converges to some w P Ml´1 weakly in D and strongly in H since the embedding D ãÑ H is compact. Then θ pwj q Ñ θ pwq by (4.28) and hence σ pwj q á σ pwq in D, so (4.51) follows from the weak lower semicontinuity of I . By Proposition 4.7.1 (i), I pσ pwj qq ě I pwj q “ 1 ´ a }wj´ }2 ´ b }wj` }2 . Passing to the limit gives a }w´ }2 ` b }w ` }2 ě 1, so w ‰ 0 and hence σ pwq ‰ 0. paq Since I p¨, a 1 , b1 q ě I p¨, a, bq, nl´1 pa 1 , b1 q ě nl´1 pa, bq “ 0. Suppose nl´1 pa 1 , b1 q “ 0. As above, there is a w1 P Ml´1 z t0u such that I pσ pw1 , a 1 , b1 q, a 1 , b1 q “ 0. Let u “ σ pw 1 , a, bq P Sl pa, bqz t0u. Then 0 ď I pu, a, bq

by (4.44)

ď I pu, a 1 , b1 q ď I pσ pw1 , a 1 , b1 q, a 1 , b1 q by Proposition 4.7.1 (i) “0 and hence equality holds throughout, so u P Kpa, bq by (4.45). Moreover, 0 ď pa ´ a 1 q }u´ }2 ` pb ´ b1 q }u` }2 “ I pu, a 1 , b1 q ´ I pu, a, bq “ 0 and hence u´ or u` is zero. Thus, either u “ u` is a nontrivial solution of Au “ bu, or u “ ´u´ is a nontrivial solution of Au “ au. Since λl is the only eigenvalue of A in pλl´1 , λl`1 q, u P El in either case. This contradicts (4.7) since l ě 2, so nl´1 pa 1 , b1 q ą 0.

Fuˇc´ık spectrum

88

Then I pσ pw, a 1 , b1 q, a 1 , b1 q ě }w}2D nl´1 pa 1 , b1 q ą 0

@w P Ml´1 z t0u

by (4.42), so (4.46) holds. Hence Kpa 1 , b1 q “ t0u by (4.41), so pa 1 , b1 q R pAq. pbq Since I p¨, a 1 , b1 q ď I p¨, a, bq, nl´1 pa 1 , b1 q ď nl´1 pa, bq “ 0. If nl´1 pa 1 , b1 q “ 0, then nl´1 pa, bq ą 0 by paq, so nl´1 pa 1 , b1 q ă 0. Then there is a w P Ml´1 z t0u such that I pσ pw, a 1 , b1 q, a 1 , b1 q ă 0 by (4.42). (ii) By (4.43), I pζ pvqq ď }v}2D ml “ 0

@v P Nl ,

(4.53)

so (4.47) holds. By (4.41), K is contained in the set C on the right-hand side of (4.48). Noting that  ( C “ ζ pvq : v P Nl , I pζ pvqq “ 0 , suppose ζ pvq P C, so I pζ pvqq “ 0.

(4.54)

To show that I 1 pζ pvqq “ 0 and hence ζ pvq P K, it suffices to check that ` 1 ˘ I pζ pvqq, y “ 0 @y P Nl (4.55) since I 1 pζ pvqq K Ml by Proposition 4.7.1 (ii). For t P R, I pv ` ty ` τ pv ` tyqq ´ I pv ` τ pv ` tyqq ď I pζ pv ` tyqq ´ I pζ pvqq ď 0 by Proposition 4.7.1 (ii), (4.53), and (4.54). So ż1 ˘ d ` I pv ` sty ` τ pv ` tyqq ds ď 0, 0 ds or

ż1 t

`

˘ I 1 pv ` sty ` τ pv ` tyqq, y ds ď 0.

0

Dividing by t ą 0 and letting t Œ 0 gives pI 1 pζ pvqq, yq ď 0, and dividing by t ă 0 and letting t Õ 0 gives pI 1 pζ pvqq, yq ě 0, so (4.55) holds.

4.7 Minimal and maximal curves

89

To show that pa, bq P pAq, it now suffices to produce a nonzero element of C. Since Nl is finite dimensional, the supremum in (4.43) is achieved at some rl , so (4.54) holds and ζ pvq ‰ 0. vPN paq Since I p¨, a 1 , b1 q ď I p¨, a, bq,

ml pa 1 , b1 q ď ml pa, bq “ 0. Suppose ml pa 1 , b1 q “ 0. As above, there is a v 1 P Nl z t0u such that I pζ pv 1 , a 1 , b1 q, a 1 , b1 q “ 0. Let u “ ζ pv 1 , a, bq P S l pa, bqz t0u. Then 0 ě I pu, a, bq 1

by (4.47)

1

ě I pu, a , b q ě I pζ pv 1 , a 1 , b1 q, a 1 , b1 q by Proposition 4.7.1 (ii) “0 and hence equality holds throughout, so u P Kpa, bq by (4.48). Moreover, 0 ď pa 1 ´ aq }u´ }2 ` pb1 ´ bq }u` }2 “ I pu, a, bq ´ I pu, a 1 , b1 q “ 0 and hence u´ or u` is zero. Thus, either u “ u` is a nontrivial solution of Au “ bu, or u “ ´u´ is a nontrivial solution of Au “ au. Since λl is the only eigenvalue of A in pλl´1 , λl`1 q, u P El in either case. This contradicts (4.7) since l ě 2, so ml pa 1 , b1 q ă 0. Then I pζ pv, a 1 , b1 q, a 1 , b1 q ď }v}2D ml pa 1 , b1 q ă 0 @v P Nl z t0u by (4.43), so (4.49) holds. Hence Kpa 1 , b1 q “ t0u by (4.41), so pa 1 , b1 q R pAq. pbq Since I p¨, a 1 , b1 q ě I p¨, a, bq, ml pa 1 , b1 q ě ml pa, bq “ 0. If ml pa 1 , b1 q “ 0, then ml pa, bq ă 0 by paq, so ml pa 1 , b1 q ą 0. Then there is a v P Nl z t0u such that I pζ pv, a 1 , b1 q, a 1 , b1 q ą 0 by (4.43).



Fuˇc´ık spectrum

90

Next we show that nl´1 , ml depend continuously on pa, bq. Lemma 4.7.6 Given pa0 , b0 q P Ql and a neighborhood N ĂĂ Ql of pa0 , b0 q, there is a constant C ą 0 such that (i) for all pa, bq P N, |nl´1 pa, bq ´ nl´1 pa0 , b0 q| ď C p|a ´ a0 | ` |b ´ b0 |q,

(4.56)

(ii) for all pa, bq P N, |ml pa, bq ´ ml pa0 , b0 q| ď C p|a ´ a0 | ` |b ´ b0 |q.

(4.57)

Proof (i) By Lemmas 4.5.1 and 4.7.2 (i), there is a constant C ą 0 such that for all pa, bq P N, |I pσ pw, a, bq, a, bq ´ I pσ pw, a0 , b0 q, a0 , b0 q| ď C p|a ´ a0 | ` |b ´ b0 |q }w}2D

@w P Ml´1 ,

which together with (4.42) gives (4.56). (ii) By Lemmas 4.5.1 and 4.7.2 (ii), there is a constant C ą 0 such that for all pa, bq P N , |I pζ pv, a, bq, a, bq ´ I pζ pv, a0 , b0 q, a0 , b0 q| ď C p|a ´ a0 | ` |b ´ b0 |q }v}2

@v P Nl ,

which together with (4.43) gives (4.57).



When a “ b “ λl , we have the following proposition. Proposition 4.7.7 We have (i) nl´1 pλl , λl q “ 0, (ii) ml pλl , λl q “ 0. Proof (i) By Proposition 4.7.4 (i), σ pw, λl , λl q “ w for w P Ml´1 , so (4.42) gives nl´1 pλl , λl q “ inf I pw, λl , λl q. r l´1 wPM

We have I pw, λl , λl q “ }w}2D ´ λl }w}2 ě 0 rl . by (4.6) and equality holds for w P E

4.7 Minimal and maximal curves

91

(ii) By Proposition 4.7.4 (ii), ζ pv, λl , λl q “ v for v P Nl , so (4.43) gives ml pλl , λl q “ sup I pv, λl , λl q. rl vPN

We have I pv, λl , λl q “ }v}2D ´ λl }v}2 ď 0 rl . by (4.5) and equality holds for v P E



For a P pλl´1 , λl`1 q, set

 ( νl´1 paq “ sup b P pλl´1 , λl`1 q : nl´1 pa, bq ě 0 ,  ( μl paq “ inf b P pλl´1 , λl`1 q : ml pa, bq ď 0 .

Lemma 4.7.8 Let pa, bq P Ql . (i) b “ νl´1 paq ðñ nl´1 pa, bq “ 0. (ii) b “ μl paq ðñ ml pa, bq “ 0. Proof (i) Since nl´1 pa, ¨q is continuous on pλl´1 , λl`1 q by Lemma 4.7.6 (i), forward implication holds. Reverse implication follows from Proposition 4.7.5 (i) pbq. (ii) Since ml pa, ¨q is continuous on pλl´1 , λl`1 q by Lemma 4.7.6 (ii), forward implication holds. Reverse implication follows from Proposition 4.7.5 (ii) pbq.  The main theorem of this section is as follows. Theorem 4.7.9 Let pa, bq P Ql . (i) The function νl´1 is continuous, strictly decreasing, and satisfies (a) νl´1 pλl q “ λl , (b) b “ νl´1 paq ùñ pa, bq P pAq, (c) b ă νl´1 paq ùñ pa, bq R pAq. (ii) The function μl is continuous, strictly decreasing, and satisfies (a) μl pλl q “ λl , (b) b “ μl paq ùñ pa, bq P pAq, (c) b ą μl paq ùñ pa, bq R pAq. (iii) νl´1 paq ď μl paq. Proof (i) To see that νl´1 is continuous, let pa0 , νl´1 pa0 qq P Ql . Given ε ą 0, let 0 ă r ε ď ε be so small that νl´1 pa0 q ˘ r ε P pλl´1 , λl`1 q. Since nl´1 pa0 , νl´1 pa0 qq “ 0 by Lemma 4.7.8 (i), nl´1 pa0 , νl´1 pa0 q ˘ r εq ž 0

92

Fuˇc´ık spectrum

by Proposition 4.7.5 (i). Then there is a δ ą 0 such that |a ´ a0 | ă δ ùñ nl´1 pa, νl´1 pa0 q ˘ r εq ž 0 since nl´1 p¨, νl´1 pa0 q ˘ r εq are continuous by Lemma 4.7.6 (i). Since nl´1 pa, ¨q is nonincreasing and nl´1 pa, νl´1 paqq “ 0, then |a ´ a0 | ă δ ùñ |νl´1 paq ´ νl´1 pa0 q| ă r ε. To see that νl´1 is strictly decreasing, let pa1 , νl´1 pa1 qq, pa2 , νl´1 pa2 qq P Ql with a1 ă a2 . Since nl´1 pa1 , νl´1 pa1 qq “ 0 by Lemma 4.7.8 (i), nl´1 pa2 , νl´1 pa1 qq ă 0 by Proposition 4.7.5 (i) pbq. Then νl´1 pa2 q ă νl´1 pa1 q since nl´1 pa2 , ¨q is nonincreasing and nl´1 pa2 , νl´1 pa2 qq “ 0. paq Follows from Proposition 4.7.7 (i) and Lemma 4.7.8 (i). pbq We have nl´1 pa, bq “ 0 by Lemma 4.7.8 (i), so pa, bq P pAq by Proposition 4.7.5 (i). pcq We have nl´1 pa, bq ě 0, and since b ă νl´1 paq, nl´1 pa, bq ą 0 by Lemma 4.7.8 (i). Then pa, bq R pAq as in the proof of Proposition 4.7.5 (i) paq. (ii) To see that μl is continuous, let pa0 , μl pa0 qq P Ql . Given ε ą 0, let 0ăr ε ď ε be so small that μl pa0 q ˘ r ε P pλl´1 , λl`1 q. Since ml pa0 , μl pa0 qq “ 0 by Lemma 4.7.8 (ii), ml pa0 , μl pa0 q ˘ r εq ž 0 by Proposition 4.7.5 (ii). Then there is a δ ą 0 such that |a ´ a0 | ă δ ùñ ml pa, μl pa0 q ˘ r εq ž 0 since ml p¨, μl pa0 q ˘ r εq are continuous by Lemma 4.7.6 (ii). Since ml pa, ¨q is nonincreasing and ml pa, μl paqq “ 0, then |a ´ a0 | ă δ ùñ |μl paq ´ μl pa0 q| ă r ε. To see that μl is strictly decreasing, let pa1 , μl pa1 qq, pa2 , μl pa2 qq P Ql with a1 ă a2 . Since ml pa1 , μl pa1 qq “ 0 by Lemma 4.7.8 (ii), ml pa2 , μl pa1 qq ă 0 by Proposition 4.7.5 (ii) paq. Then μl pa2 q ă μl pa1 q

4.8 Null manifold

93

since ml pa2 , ¨q is nonincreasing and ml pa2 , μl pa2 qq “ 0. paq Follows from Proposition 4.7.7 (ii) and Lemma 4.7.8 (ii). pbq We have ml pa, bq “ 0 by Lemma 4.7.8 (ii), so pa, bq P pAq by Proposition 4.7.5 (ii). pcq We have ml pa, bq ď 0, and since b ą μl paq, ml pa, bq ă 0 by Lemma 4.7.8 (ii). Then pa, bq R pAq as in the proof of Proposition 4.7.5 (ii) paq. (iii) By (i) paq and (ii) paq, νl´1 pλl q “ λl “ μl pλl q. If a P pλl´1 , λl q and νl´1 paq ą μl paq, then λl ă μl paq ă νl´1 paq ď λl`1 since μl is strictly decreasing, so pa, μl paqq P pAq by (ii) pbq, contradicting (i) pcq. If a P pλl , λl`1 q and νl´1 paq ą μl paq, then λl´1 ď μl paq ă νl´1 paq ă λl since νl´1 is strictly decreasing, so pa, νl´1 paqq P pAq by (i) pbq, contradicting (ii) pcq.  Thus, Cl : b “ νl´1 paq,

C l : b “ μl paq

are strictly decreasing curves in Ql that belong to pAq. They both pass through the point pλl , λl q and may coincide. The region  ( Il “ pa, bq P Ql : b ă νl´1 paq below the lower curve Cl and the region  ( Il “ pa, bq P Ql : b ą μl paq above the upper curve C l are free of pAq. They are the minimal and maximal curves of pAq in Ql in this sense. Points in the region  ( IIl “ pa, bq P Ql : νl´1 paq ă b ă μl paq between Cl , C l , when it is nonempty, may or may not belong to pAq.

4.8 Null manifold For pa, bq P Ql , let N l pa, bq “ Sl pa, bq X S l pa, bq

Fuˇc´ık spectrum

94

(see p. 84). Since Sl , S l are radial sets, so is Nl . We will see that Nl is a topological manifold modeled on El and hence dim Nl “ dl ´ dl´1 . We will call it the null manifold of I . Proposition 4.8.1 Let pa, bq P Ql . (i) There is a positive homogeneous map ηp¨, a, bq P CpEl , Nl´1 q such that v “ ηpyq is the unique solution of I pζ pv ` yqq “ sup I pζ pv 1 ` yqq, v 1 PNl´1

y P El .

(4.58)

Moreover, I 1 pζ pv ` yqq K Nl´1 ðñ v “ ηpyq.

(4.59)

(ii) There is a positive homogeneous map ξ p¨, a, bq P CpEl , Ml q such that w “ ξ pyq is the unique solution of I pσ py ` wqq “ inf I pσ py ` w1 qq, 1 w PMl

y P El .

(4.60)

Moreover, I 1 pσ py ` wqq K Ml ðñ w “ ξ pyq.

(4.61)

ζ pηpyq ` yq “ σ py ` ξ pyqq,

(4.62)

(iii) For all y P El ,

i.e., ηpyq “ θ py ` ξ pyqq,

ξ pyq “ τ pηpyq ` yq.

First we prove a lemma. Lemma 4.8.2 If ui “ vi ` y ` wi P Nl´1 ‘ El ‘ Ml , I 1 pui q K ElK , then u1 “ u2 . Proof By Proposition 4.7.1, u1 “ σ py ` w1 q since I 1 pu1 q K Nl´1 and u2 “ ζ pv2 ` yq

i “ 1, 2,

(4.63)

4.8 Null manifold

95

since I 1 pu2 q K Ml , so I pu1 q ě I pv2 ` y ` w1 q ě I pu2 q. First inequality is strict if v1 ‰ v2 and the second is strict if w1 ‰ w2 , so I pu1 q ą I pu2 q if u1 ‰ u2 . This is impossible since interchanging u1 and u2 then gives the reverse inequality.  Proof of Proposition 4.8.1 (i) For v P Nl´1 , y P El , I pζ pv ` yqq ď I pv ` yq ď }v `

y}2D

by Prop. 4.7.1 (ii) ´ λ }v ` y}

2

by (4.19)

ď ´pλ ´ λl´1 q }v}2 ` }y}2D ´ λ }y}2

by (4.5).

(4.64) Since λ ą λl´1 , this implies that I pζ p¨ ` yqq is bounded from above and anticoercive on the finite-dimensional space Nl´1 , so (4.58) has a solution v “ ηpyq satisfying I 1 pζ pηpyq ` yqq K Nl´1 . Since any solution v of (4.58) satisfies I 1 pζ pv ` yqq K Nl´1 , to prove uniqueness and (4.59) it only remains to show that I 1 pζ pvi ` yqq K Nl´1 , i “ 1, 2 ùñ v1 “ v2 . Apply Lemma 4.8.2 with ui “ vi ` y ` τ pvi ` yq and note that I 1 pui q K Ml by Proposition 4.7.1 (ii). For s ě 0, by (4.59), I 1 pζ ps ηpyq ` syqq “ s I 1 pζ pηpyq ` yqq K Nl´1 and hence ηpsyq “ s ηpyq. (ii) For y P El , w P Ml , I pσ py ` wqq ě I py ` wq

by Prop. 4.7.1 (i)

ě }y `

w}2D

ě p1 ´

λ{λl`1 q }w}2D

´ λ }y ` w} `

2

}y}2D

by (4.19) 2

´ λ }y}

by (4.6).

Since λ ă λl`1 , this implies that I pσ py ` ¨qq is bounded from below and coercive on Ml . It is also weakly lower semicontinuous since the embedding

Fuˇc´ık spectrum

96

D ãÑ H is compact. So (4.60) has a solution w “ ξ pyq satisfying I 1 pσ py ` ξ pyqqq K Ml . Since any solution w of (4.60) satisfies I 1 pσ py ` wqq K Ml , to prove uniqueness and (4.61) it only remains to show that I 1 pσ py ` wi qq K Ml , i “ 1, 2 ùñ w1 “ w2 . Apply Lemma 4.8.2 with ui “ θ py ` wi q ` y ` wi and note that I 1 pui q K Nl´1 by Proposition 4.7.1 (i). For s ě 0, by (4.61), I 1 pσ psy ` s ξ pyqqq “ s I 1 pσ py ` ξ pyqqq K Ml and hence ξ psyq “ s ξ pyq. (iii) Applying Lemma 4.8.2 with u1 “ ηpyq ` y ` τ pηpyq ` yq, u2 “ θpy ` ξ pyqq ` y ` ξ pyq and noting that I 1 pu1 q K Ml , I 1 pu2 q K Nl´1 by Proposition 4.7.1 gives (4.62). Continuity of η, ξ follows from our next lemma.  Lemma 4.8.3 We have (i) η is continuous on El ˆ Ql , (ii) ξ is continuous on El ˆ Ql . Proof (i) Let pyj , aj , bj q Ñ py, a, bq in El ˆ Ql and suppose that vj “ ηpyj , aj , bj q Û ηpy, a, bq, so inf }vj ´ ηpy, a, bq} ą 0 j

(4.65)

for a renamed subsequence. Since I pζ pvj ` yj , aj , bj q, aj , bj q ě I pζ pyj , aj , bj q, aj , bj q, (4.64) gives }vj }2 ď p}yj }2D ´ λj }yj }2 ´ I pζ pyj , aj , bj q, aj , bj qq{pλj ´ λl´1 q Ñ p}y}2D ´ λ }y}2 ´ I pζ py, a, bq, a, bqq{pλ ´ λl´1 q where λj “ min taj , bj u , λ “ min ta, bu. So pvj q is bounded and hence converges to some v P Nl´1 for a renamed subsequence since Nl´1 is finite dimensional. Then ζ pvj ` yj , aj , bj q Ñ ζ pv ` y, a, bq by Corollary 4.7.3 (ii) and hence I 1 pζ pvj ` yj , aj , bj q, aj , bj q Ñ I 1 pζ pv ` y, a, bq, a, bq.

4.8 Null manifold

97

Since I 1 pζ pvj ` yj , aj , bj q, aj , bj q K Nl´1 , then I 1 pζ pv ` y, a, bq, a, bq K Nl´1 , but v ‰ ηpy, a, bq by (4.65). This contradicts (4.59). (ii) If pyj , aj , bj q Ñ py, a, bq in El ˆ Ql , then ηpyj , aj , bj q Ñ ηpy, a, bq by (i) and hence ξ pyj , aj , bj q “ τ pηpyj , aj , bj q ` yj , aj , bj q Ñ τ pηpy, a, bq ` y, a, bq “ ξ py, a, bq 

by (4.63) and Corollary 4.7.3 (ii). When a “ b “ λl , we have the following proposition. Proposition 4.8.4 We have (i) ηpy, λl , λl q “ 0 @y P El , (ii) ξ py, λl , λl q “ 0 @y P El . Proof (i) Follows from (4.59) since Proposition 4.7.4 (ii) gives I 1 pζ py, λl , λl q, λl , λl q “ I 1 py, λl , λl q “ 2pA ´ λl q y “ 0. (ii) Follows from (4.61) since Proposition 4.7.4 (i) gives I 1 pσ py, λl , λl q, λl , λl q “ I 1 py, λl , λl q “ 2pA ´ λl q y “ 0.



Referring to Proposition 4.8.1 (iii), let ϕpyq “ ζ pηpyq ` yq “ σ py ` ξ pyqq,

y P El .

Proposition 4.8.5 Let pa, bq P Ql . (i) ϕp¨, a, bq P CpEl , Dq is a positive homogeneous map such that I pϕpyqq “ inf sup I pv`y ` wq “ sup inf I pv ` y ` wq, wPMl vPNl´1

vPNl´1 wPMl

and I 1 pϕpyqq P El

@y P El .

(ii) If pa 1 , b1 q P Ql with a 1 ě a and b1 ě b, then I pϕpy, a 1 , b1 q, a 1 , b1 q ď I pϕpy, a, bq, a, bq (iii) (iv) (v) (vi)

@y P El .

ϕ is continuous on El ˆ Ql . ϕpy, λl , λl q “ y @y P El . ( Nl pa, bq “ ϕpy, a, bq : y P El . Nl pλl , λl q “ El .

Proof (i) Follows from Propositions 4.7.1 and 4.8.1 (i) and (ii). (ii) Follows from (i) since I p¨, a 1 , b1 q ď I p¨, a, bq.

y P El

Fuˇc´ık spectrum

98

(iii) Follows from Corollary 4.7.3 and Lemma 4.8.3. (iv) Follows from Propositions 4.7.4 and 4.8.4. (v) Clearly, ϕpyq is in Sl , S l and hence in Nl for each y P El . Conversely, let u P Nl , and write u “ v ` y ` w P Nl´1 ‘ El ‘ Ml . Since u P Sl , S l , u “ σ py ` wq “ ζ pv ` yq. Then I 1 pσ py ` wqq “ I 1 pζ pv ` yqq K Ml by Proposition 4.7.1 (ii) and hence w “ ξ pyq by Proposition 4.8.1 (ii). So u “ σ py ` ξ pyqq “ ϕpyq. 

(vi) Follows from (iv) and (v). By (4.40) and (4.41),  (  ( K “ u P Nl : I 1 puq K El Ă u P Nl : I puq “ 0 .

(4.66)

The following theorem shows that the curves Cl , C l are closely related to Ir “ I |Nl . Theorem 4.8.6 Let pa, bq P Ql . (i) If b ă νl´1 paq, then Irpu, a, bq ą 0 @u P Nl pa, bqz t0u . (ii) If b “ νl´1 paq, then Irpu, a, bq ě 0 @u P Nl pa, bq,  ( Kpa, bq “ u P Nl pa, bq : Irpu, a, bq “ 0 . (iii) If νl´1 paq ă b ă μl paq, then there are ui P Nl pa, bqz t0u , i “ 1, 2 such that Irpu1 , a, bq ă 0 ă Irpu2 , a, bq. (iv) If b “ μl paq, then Irpu, a, bq ď 0 @u P Nl pa, bq,  ( Kpa, bq “ u P Nl pa, bq : Irpu, a, bq “ 0 . (v) If b ą μl paq, then Irpu, a, bq ă 0 @u P Nl pa, bqz t0u .

4.8 Null manifold

99

Proof (i) As in the proof of Theorem 4.7.9 (i) pcq, nl´1 pa, bq ą 0. Then I ą 0 on Sl z t0u Ą Nl z t0u as in the proof of Proposition 4.7.5 (i) paq. (ii) By Lemma 4.7.8 (i), nl´1 pa, bq “ 0. Apply Proposition 4.7.5 (i) and note that Sl Ą Nl Ą K. (iii) We have nl´1 pa, bq ă 0 ă ml pa, bq. Then there are σ py1 ` wq P Sl z t0u , ζ pv ` y2 q P S l z t0u such that I pσ py1 ` wqq ă 0 ă I pζ pv ` y2 qq as in the proof of Proposition 4.7.5 (i) pbq and (ii) pbq. Let u1 “ σ py1 ` ξ py1 qq, u2 “ ζ pηpy2 q ` y2 q. By Proposition 4.8.1 (i) and (ii), ui P Nl and I pu1 q ď I pσ py1 ` wqq,

I pu2 q ě I pζ pv ` y2 qq.

(iv) By Lemma 4.7.8 (ii), ml pa, bq “ 0. Apply Proposition 4.7.5 (ii) and note that S l Ą Nl Ą K. (v) As in the proof of Theorem 4.7.9 (ii) pcq, ml pa, bq ă 0. Then I ă 0 on S l z t0u Ą Nl z t0u as in the proof of Proposition 4.7.5 (ii) paq.  By (4.66), solutions of (4.10) are in Nl . Next we show that the set of solutions is all of Nl exactly when pa, bq is on both Cl and C l . Theorem 4.8.7 If pa, bq P Ql , then Kpa, bq “ Nl pa, bq if and only if pa, bq P Cl X C l . Proof If pa, bq P Cl X C l , then Ir ě 0 by Theorem 4.8.6 (ii) and Ir ď 0 by (iv), so Ir “ 0 and hence K “ Nl by (ii) or (iv). If K “ Nl , then Ir “ 0 by (4.66) and hence pa, bq P Cl Y C l by Theorem 4.8.6 (i), (iii), and (v). If pa, bq P Cl , then Irp¨, a, b1 q ď Irp¨, a, bq “ 0 for all b1 P rb, λl`1 q by Proposition 4.8.5 (ii) and hence pa, bq P C l by Theorem 4.8.6 (iii). If pa, bq P C l , then Irp¨, a, b1 q ě Irp¨, a, bq “ 0 for all b1 P pλl´1 , bs by Proposition 4.8.5 (ii) and hence pa, bq P Cl by Theorem 4.8.6 (iii).  When λl is a simple eigenvalue, Nl is one-dimensional and hence Theorem 4.8.7 implies the following. Corollary 4.8.8 If λl is simple, then pa, bq P Ql is on exactly one of the curves Cl , C l if and only if  ( Kpa, bq “ t ϕpy0 , a, bq : t ě 0 for some y0 P El z t0u.

Fuˇc´ık spectrum

100

4.9 Type II regions In this section we give a sufficient condition for the region IIl to be nonempty (see p. 93). Theorem 4.9.1 If there are yi P El , i “ 1, 2 such that }y1` } ´ }y1´ } ă 0 ă }y2` } ´ }y2´ }, then there is a neighborhood N Ă Ql of pλl , λl q such that every point pa, bq P N z tpλl , λl qu with a ` b “ 2λl is in IIl . Proof It suffices to show that there are ui P Nl pa, bq, i “ 1, 2 such that Irpu1 , a, bq ă 0 ă Irpu2 , a, bq

(4.67)

by Theorem 4.8.6 (i), (ii), (iv), and (v). Since yi “ ηpyi , λl , λl q ` yi “ yi ` ξ pyi , λl , λl q by Proposition 4.8.4 and ηpyi , ¨, ¨q ` yi , yi ` ξ pyi , ¨, ¨q are continuous on Ql by Lemma 4.8.3, there is a neighborhood N Ă Ql of pλl , λl q such that for pa, bq P N , setting vi “ ηpyi , a, bq ` yi ,

wi “ yi ` ξ pyi , a, bq

we have }v1` } ´ }v1´ } ă 0 ă }v2` } ´ }v2´ },

}w1` } ´ }w1´ } ă 0 ă }w2` } ´ }w2´ }.

By Proposition 4.8.5 (v), ϕpyi q “ ζ pvi q “ σ pwi q P Nl , and if a ` b “ 2λl , then Irpζ pvi qq ď I pvi q ď

by Proposition 4.7.1 (ii)

λl p}vi` }2

}vi´ }2 q

`

´ p2λl ´

aq }vi` }2

´a

}vi´ }2 by (4.5)

“ pa ´ λl qp}vi` }2 ´ }vi´ }2 q and Irpσ pwi qq ě I pwi q ě

by Proposition 4.7.1 (i)

λl p}wi` }2 ´ p2λl ´

“ pa ´

`

}wi´ }2 q

´a

}wi´ }2

aq }wi` }2

λl qp}wi` }2

´

by (4.6) }wi´ }2 q.

4.11 Critical groups

101

If pa, bq ‰ pλl , λl q, then a ‰ λl since a ` b “ 2λl . Thus, (4.67) holds for $ $ &ϕpy2 q, a ă λl &ϕpy1 q, a ă λl u1 “ u2 “  %ϕpy q, a ą λ , %ϕpy q, a ą λ . 1 l 2 l Corollary 4.9.2 If (4.15) holds and there is a y P El such that }y ` } ‰ }y ´ }, then there is a neighborhood N Ă Ql of pλl , λl q such that every point pa, bq P N z tpλl , λl qu with a ` b “ 2λl is in IIl . Proof By (4.15), }p´yq` } ´ }p´yq´ } “ ´p}y ` } ´ }y ´ }q and hence }p˘yq` } ´ }p˘yq´ } have opposite signs.



Remark 4.9.3 For problem (4.11), Corollary 4.9.2 is due to Li et al. [69].

4.10 Simple eigenvalues In this section we show that when λl is a simple eigenvalue, the region IIl is free of pAq. Theorem 4.10.1 If λl is simple, then IIl X pAq “ H. Proof If pa, bq P IIl , then there are ui P Nl z t0u , i “ 1, 2 such that Irpu1 q ă 0 ă Irpu2 q by Theorem 4.8.6 (iii). Since Nl is one-dimensional and Ir is positive homogeneous of degree 2, then Ir ‰ 0 on Nl z t0u and hence K “ t0u by (4.66), so pa, bq R pAq.  Remark 4.10.2 For problem (4.11), Theorem 4.10.1 is due to Gallou¨et and Kavian [54].

4.11 Critical groups When pa, bq R pAq, the origin is the only critical point of I , so the critical groups Cq pI, 0q are defined. First we show that they are constant in connected components of R2 zpAq.

Fuˇc´ık spectrum

102

Proposition 4.11.1 If pa0 , b0 q and pa1 , b1 q belong to the same connected component of R2 zpAq, then Cq pI p¨, a0 , b0 q, 0q « Cq pI p¨, a1 , b1 q, 0q

@q.

Proof Since R2 zpAq is open by Proposition 4.4.3, so are its connected components, which are then path connected. Let r0, 1s Ñ R2 zpAq, t ÞÑ pat , bt q be a path joining pa0 , b0 q and pa1 , b1 q. Since pat , bt q R pAq, the origin is the only critical point of I p¨, at , bt q, which satisfies pPSq by Proposition 4.4.2. We apply Theorem 1.4.4 in a closed and bounded neighborhood U of the origin. Clearly, (i) holds. For t, t0 P r0, 1s, u P U , › › |I pu, at , bt q ´ I pu, at0 , bt0 q| ` ›I 1 pu, at , bt q ´ I 1 pu, at0 , bt0 q›D › › “ |pat ´at q }u´ }2 ` pbt ´bt q }u` }2 | ` 2 ›pat ´ at q u´ ´ pbt ´ bt q u` › 0

0

0

0

D

ď C p|at ´ at0 | ` |bt ´ bt0 |q for some constant C ą 0, so the continuity of at , bt gives (ii).



For B Ă D, set  ( B ´ “ u P B : I puq ă 0 ,

B ` “ BzB ´ .

The main theorem of this section is as follows. Theorem 4.11.2 Let pa, bq P Ql zpAq. (i) If pa, bq P Il , then Cq pI, 0q « δqdl´1 Z2 . (ii) If pa, bq P Il , then Cq pI, 0q « δqdl Z2 . (iii) If pa, bq P IIl , then Cq pI, 0q “ 0,

q ď dl´1 or q ě dl

and r ´ q, Cq pI, 0q « Hrq´dl´1 ´1 pN l

dl´1 ă q ă dl .

In particular, Cq pI, 0q “ 0 for all q when λl is simple.

(4.68)

4.11 Critical groups

103

Proof By Proposition 4.4.2, I satisfies pPSq since pa, bq R pAq, so applying Proposition 1.4.1 with a ă 0 “ I p0q, b “ `8 gives Cq pI, 0q « Hq pD, I a q.

(4.69)

Since D is contractible, Lemma 1.4.6 (ii) gives Hq pD, I a q « Hrq´1 pI a q.

(4.70)

By Remark 1.3.8, I a is a strong deformation retract of D ´ and hence Hrq´1 pI a q « Hrq´1 pD ´ q.

(4.71)

Writing u P D ´ as v ` w P Nl ‘ Ml , let η1 pu, tq “ v ` p1 ´ tq w ` t τ pvq,

pu, tq P D ´ ˆ r0, 1s.

We have I pη1 pu, tqq ď p1 ´ tq I pv ` wq ` t I pv ` τ pvqq by Proposition 4.6.1 (ii) ď I puq

by Proposition 4.7.1 (ii)

ă 0, so η1 is a strong deformation retraction of D ´ onto S l´ . On the other hand, η2 pu, tq “ p1 ´ tq u ` t πpuq,

pu, tq P S l´ ˆ r0, 1s,

where π : Dz t0u Ñ S,

u ÞÑ

u }u}D

is the radial projection onto S, is a strong deformation retraction of S l´ onto Srl´ by the positive homogeneity of ζ and I . Thus, Hrq´1 pD ´ q « Hrq´1 pSrl´ q.

(4.72)

Combining (4.69)–(4.72) gives Cq pI, 0q « Hrq´1 pSrl´ q.

(4.73)

If pa, bq P Il , then I ă 0 on S l z t0u by the proof of Theorem 4.8.6 (v) and rl , (ii) follows. Since Il hence Srl´ “ Srl . Since the latter is homeomorphic to N and Il´1 are subsets of the same connected component of R2 zpAq, (i) follows from Proposition 4.11.1 and (ii). Now let pa, bq P IIl . By Theorem 4.8.6 (iii), Srl´ is a proper subset of Srl and hence Cq pI, 0q « Hrq´1 pSrl´ q “ 0 for q ě dl . Since Hrq pSrl q “ δqpdl ´1q Z2 , the

Fuˇc´ık spectrum

104

exact sequence ¨ ¨ ¨ ÝÝÝÝÑ Hrq pSrl q ÝÝÝÝÑ Hq pSrl , Srl´ q ÝÝÝÝÑ Hrq´1 pSrl´ q ÝÝÝÝÑ Hrq´1 pSrl q ÝÝÝÝÑ

¨¨¨

of the pair pSrl , Srl´ q now gives Hrq´1 pSrl´ q « Hq pSrl , Srl´ q{δqpdl ´1q Z2 .

(4.74)

By the Poincar´e–Lefschetz duality theorem, Hq pSrl , Srl´ q « Hq dl ´1´q pSrl` q.

(4.75)

Writing u P Srl` as ζ pv ` yq with v ` y P Nl´1 ‘ El , let η3 pu, tq “ ζ pp1 ´ tq v ` t ηpyq ` yq,

pu, tq P Srl` ˆ r0, 1s.

We have I pη3 pu, tqq “ inf I pp1 ´ tq v ` t ηpyq ` y ` wq wPMl

by Proposition 4.7.1 (ii)

ě inf rp1 ´ tq I pv ` y ` wq wPMl

`t I pηpyq ` y ` wqs

by Proposition 4.6.1 (i)

ě p1 ´ tq I pζ pv ` yqq ` t I pζ pηpyq ` yqq by Proposition 4.7.1 (ii) ě I puq

by Proposition 4.8.1 (i)

ě 0. If η3 pu, tq “ 0, then p1 ´ tq v ` t ηpyq ` y “ 0 and hence y “ 0, so u “ ζ pvq. Since u ‰ 0, then v ‰ 0, so I puq ď I pvq ď

}v}2D

by Proposition 4.7.1 (ii) ´ λ }v}

2

ď ´pλ ´ λl´1 q }v}2

by (4.19) by (4.5)

ă 0, contrary to assumption. So η3 pu, tq ‰ 0. Thus, η4 “ π ˝ η3 r ` , and hence is a strong deformation retraction of Srl` onto N l r ` q. Hq dl ´1´q pSrl` q « Hq dl ´1´q pN l

(4.76)

4.11 Critical groups

105

Applying the Poincar´e–Lefschetz duality theorem again gives r ` q « Hq´d pN rl , N r ´ q. Hq dl ´1´q pN l´1 l l

(4.77)

r ´ is a proper subset of N rl and hence Hrq´d ´1 pN r ´q By Theorem 4.8.6 (iii), N l´1 l l rl q “ δqd Z2 , the exact sequence “ 0 for q ě dl . Since Hrq´dl´1 ´1 pN l rl q ÝÝÝÝÑ Hq´d pN rl , N r ´ q ÝÝÝÝÑ Hrq´d ´1 pN r ´q ¨ ¨ ¨ ÝÝÝÝÑ Hrq´dl´1 pN l´1 l´1 l l rl q ÝÝÝÝÑ ÝÝÝÝÑ Hrq´dl´1 ´1 pN

¨¨¨

rl , N r ´ q then gives of the pair pN l rl , N r ´ q{δqpd ´1q Z2 « Hrq´d ´1 pN r ´ q. Hq´dl´1 pN l´1 l l l

(4.78)

r ´ q, from which the Combining (4.73)–(4.78) gives Cq pI, 0q « Hrq´dl´1 ´1 pN l rest of (iii) follows.  Remark 4.11.3 Note that (4.72) holds for all pa, bq P Ql and (4.74)–(4.78) hold for all pa, bq P IIl , so r ´ q @pa, bq P pAq X IIl . Hrq´1 pD ´ q « Hrq´dl´1 ´1 pN l r ´ “ H by Theorem 4.8.6 (ii), so If pa, bq P Cl , (4.74)–(4.77) hold and N l rl q{δqpd ´1q Z2 “ δqd Z2 . Hrq´1 pD ´ q « Hq´dl´1 pN l´1 l r` “ K r by Theorem If pa, bq P C l , (4.74)–(4.76) hold and, referring to (4.37), N l 4.8.6 (iv), so r qpd ´1q Z2 . Hrq´1 pD ´ q « Hq dl ´1´q pKq{δ l Let Ol “

$ &I2 ,

l“2

%I Y Il´1 , l ě 3. l

(4.79)

Then Cq pI, 0q « δqdl´1 Z2 ,

pa, bq P Ol

(4.80)

by Theorem 4.11.2 (i) and (ii), so the following corollary is now immediate from Proposition 4.11.1 and Theorem 4.11.2 (iii). Corollary 4.11.4 The points pa, bq and pa 1 , b1 q belong to different connected components of R2 zpAq in the following cases: (i) pa, bq P Ol and pa 1 , b1 q P Ol 1 for some l ‰ l 1 ,

106

Fuˇc´ık spectrum

(ii) pa, bq P Ol and pa 1 , b1 q P IIl1 for some l, l 1 . For example, there is no path in R2 zpAq joining a point in IIl to the diagonal a “ b. Remark 4.11.5 For problem (4.11), Theorem 4.11.2 is due to Dancer [39, 40] and Perera and Schechter [115, 116, 117].

5 Jumping nonlinearities

5.1 Introduction Consider the equation Au “ f puq,

uPD

(5.1)

where A is as in Chapter 4 and f P CpD, H q is a potential operator. Solutions of (5.1) coincide with critical points of the C 1 -functional Gpuq “ }u}2D ´ 2F puq,

uPD

where F is the potential of f with F p0q “ 0. We will first consider the solvability of (5.1) when f puq “ bu` ´ au´ ´ ppuq

(5.2)

for some pa, bq P R2 and a bounded potential operator p P CpD, H q with ppuq “ op}u}D q as }u}D Ñ 8.

(5.3)

We say that (5.1) is resonant at infinity if pa, bq P pAq, otherwise it is nonresonant. Now we have Gpuq “ I pu, a, bq ` 2P puq,

uPD

where P is the potential of p with P p0q “ 0, and P is bounded and ż1 P puq “ pppsuq, uq ds “ op}u}2D q as }u}D Ñ 8 0

by Proposition 4.3.2 and (5.1.3). 107

(5.4)

(5.5)

108

Jumping nonlinearities

We will also consider the existence of nontrivial solutions when f puq “ b0 u` ´ a0 u´ ´ p0 puq

(5.6)

for some pa0 , b0 q P R2 and a bounded potential operator p0 P CpD, H q with p0 puq “ op}u}D q as }u}D Ñ 0.

(5.7)

We say that (5.1) is resonant at zero if pa0 , b0 q P pAq, otherwise it is nonresonant. We have Gpuq “ I pu, a0 , b0 q ` 2P0 puq,

uPD

where P0 is the potential of p0 with P0 p0q “ 0, and P0 is bounded and ż1 P0 puq “ pp0 psuq, uq ds “ op}u}2D q as }u}D Ñ 0

(5.8)

0

by Proposition 4.3.2 and (5.1.7). Example 5.1.1 In problem (4.2.2), 3.1.2 and (5.3) hold if f px, tq “ bt ` ´ at ´ ´ ppx, tq for some Carath´eodory function p on ˆ R with ppx, tq “ optq as |t| Ñ 8, uniformly a.e., and (5.6) and (5.7) also hold when f px, tq “ b0 t ` ´ a0 t ´ ´ p0 px, tq

(5.9)

p0 px, tq “ optq as t Ñ 0, uniformly a.e.

(5.10)

with

Here

ż

|∇u|2 ´ a pu´ q2 ´ b pu` q2 ` 2P px, uq,

Gpuq “

u P H01 p q

where the primitive żt P px, tq “ ppx, sq ds “ opt 2 q as |t| Ñ 8, uniformly a.e. 0

When (5.9) and (5.10) hold, we also have ż Gpuq “ |∇u|2 ´ a0 pu´ q2 ´ b0 pu` q2 ` 2P0 px, uq

where P0 px, tq “

żt 0

p0 px, sq ds “ opt 2 q as t Ñ 0, uniformly a.e.

(5.11)

5.2 Compactness

109

5.2 Compactness In this section we prove some results on the convergence of pPSq and pCq sequences of G. We assume (5.2) and (5.3), so that G is given by (5.4). Our first lemma implies that every bounded pPSq sequence has a convergent subsequence. Lemma 5.2.1 Every bounded sequence puj q Ă D such that G1 puj q Ñ 0 has a convergent subsequence. Proof Since ´ 1 ´1 1 uj “ A´1 pbu` j ´ auj ´ ppuj q ` G puj q{2q “ A puj q

where pu1j q is bounded and A´1 is compact, uj converges in D for a renamed subsequence.  The following lemma is useful for verifying the boundedness of pPSq sequences. Lemma 5.2.2 If G1 puj q Ñ 0 and ρj :“ }uj }D Ñ 8, then a subsequence of rj :“ uj {ρj converges to a nontrivial critical point of I , in particular, pa, bq P u pAq. Proof We have I 1 pr uj q “

I 1 puj q G1 puj q ppuj q “ ´2 Ñ0 ρj ρj }uj }D

by (5.3) and }r uj }D “ 1, so the conclusion follows from Lemma 4.4.1.



We can now prove the pPSq condition in the nonresonant case. Proposition 5.2.3 If pa, bq R pAq and (5.2) and (5.3) hold, then every sequence puj q Ă D such that G1 puj q Ñ 0 has a convergent subsequence, in particular, G satisfies pPSq. Proof Since pa, bq R pAq, puj q is bounded by Lemma 5.2.2, so the conclusion follows from Lemma 5.2.1.  Finally we give sufficient conditions for the pCq condition to hold in the resonant case. The nonquadratic part of G is given by H puq “ Gpuq ´

˘ 1` 1 G puq, u “ 2P puq ´ pppuq, uq 2

110

Jumping nonlinearities

by (4.38). Note that pH puj qq is bounded for every pCq sequence puj q. Denoting rj :“ by N the class of sequences puj q Ă D such that ρj :“ }uj }D Ñ 8 and u r ‰ 0, we assume one of uj {ρj converges weakly to some u pH˘ q Every sequence puj q P N has a subsequence such that H puj q Ñ ˘8. In particular, no pCq sequence can belong to N . Proposition 5.2.4 If (5.2), (5.3), and pH` q or pH´ q hold, then G satisfies pCq. Proof If a pCq sequence puj q is unbounded, then Lemma 5.2.2 gives a subsequence that belongs to N , contradicting pH˘ q, so puj q is bounded and hence has a convergent subsequence by Lemma 5.2.1.  Example 5.2.5 In Example 5.1.1,

ż

H puq “

H px, uq

where H px, tq “ 2P px, tq ´ tppx, tq, and pH˘ q holds if H px, tq ě (resp. ď) Cpxq a.e.

(5.12)

for some C P L1 p q and H px, tq Ñ ˘8 a.e. as |t| Ñ 8. rj Ñ u r a.e. and hence Indeed, if puj q P N , for a subsequence, u ż ż rj pxqq ` H puj q ě (resp. ď) H px, ρj u Cpxq Ñ ˘8 rpxq‰0 u

rpxq“0 u

by Fatou’s lemma.

5.3 Critical groups at infinity In this section we consider the problem of computing the critical groups of G “ I p¨, a, bq ` 2P at infinity when (5.3) holds. First we show that in the nonresonant case the lower-order term 2P can be deformed away outside a large ball without changing the critical set of G. Let S be the unit sphere in D. Since pa, bq R pAq, δ :“ inf uPS }I 1 puq}D ą 0 by Lemma 4.4.1, and then inf uPS }I 1 pRuq}D “ δR for R ą 0 by the positive homogeneity of I 1 . Since supuPS }ppRuq}D “ opRq by (5.3), it follows that › › inf ›G1 pRuq›D “ pδ ` op1qq R as R Ñ 8. (5.13) uPS

5.3 Critical groups at infinity

111

Take a smooth function ϕ : r0, 8q Ñ r0, 2s such that ϕ “ 2 on r0, 1s and ϕ “ 0 on r4, 8q and set r Gpuq “ I puq ` ϕp}u}2D {R 2 q P puq, so that r Gpuq “

$ &Gpuq,

}u}D ď R

%I puq,

}u}D ě 2R.

(5.14)

› › › › Since supuPS ›ϕ 1 p}Ru}2D {R 2 q 2Ru{R 2 › “ OpR ´1 q and supuPS |P pRuq| “ D r also. So for sufficiently opR 2 q by (5.5), (5.13) holds with G replaced by G large R, › › r 1 puq}D ą 0 inf ›G1 puq›D ą 0, inf }G (5.15) }u}D ěR

}u}D ěR

˝

r are in BR , so solutions of (5.1) and hence the critical sets of both G and G r by (5.14). coincide with critical points of G r satisfies pPSq and Proposition 5.3.1 If pa, bq R pAq and (5.3) holds, then G r 8q « Cq pI, 0q Cq pG,

@q. ˝

r has a subsequence in BR by (5.15), which Proof Every pPSq sequence of G then is a pPSq sequence of G by (5.14) and hence has a convergent subsequence by Proposition 5.2.3. r The Since I , P , and ϕ all map bounded sets into bounded sets, so does G. r r critical values of G are bounded from below by inf GpBR q by (5.15). Taking r 2R q and inf I pB2R q, say a 1 , gives the a in (1.27) to be less than both inf GpB r 8q “ Hq pD, G r a 1 q “ Hq pD, I a 1 q Cq pG, r a and I a lie outside B2R , where G r “ I by (5.14). Since the origin is since G the only critical point of I and a 1 ă I p0q, 1

1

1

Hq pD, I a q « Cq pI, 0q 

by Proposition 1.4.1. In the resonant case we strengthen the conditions pH˘ q of Section 5.2 to

pH˘ q H is bounded from below (resp. above) and every sequence puj q P N has a subsequence such that H ptuj q Ñ ˘8 @t ě 1.

112

Jumping nonlinearities

Example 5.3.2 In Example 5.2.5, H is bounded from below (resp. above) by rj Ñ u r a.e. and hence (5.12), and if puj q P N , for a subsequence, u ż ż rj pxqq ` H ptuj q ě (resp. ď) H px, tρj u Cpxq Ñ ˘8 @t ě 1 rpxq‰0 u

rpxq“0 u

by Fatou’s lemma. Lemma 5.3.3 If pH˘ q holds, then P is bounded from below (resp. above) and every sequence puj q P N has a subsequence such that P puj q Ñ ˘8.

(5.16)

Proof We have d dt

ˆ ˙ P ptuq H ptuq ´ 2 “ , t t3

and lim

tÑ8

P ptuq “0 t2

by (5.5), so P puq “ ş8

ż8 1

H ptuq dt. t3

Since 1 dt{t “ 1{2, we have inf H {2 ď P ď sup H {2, and (5.16) for the subsequence in pH˘ q follows from Fatou’s lemma.  3

We can now prove the following. Proposition 5.3.4 Let pa, bq P Ql and assume (5.3). (i) If pa, bq P Cl and pH` q holds, then Cdl´1 pG, 8q ‰ 0. (ii) If pa, bq P C l and pH´ q holds, then Cdl pG, 8q ‰ 0. Proof (i) Since pa, bq P Cl , b “ νl´1 paq and hence nl´1 pa, bq “ 0 by Lemma 4.7.8 (i), so I ě 0 on B “ Sl pa, bq by Proposition 4.7.5 (i). Since pH` q holds, P is bounded from below by Lemma 5.3.3, so it follows that G is bounded from below on B. Let a 1 ă G|B be less than all critical values. For v P Nl´1 , Gpvq ď ´pλ{λl´1 ´ 1 ` op1qq }v}2D as }v}D Ñ 8 by (4.19), (4.5), and (5.5),( and λ ą λl´1 since pa, bq P Ql , so G ď a 1 on  A “ v P Nl´1 : }v}D “ R for sufficiently large R ą 0.

5.3 Critical groups at infinity

113

By Proposition 2.4.3, A homologically links Ml´1 in dimension q “ dl´1 ´ 1. Define a homeomorphism of D by hpuq “ v ` σ pw, a, bq,

u “ v ` w P Nl´1 ‘ Ml´1 .

Then, noting that h|Nl´1 “ id Nl´1 since σ p0q “ 0 by positive homogeneity, hpAq “ A homologically links hpMl´1 q “ σ pMl´1 q “ B in dimension q by Proposition 2.4.5. Since G satisfies pCq by Proposition 5.2.4, the conclusion now follows from Proposition 2.4.7. (ii) For w P Ml , Gpwq ě p1 ´ λ{λl`1 ` op1qq }w}2D as }w}D Ñ 8 by (4.19), (4.6), and (5.5), and λ ă λl`1 since pa, bq P Ql . Since G maps bounded sets into bounded sets, it follows that G is bounded from below on B “ Ml .Let a 1 ă G|B be less than all (critical values. We claim that G ď a 1 on A “ ζ pv, a, bq : v P Nl , }v}D “ R for sufficiently large R ą 0. If not, there is a sequence pvj q Ă Nl , }vj }D “ Rj Ñ 8 such that setting uj “ ζ pvj q we have Gpuj q ą a 1 . Since Nl is finite dimensional, vrj :“ vj {Rj converges to some vr ‰ 0 in Nl for a renamed subsequence. By the continuity of ζ , ζ pr vj q Ñ ζ pr v q “ vr ` τ pr v q ‰ 0. Then vj q}D Ñ 8 ρj :“ }uj }D “ Rj }ζ pr and rj :“ u

uj ζ pr vj q ζ pr vq “ Ñ ‰ 0, ρj }ζ pr vj q}D }ζ pr v q}D

so puj q P N . Since pH´ q holds, then P puj q Ñ ´8 for a renamed subsequence by Lemma 5.3.3. Since pa, bq P C l , b “ μl paq and hence ml pa, bq “ 0 by Lemma 4.7.8 (ii), so I puj q ď 0 by Proposition 4.7.5 (ii) and it follows that Gpuj q Ñ ´8, a contradiction.  ( By Proposition 2.4.3, v P Nl : }v}D “ R homologically links B in dimension q “ dl ´ 1. Define a homeomorphism of D by hpuq “ ζ pv, a, bq ` w,

u “ v ` w P Nl ‘ Ml .

Then,  noting that h|M ( l “ id Ml since ζ p0q “ 0 by positive homogeneity, hp v P Nl : }v}D “ R q “ A homologically links hpBq “ B in dimension q by Proposition 2.4.5. Since G satisfies pCq by Proposition 5.2.4, the conclusion now follows from Proposition 2.4.7. 

114

Jumping nonlinearities

Critical groups can be computed more precisely when pH` q holds. Referring to (4.37) and (4.68), first we prove a lemma. Lemma 5.3.5 If pa, bq P Ql and (5.3) and pH` q hold, then Cq pG, 8q « Hrq´1 pD ´ q

@q.

Proof For u P S and t ą 0, Gptuq “ t 2 I puq ` 2P ptuq

(5.17)

since I is positive homogeneous of degree 2, so ˘ ` ˘ d ` Gptuq “ 2 t I puq ` ppptuq, uq dt ˘ 2` “ Gptuq ´ H ptuq t ˘ 2` ď Gptuq ´ inf H (5.18) t and hence all critical values of G are greater than or equal to inf H (note that inf H ď 0 since H p0q “ 0). So Cq pG, 8q « Hrq´1 pGa q for any a ă inf H by Proposition 1.4.5 (ii). On D ` , G “ I ` 2P ě inf H by the proof of Lemma 5.3.3, so Ga Ă D ´ . We will show that Ga is a strong deformation retract of D ´ . For u P S ´ and t ą 0, ˆ ˙ 2P ptuq Gptuq “ t 2 I puq ` Ñ ´8 as t Ñ 8 t2 by (5.17) and (5.5), so Gptuq ď a for all sufficiently large t. By (5.18), Gptuq ď a ùñ

˘ d ` Gptuq ă 0. dt

Thus, there is a unique Ta puq ą 0 such that t ă (resp. “, ą) Ta puq ùñ Gptuq ą (resp. “, ă) a and the map Ta : S ´ Ñ p0, 8q is C 1 by the implicit function theorem. Then  ( Ga “ tu : u P S ´ , t ě Ta puq and D ´ ˆ r0, 1s Ñ D ´ , pu, tq ÞÑ

$ &p1 ´ tq u ` t Ta pπ puqq π puq,

u P D ´ zGa

%u,

u P Ga ,

5.4 Solvability

115

where π is the radial projection onto S, is a strong deformation retraction of D ´ onto Ga .  We can now prove the following. Proposition 5.3.6 Let pa, bq P Ql X pAq and assume (5.3) and pH` q. (i) If pa, bq P Cl , then Cq pG, 8q « δqdl´1 Z2 . (ii) If pa, bq P C l , then r qpd ´1q Z2 Cq pG, 8q « Hq dl ´1´q pKq{δ l

@q.

In particular, Cq pG, 8q “ 0,

q ă dl´1 or q ě dl

and Cdl´1 pG, 8q “ 0 when pa, bq R Cl . (iii) If pa, bq P IIl , then r ´q Cq pG, 8q « Hrq´dl´1 ´1 pN l

@q.

In particular, Cq pG, 8q “ 0,

q ď dl´1 or q ě dl

and Cq pG, 8q “ 0 for all q when λl is simple. Proof Part (i) and the first parts of (ii) and (iii) are immediate from Lemma 5.3.5 r is a subset of the and Remark 4.11.3. The second part of (ii) follows since K rl , and a proper subset when pa, bq R Cl pdl ´ dl´1 ´ 1q-dimensional sphere N r ´ is a nonempty by Theorem 4.8.7. The second part of (iii) also follows since N l rl by Theorem 4.8.6 (iii). proper subset of N 

5.4 Solvability In this section we consider the solvability of equation (5.1) when (5.2) and (5.3) hold. rl be the First we consider the nonresonant case. Referring to (4.79), let O 2 connected component of R zpAq containing Ol . rl for some l and (5.2) and (5.3) hold, then (5.1) Theorem 5.4.1 If pa, bq P O has a solution. In particular, (5.1) has a solution if there is a path in R2 zpAq joining pa, bq to the diagonal.

116

Jumping nonlinearities

r be the functional constructed in Section 5.3. By Propositions 5.3.1 Proof Let G r satisfies pPSq and and 4.11.1, and (4.80), G r 8q « Cq pI, 0q « δqd Z2 . Cq pG, l´1 r 8q ‰ 0, then G r has a critical point by Proposition 1.4.7. Since Cdl´1 pG,



In the resonant case we assume the conditions pH˘ q of Section 5.3. Theorem 5.4.2 If pa, bq P Ql and (5.2) and (5.3) hold, then (5.1) has a solution in the following cases: (i) pa, bq P Cl and pH` q holds, (ii) pa, bq P C l and pH´ q holds. Proof By Propositions 5.2.4 and 5.3.4, G satisfies pCq and Cdl´1 pG, 8q ‰ 0 in Case (i) and Cdl pG, 8q ‰ 0 in Case (ii), so G has a critical point by Proposition 1.4.7. 

5.5 Critical groups at zero In this section we consider the problem of computing the critical groups of G “ I p¨, a0 , b0 q ` 2P0 at zero when (5.7) holds. First we show that in the nonresonant case zero is an isolated critical point and the higher-order term 2P0 can be deformed away without changing the critical groups there. Proposition 5.5.1 If pa0 , b0 q R pAq and (5.7) holds, then Cq pG, 0q « Cq pI p¨, a0 , b0 q, 0q

@q.

Proof We apply Theorem 1.4.4 with Gt puq “ I pu, a0 , b0 q ` 2 p1 ´ tq P0 puq,

u P D, t P r0, 1s,

which satisfies pPSq by Proposition 5.2.3 since pa0 , b0 q R pAq. We claim that (ii) holds for a sufficiently small closed and bounded neighborhood U of the origin. If not, there are sequences ptj q Ă r0, 1s and puj q Ă Dz t0u , ρj :“ rj :“ uj {ρj we have }uj }D Ñ 0 such that G1tj puj q “ 0. Then setting u I 1 pr uj , a0 , b0 q “

G1tj puj q I 1 puj , a0 , b0 q p0 puj q “ ´ 2 p1 ´ tj q Ñ0 ρj ρj }uj }D

by (5.7) and }r uj }D “ 1, so a subsequence of pr uj q converges to a nontrivial critical point of I p¨, a0 , b0 q by Lemma 4.4.1, contradicting pa0 , b0 q R pAq.

5.5 Critical groups at zero

117

For t, t0 P r0, 1s, u P U ,

› › |Gt puq ´ Gt0 puq| ` ›G1t puq ´ G1t0 puq›D “ 2 |t ´ t0 | p|P0 puq| ` }p0 puq}D q ď C |t ´ t0 |

for some constant C ą 0 since both P0 and p0 are bounded on U , so (ii) also holds.  In the resonant case we assume that zero is an isolated critical point and use a generalized local linking to obtain a nontrivial critical group. Proposition 5.5.2 Let pa0 , b0 q P Ql and assume (5.8). (i) If pa0 , b0 q P Cl and there is an r ą 0 such that P0 pσ pw, a0 , b0 qq ą 0 @w P Ml´1 , 0 ă }w}D ď r,

(5.19)

then Cdl´1 pG, 0q ‰ 0. (ii) If pa0 , b0 q P C l and there is an r ą 0 such that P0 pζ pv, a0 , b0 qq ď 0

@v P Nl , }v}D ď r,

(5.20)

then Cdl pG, 0q ‰ 0. Proof (i) We have the direct sum decomposition E “ Nl´1 ‘ Ml´1 , u “ v ` w. Define an(origin preserving homeomorphism h from C “ u P E : }v}D ď r, }w}D ď r onto a neighborhood U of zero by hpuq “ v ` σ pw, a0 , b0 q. Since pa0 , b0 q P Cl , b0 “ νl´1 pa0 q and hence nl´1 pa0 , b0 q “ 0 by Lemma 4.7.8 (i), so I pσ p¨q, a0 , b0 q ě 0 on Ml´1 by Proposition 4.7.5 (i). This together with (5.19) gives G|hpCXMl´1 qzt0u ą 0. By (4.19), (4.5), and (5.8), for v P Nl´1 , Gpvq ď ´pλ{λl´1 ´ 1 ` op1qq }v}2D as }v}D Ñ 0 where λ “ min ta0 , b0 u ą λl´1 since pa0 , b0 q P Ql , so G ď 0 on C X Nl´1 if r is sufficiently small. Since σ p0q “ 0 by positive homogeneity, h|CXNl´1 “ id CXNl´1 , so this gives G|hpCXNl´1 q ď 0. Thus, G has a generalized local linking near zero in dimension q “ dl´1 , and then Proposition 1.9.2 gives the conclusion. (ii) We have the direct sum decomposition E “ Nl ‘Ml , u “ v ` w. Define an origin preserving homeomorphism h from C “ u P E : }v}D ď

118

Jumping nonlinearities

( r, }w}D ď r onto a neighborhood U of zero by hpuq “ ζ pv, a0 , b0 q ` w. Since pa0 , b0 q P C l , b0 “ μl pa0 q and hence ml pa0 , b0 q “ 0 by Lemma 4.7.8 (ii), so I pζ p¨q, a0 , b0 q ď 0 on Nl by Proposition 4.7.5 (ii). This together with (5.20) gives G|hpCXNl q ď 0. By (4.19), (4.6), and (5.8), for w P Ml , Gpwq ě p1 ´ λ{λl`1 ` op1qq }w}2D as }w}D Ñ 0 where λ “ max ta0 , b0 u ă λl`1 since pa0 , b0 q P Ql , so G ą 0 on C X Ml z t0u if r is sufficiently small. Since ζ p0q “ 0 by positive homogeneity, h|CXMl “ id CXMl , so this gives G|hpCXMl qzt0u ą 0. Thus, G has a generalized local linking near zero in dimension q “ dl , and then Proposition 1.9.2 gives the conclusion.  Example 5.5.3 In Example 5.1.1, (5.11) implies (5.8). Clearly, (5.19) holds if P0 ą 0 on ˆ pRz t0uq and (5.20) holds if P0 ď 0 on ˆ R. However, local sign conditions are sufficient to obtain the conclusions of Proposition 5.5.2. If there is a δ ą 0 such that P0 px, tq ą 0 @x P , 0 ă |t| ď δ,

(5.21)

then Cdl´1 pG, 0q ‰ 0, and if there is a δ ą 0 such that P0 px, tq ď 0 @x P , |t| ď δ,

(5.22)

then Cdl pG, 0q ‰ 0. To see this, take a smooth nondecreasing function ϑ : R Ñ r´δ, δs such that ϑptq “ ´δ for t ď ´δ, ϑptq “ t for ´δ{2 ď t ď δ{2, and ϑptq “ δ for t ě δ, set Pr0 px, tq “ P0 px, ϑptqq, and apply Theorem 3.2.2 to G and ż r Gpuq “ |∇u|2 ´ a0 pu´ q2 ´ b0 pu` q2 ` 2Pr0 px, uq

r 0q. Since (5.21) implies Pr0 ą 0 on ˆ pRz t0uq and to get C˚ pG, 0q « C˚ pG, r (5.22) implies P0 ď 0 on ˆ R, the conclusions follow. Note that H puq “ 2P0 puq ´ pp0 puq, uq .

(5.23)

Critical groups can be computed more precisely when H puq ă 0

@u P Br z t0u

for some r ą 0. First we prove a lemma.

(5.24)

5.5 Critical groups at zero

119

Lemma 5.5.4 If (5.24) holds, then P0 puq ą 0 Proof We have d dt

ˆ

P0 ptuq t2

@u P Br z t0u . ˙ “´

H ptuq t3

by (5.23), and lim

tÑ0

by (5.8), so P0 puq “ ´

ż1 0

P0 ptuq “0 t2

H ptuq dt ą 0 @u P Br z t0u t3 

by (5.24). Setting

 ( B´ “ u P B : I pu, a0 , b0 q ă 0 ,

B` “ BzB´

for B Ă D, next we prove a lemma. Lemma 5.5.5 If pa0 , b0 q P Ql and (5.8) and (5.24) hold, then Cq pG, 0q « Hrq´1 pD´ q @q. Proof We have Cq pG, 0q “ Hq pG0 X Br , G0 X Br z t0uq. On Br X D` z t0u, G “ I p¨, a0 , b0 q ` 2P0 ą 0 by Lemma 5.5.4, so G0 X Br z t0u Ă D´ . We will show that G0 X Br is contractible to 0 and G0 X Br z t0u is a strong deformation retract of D´ . The conclusion will then follow from Lemma 1.4.6 (ii). For u P Sr “ BBr and 0 ă t ď 1, Gptuq “ t 2 I pu, a0 , b0 q ` 2P0 ptuq since I is positive homogeneous of degree 2, so ˘ ` ˘ d ` Gptuq “ 2 t I pu, a0 , b0 q ` pp0 ptuq, uq dt ˘ 2` Gptuq ´ H ptuq “ t 2 ą Gptuq t

(5.25)

(5.26)

120

Jumping nonlinearities

by (5.24). So for u P pSr q´ ,

ˆ ˙ 2P0 ptuq Gptuq “ t 2 I pu, a0 , b0 q ` ă0 t2

for all sufficiently small t by (5.25) and (5.8), Gptuq ě 0 ùñ

˘ d ` Gptuq ą 0 dt

by (5.26), and hence there is a unique 0 ă T puq ď 1 such that Gptuq ă 0 for 0 ă t ă T puq, GpT puq uq ď 0, and Gptuq ą 0 for T puq ă t ď 1. We claim that the map T : pSr q´ Ñ p0, 1s is continuous. By( (5.26) and the implicit function theorem, T is C 1 on u P pSr q´ : T puq ă 1 , so it suffices to show that if uj Ñ u and T puq “ 1, then T puj q Ñ 1. But for any t ă 1, Gptuj q Ñ Gptuq ă 0, so T puj q ą t for j sufficiently large. Thus,  ( G0 X Br “ tu : u P pSr q´ , 0 ď t ď T puq and is radially contractible to 0, and D´ ˆ r0, 1s Ñ D´ , $ &p1 ´ tq u ` t T pπr puqq πr puq, pu, tq ÞÑ %u,

u P D´ zpG0 X Br z t0uq u P G0 X Br z t0u ,

where πr is the radial projection onto Sr , is a strong deformation retraction of D´ onto G0 X Br z t0u.  Referring to (4.37), we can now prove Proposition 5.5.6 Let pa0 , b0 q P Ql X pAq and assume (5.8) and (5.24). (i) If pa0 , b0 q P Cl , then Cq pG, 0q « δqdl´1 Z2 . (ii) If pa0 , b0 q P C l , then r 0 , b0 qq{δqpd ´1q Z2 Cq pG, 0q « Hq dl ´1´q pKpa l In particular, Cq pG, 0q “ 0,

q ă dl´1 or q ě dl

and Cdl´1 pG, 0q “ 0 when pa0 , b0 q R Cl .

@q.

5.5 Critical groups at zero

121

(iii) If pa0 , b0 q P IIl , then rl pa0 , b0 q´ q @q. Cq pG, 0q « Hrq´dl´1 ´1 pN In particular, Cq pG, 0q “ 0,

q ď dl´1 or q ě dl

and Cq pG, 0q “ 0 for all q when λl is simple. Proof Part (i) and the first parts of (ii) and (iii) are immediate from Lemma r 0 , b0 q is a 5.5.5 and Remark 4.11.3. The second part of (ii) follows since Kpa rl pa0 , b0 q, and a proper subsubset of the pdl ´ dl´1 ´ 1q-dimensional sphere N set when pa0 , b0 q R Cl by Theorem 4.8.7. The second part of (iii) also follows rl pa0 , b0 q´ is a nonempty proper subset of N rl pa0 , b0 q by Theorem 4.8.6 since N (iii).  Example 5.5.7 In Example 5.1.1, (5.24) holds if H ă 0 on ˆ pRz t0uq, but the conclusions of Proposition 5.5.6 also hold under (5.11) and the local sign condition H px, tq ă 0 @x P , 0 ă |t| ď δ for some δ ą 0. To see this, set $ t ’ ´p0 px, ´δq , t ă ´δ ’ ’ ’ δ & p px, tq, |t| ď δ r p0 px, tq “ 0 ’ ’ ’ ’ %p0 px, δq t , t ą δ, δ

Pr0 px, tq “

(5.27)

żt

pr0 px, sq ds

0

and apply Theorem 3.2.2 to G and ż r Gpuq “ |∇u|2 ´ a0 pu´ q2 ´ b0 pu` q2 ` 2Pr0 px, uq

r 0q. A simple calculation shows that to get C˚ pG, 0q « C˚ pG, $ ’ H px, ´δq, t ă ´δ ’ ’ & Hr px, tq “ 2Pr0 px, tq ´ t pr0 px, tq “ H px, tq, |t| ď δ ’ ’ ’ % H px, δq, t ą δ, so (5.27) implies Hr ă 0 on ˆ pRz t0uq.

122

Jumping nonlinearities

5.6 Nonlinearities crossing the Fuˇc´ık spectrum In this section we consider the existence of nontrivial solutions of the equation (5.1) when (5.2), (5.3), (5.6), and (5.7) or (5.8) hold. rl be as in Section 5.4 and let First we consider the nonresonant case. Let O r l be the set of points that can be joined to IIl by a path in R2 zpAq. II Theorem 5.6.1 If (5.2), (5.3), (5.6), and (5.7) hold, then (5.1) has a nontrivial solution in the following cases: rl and pa, bq P O rl 1 for some l ‰ l 1 , (i) pa0 , b0 q P O rl and pa, bq P II r l 1 for some l, l 1 , (ii) pa0 , b0 q P O r rl 1 for some l, l 1 . (iii) pa0 , b0 q P IIl and pa, bq P O r be the functional constructed in Section 5.3. By (5.14) and PropoProof Let G sition 5.5.1, r 0q “ Cq pG, 0q « Cq pI p¨, a0 , b0 q, 0q. Cq pG, By Proposition 5.3.1, r 8q « Cq pI p¨, a, bq, 0q Cq pG, r satisfies pPSq. So G r has a nontrivial critical point if Cq pI p¨, a0 , b0 q, 0q ­« and G Cq pI p¨, a, bq, 0q for some q by Proposition 1.6.1. (i) Cq pI p¨, a0 , b0 q, 0q « δqdl´1 Z2 and Cq pI p¨, a, bq, 0q « δqdl 1 ´1 Z2 by Proposition 4.11.1 and (4.80). (ii) Cq pI p¨, a0 , b0 q, 0q « δqdl´1 Z2 and Cdl´1 pI p¨, a, bq, 0q “ 0 by Proposition 4.11.1 and Theorem 4.11.2 (iii). (iii) Cdl 1 ´1 pI p¨, a0 , b0 q, 0q “ 0 and Cq pI p¨, a, bq, 0q « δqdl 1 ´1 Z2 .  Remark 5.6.2 In particular, there is a nontrivial solution when pa0 , b0 q and pa, bq are on opposite sides of Cl or C l in Ql zpAq. For problem (4.11), this was proved by Perera and Schechter [116]. It generalizes a well-known result of Amann and Zehnder [3] on the existence of nontrivial solutions for problems crossing an eigenvalue. In the case of resonance at zero we have the following theorem. Theorem 5.6.3 If (5.2), (5.3), (5.6), and (5.8) hold, then (5.1) has a nontrivial solution in the following cases: (i) pa0 , b0 q P Cl , there is an r ą 0 such that P0 pσ pw, a0 , b0 qq ą 0

@w P Ml´1 , 0 ă }w}D ď r,

r l 1 for some l 1 , rl 1 for some l 1 ‰ l or pa, bq P II and either pa, bq P O

5.6 Nonlinearities crossing the Fuˇc´ık spectrum

123

(ii) pa0 , b0 q P Ql X pAqzCl , there is an r ą 0 such that H puq ă 0 @u P D, 0 ă }u}D ď r, rl 1 for some l 1 , and pa, bq P O l (iii) pa0 , b0 q P C , there is an r ą 0 such that P0 pζ pv, a0 , b0 qq ď 0 @v P Nl , }v}D ď r, rl 1 for some l 1 ‰ l ` 1 or pa, bq P II r l 1 for some l 1 . and either pa, bq P O r be the functional constructed in Section 5.3. By (5.14), Proof Let G r 0q “ Cq pG, 0q. Cq pG, By Proposition 5.3.1, r 8q « Cq pI p¨, a, bq, 0q Cq pG, r satisfies pPSq. So G r has a nontrivial critical point if Cq pG, 0q ­« and G Cq pI p¨, a, bq, 0q for some q by Proposition 1.6.1. (i) Cdl´1 pG, 0q ‰ 0 by Proposition 5.5.2 (i) and Cdl´1 pI p¨, a, bq, 0q “ 0 by Proposition 4.11.1, (4.80), and Theorem 4.11.2 (iii). (ii) Cdl1 ´1 pG, 0q “ 0 by Proposition 5.5.6 (ii) and (iii), and Cq pI p¨, a, bq, 0q « δqdl 1 ´1 Z2 . (iii) Cdl pG, 0q ‰ 0 by Proposition 5.5.2 (ii) and Cdl pI p¨, a, bq, 0q “ 0.  In the case of resonance at infinity we assume the conditions pH˘ q of Section 5.3. Theorem 5.6.4 If (5.2), (5.3), (5.6), and (5.7) hold, then (5.1) has a nontrivial solution in the following cases: rl1 for some l 1 ‰ l or (i) pa, bq P Cl , pH` q holds, and either pa0 , b0 q P O 1 r pa0 , b0 q P IIl 1 for some l , rl1 for some l 1 , (ii) pa, bq P Ql X pAqzCl , pH` q holds, and pa0 , b0 q P O l rl 1 for some l 1 ‰ l ` 1 or (iii) pa, bq P C , pH´ q holds, and either pa0 , b0 q P O r l 1 for some l 1 . pa0 , b0 q P II Proof By Proposition 5.5.1, Cq pG, 0q « Cq pI p¨, a0 , b0 q, 0q. By Proposition 5.2.4, G satisfies pCq. So G has a nontrivial critical point if Cq pG, 8q ­« Cq pI p¨, a0 , b0 q, 0q for some q by Proposition 1.6.1. (i) Cdl´1 pG, 8q ‰ 0 by Proposition 5.3.4 (i) and Cdl´1 pI p¨, a0 , b0 q, 0q “ 0 by Proposition 4.11.1, (4.80), and Theorem 4.11.2 (iii).

124

Jumping nonlinearities

(ii) Cdl 1 ´1 pG, 8q “ 0 by Proposition 5.3.6 (ii) and (iii), and Cq pI p¨, a0 , b0 q, 0q « δqdl1 ´1 Z2 . (iii) Cdl pG, 8q ‰ 0 by Proposition 5.3.4 (ii) and Cdl pI p¨, a0 , b0 q, 0q “ 0.  In the double resonance case we have the following. Theorem 5.6.5 If (5.2), (5.3), pH` q, (5.6), and (5.8) hold, then (5.1) has a nontrivial solution in the following cases: (i) pa0 , b0 q P Cl , there is an r ą 0 such that P0 pσ pw, a0 , b0 qq ą 0

@w P Ml´1 , 0 ă }w}D ď r,

and either pa, bq P Cl 1 for some l 1 ‰ l or pa, bq P Ql 1 X pAqzCl 1 for some l 1 , (ii) pa0 , b0 q P C l , there is an r ą 0 such that P0 pζ pv, a0 , b0 qq ď 0 @v P Nl , }v}D ď r, and either pa, bq P Cl 1 for some l 1 ‰ l ` 1 or pa, bq P Ql 1 X pAqzCl 1 for some l 1 . Proof By Proposition 5.2.4, G satisfies pCq. We apply Proposition 1.6.1. (i) Cdl´1 pG, 0q ‰ 0 by Proposition 5.5.2 (i) and Cdl´1 pG, 8q “ 0 by Proposition 5.3.6. (ii) Cdl pG, 0q ‰ 0 by Proposition 5.5.2 (ii) and Cdl pG, 8q “ 0.  Theorem 5.6.6 If (5.2), (5.3), (5.6), and (5.8) hold and there is an r ą 0 such that H puq ă 0

@u P D, 0 ă }u}D ď r,

then (5.1) has a nontrivial solution in the following cases: (i) pa, bq P Cl , pH` q holds, and either pa0 , b0 q P Cl1 for some l 1 ‰ l or pa0 , b0 q P Ql 1 X pAqzCl 1 for some l 1 , (ii) pa, bq P C l , pH´ q holds, and either pa0 , b0 q P Cl 1 for some l 1 ‰ l ` 1 or pa0 , b0 q P Ql 1 X pAqzCl 1 for some l 1 . Proof By Proposition 5.2.4, G satisfies pCq. We apply Proposition 1.6.1. (i) Cdl´1 pG, 8q ‰ 0 by Proposition 5.3.4 (i) and Cdl´1 pG, 0q “ 0 by Proposition 5.5.6. (ii) Cdl pG, 8q ‰ 0 by Proposition 5.3.4 (ii) and Cdl pG, 0q “ 0. 

6 Sandwich pairs

6.1 Introduction The notion of sandwich pairs is a useful tool for finding critical points of a functional. It was introduced by Schechter [145, 146] and was based on the sandwich theorem for complementing subspaces by Schechter [137, 138] and Silva [150]. Definition 6.1.1 We say that a pair of subsets A, B of a Banach space E is a sandwich pair if every G P C 1 pE, Rq satisfying ´8 ă b :“ inf G ď sup G “: a ă `8 B

(6.1)

A

and pPSqc for all c P rb, as has a critical point u with b ď Gpuq ď a. Example 6.1.2 If E “ N ‘ M is a direct sum decomposition with N nontrivial and finite dimensional, then N, M form a sandwich pair. In fact, we will see in Section 6.3 that this is a special case of a much more general class of sandwich pairs.

6.2 Flows First we give a criterion, involving a certain class of flows on E, for a pair of subsets to form a sandwich pair. Denote by  the set of all maps σ P CpE ˆ r0, 1s, Eq such that, writing σ pu, tq “ σ ptq u, (i) σ p0q “ id E , (ii) sup }σ ptq u ´ u} ă 8. pu,tqPEˆr0,1s

125

126

Sandwich pairs

Theorem 6.2.1 A, B Ă E form a sandwich pair if σ p1q A X B ‰ H

@σ P .

(6.2)

Proof Let G P C 1 pE, Rq satisfy (6.1) and set c :“ inf sup Gpσ p1q uq. σ P uPA

Then c ě b by (6.2) and c ď a since the identity σ ptq u ” u is in , so G satisfies pPSqc . We claim that c is a critical value of G. If not, there are ε ą 0 and η P  such that ηp1q Gc`ε Ă Gc´ε by Lemma 1.3.5. Take a σ P  such that σ p1q A Ă Gc`ε and define σr P  by # σ p2tq u, 0 ď t ď 1{2 σr ptq u “ ηp2t ´ 1q σ p1q u, 1{2 ă t ď 1. r p1q A Ă Gc´ε , contradicting the definition of c. Then σ



Remark 6.2.2 Theorem 6.2.1 was proved by Perera and Schechter [122]; see also [120, 121].

6.3 Cohomological index Next we construct a class of sandwich pairs, based on the cohomological index of Fadell and Rabinowitz [51], applicable to elliptic boundary value problems. Let us recall the construction and some properties of the cohomological index. Writing the group Z2 multiplicatively as t1, ´1u, a paracompact Z2 -space is a paracompact space X together with a continuous mapping μ : Z2 ˆ X Ñ X, called a Z2 -action on X, such that μp1, xq “ x, ´p´xq “ x

@x P X

where ´x :“ μp´1, xq. The action is fixed-point free if ´x ‰ x A subset A of X is invariant if ´A :“



@x P X.

( ´ x : x P A “ A,

and a map f : X Ñ X 1 between two paracompact Z2 -spaces is equivariant if f p´xq “ ´f pxq

@x P X.

Two spaces X and X1 are equivalent if there is an equivariant homeomorphism f : X Ñ X1 . We denote by F the set of all paracompact free Z2 -spaces, identifying equivalent ones.

6.3 Cohomological index

127

A principal Z2 -bundle with paracompact base is a triple ξ “ pE, p, Bq consisting of an E P F, called the total space, a paracompact space B, called the base space, and a map p : E Ñ B, called the bundle projection, such that there are (1) an open covering tUλ uλP of B, (2) for each λ P , a homeomorphism ϕλ : Uλ ˆ Z2 Ñ p ´1 pUλ q satisfying ϕλ pb, ´1q “ ´ϕλ pb, 1q, p ˝ ϕλ pb, ˘1q “ b

@b P B.

Then each p´1 pbq, called a fiber, is some pair te, ´eu , e P E. A bundle map f : ξ Ñ ξ 1 consists of an equivariant map f : E Ñ E 1 and a map f : B Ñ B 1 such that p1 ˝ f “ f ˝ p, i.e. the diagram f

E ÝÝÝÝÑ E 1 § § § § pđ p1 đ f

B ÝÝÝÝÑ B 1 commutes. Two bundles ξ and ξ 1 are equivalent if there are bundle maps f : ξ Ñ ξ 1 and f 1 : ξ 1 Ñ ξ such that f 1 ˝ f and f ˝ f 1 are the identity bundle maps on ξ and ξ 1 , respectively. We denote by PrinZ2 B the set of principal Z2 bundles over B and Prin Z2 the set of all principal Z2 -bundles with paracompact base, identifying equivalent ones. Each X P F can be identified with a ξ P Prin Z2 as follows. Let X “ X{Z2 be the (paracompact) quotient space of X P F with each x and ´x identified, called the orbit space of X, and π : X Ñ X the quotient map. Then P : F Ñ Prin Z2 ,

X ÞÑ ξ :“ pX, π, Xq

is a one-to-one correspondence. A map f : B Ñ B 1 induces a bundle f ˚ ξ 1 “ pf ˚ pE 1 q, p, Bq P Prin Z2 , called the pullback, where  ( f ˚ pE 1 q “ pb, e1 q P B ˆ E 1 : f pbq “ p 1 pe1 q , ´pb, e1 q “ pb, ´e1 q and ppb, e1 q “ b. Homotopic maps induce equivalent bundles, so for each ξ 1 P Prin Z2 , we have the mapping T : rB, B 1 s Ñ PrinZ2 B,

rf s ÞÑ f ˚ ξ 1

128

Sandwich pairs

where rB, B 1 s is the set of homotopy classes of maps from B to B 1 . For the bundle ξ 1 “ pS 8 , π, RP8 q, called the universal principal Z2 -bundle, where S 8 is the unit sphere in R8 , RP8 is the infinite-dimensional real projective space, and π identifies antipodal points ˘x, T is a one-to-one correspondence (see Dold [47]). Thus for each X P F, there is a map f : X Ñ RP8 , unique up to homotopy and called the classifying map, such that T prf sq “ PpXq. Let f ˚ : H ˚ pRP8 q Ñ H ˚ pXq be the induced homomorphism of the Alexander–Spanier cohomology rings. The cohomological index of X is defined by #  ( sup k ě 1 : f ˚ pωk´1 q ‰ 0 , X ‰ H ipXq “ 0, X“H where ω P H 1 pRP8 q is the generator of the polynomial ring H ˚ pRP8 q “ Z2 rωs. The index i : F Ñ N Y t0, 8u has the usual properties of an index theory. (1) Definiteness: ipXq “ 0 if and only if X “ H. (2) Monotonicity: If f : X Ñ Y is an equivariant map, in particular, if X Ă Y , then ipXq ď ipY q. Thus, equality holds when f is an equivariant homeomorphism. (3) Subadditivity: If X P F and A, B are closed invariant subsets of X such that X “ A Y B, then ipXq ď ipAq ` ipBq. (4) Continuity: If X P F and A is a closed invariant subset of X, then there is a closed invariant neighborhood N of A in X such that ipN q “ ipAq. (5) Neighborhood of zero: If U is a bounded symmetric neighborhood of 0 in a Banach space E, then ipBU q “ dim E. Recall that the suspension SA of a nonempty subset A of a Banach space E is the quotient space of A ˆ r´1, 1s with A ˆ t1u and A ˆ t´1u collapsed to different points, which can be realized in E ‘ R as the union of all line segments joining the two points p0, ˘1q P E ‘ R to points of A. The cohomological

6.3 Cohomological index

129

index also has the following important stability property: If A is a closed symmetric subset of Ez t0u, then ipSAq “ ipAq ` 1. Let

 ( S “ u P E : }u} “ 1

be the unit sphere in E and π : Ez t0u Ñ S,

u ÞÑ

u }u}

the radial projection onto S. Now let M be a bounded symmetric subset of Ez t0u radially homeomorphic to S, i.e. g “ π |M : M Ñ S is a homeomorphism. Then the radial projection from Ez t0u onto M is given by πM “ g ´1 ˝ π . For A Ă M and r ě 0, we set ď  ( r “ π ´1 pAq Y t0u “ rA “ ru : u P A , A rA. M rě0

Theorem 6.3.1 If A0 , B0 is a pair of disjoint nonempty closed symmetric subsets of M such that ipA0 q “ ipMzB0 q ă 8

(6.3)

and h is an odd homeomorphism of E such that r0 qq Ñ 8 as r Ñ 8, distphprA0 q, hpB

(6.4)

r0 q, B “ hpB r0 q form a sandwich pair. then A “ hpA Proof By Theorem 6.2.1, it suffices to verify (6.2), so suppose there is a σ P  with σ p1q A X B “ H.

(6.5)

By (6.4), there is an R ą 1 such that distphpRA0 q, Bq ą

sup pu,tqPEˆr0,1s

}σ ptq u ´ u}

and hence σ ptq hpRA0 q X B “ H

@t P r0, 1s.

By (6.5) and (6.6), we can define a map η P CpA0 ˆ r0, 1s, EzBq by $ ’ hpp1 ´ 3t ` 3Rtq uq, 0 ď t ď 1{3 ’ & ηpu, tq “ σ p3t ´ 1q hpRuq, 1{3 ă t ď 2{3 ’ ’ % σ p1q hp3p1 ´ tq Ruq, 2{3 ă t ď 1.

(6.6)

130

Sandwich pairs

Since η|A0 ˆt0u “ h|A0 is odd and ηpA0 ˆ t1uq is the single point σ p1q hp0q, η can be extended to an odd map ηr P CpSA0 , EzBq. Then πM ˝ h´1 ˝ ηr is an odd continuous map from SA0 into MzB0 and hence ipMzB0 q ě ipSA0 q “ ipA0 q ` 1 by the monotonicity and the stability of the index, contradicting (6.3).



Corollary 6.3.2 If A0 , B0 is a pair of disjoint nonempty closed symmetric subsets of S such that ipA0 q “ ipSzB0 q ă 8,

distpA0 , B0 q ą 0,

then A “ π ´1 pA0 q Y t0u , B “ π ´1 pB0 q Y t0u form a sandwich pair. Proof Take h to be the identity in Theorem 6.3.1. Since distprA0 , Bq “ r distpA0 , Bq, it suffices to show that distpA0 , Bq ą 0. If not, there are sequences uj P A0 , vj P B0 , and sj ě 0 such that }uj ´ sj vj } Ñ 0. Since ˇ ˇ }uj ´ sj vj } ě ˇ}uj } ´ }sj vj } ˇ “ |1 ´ sj |, sj Ñ 1. Then distpA0 , B0 q ď }uj ´ vj } ď }uj ´ sj vj } ` }psj ´ 1q vj } “ }uj ´ sj vj } ` |sj ´ 1| Ñ 0, 

a contradiction.

Corollary 6.3.3 If E “ N ‘ M, u “ v ` w is a direct sum decomposition with 1 ď d :“ dim N ă 8, then N, M form a sandwich pair. Proof Take A0 “ S X N, B0 “ S X M in Corollary 6.3.2 and note that ipA0 q “ d. Since A0 Ă SzB0 and SzB0 Ñ A0 ,

u ÞÑ

v }v}

is an odd continuous map, ipA0 q “ ipSzB0 q by the monotonicity of the index.  Remark 6.3.4 Theorem 6.3.1 and Corollary 6.3.2 were proved by Perera and Schechter in [122] and [121], respectively.

6.4 Semilinear problems

131

6.4 Semilinear problems Now we use sandwich pairs based on the eigenspaces of the Laplacian to obtain an existence result for the semilinear elliptic boundary value problem $ & ´ u “ f px, uq in (6.7) % u“0 on B where is a bounded domain in Rn , n ě 1 and f is a Carath´eodory function on ˆ R satisfying the subcritical growth condition ` ˘ |f px, tq| ď C |t|r´1 ` 1 @px, tq P ˆ R (6.8) for some r P p1, 2˚ q and a constant C ą 0. Weak solutions of (6.7) coincide with critical points of ż Gpuq “ |∇u|2 ´ 2F px, uq, u P E “ H01 p q where F px, tq “

żt

f px, sq ds.

0

Recall that the Dirichlet spectrum of ´ consists of isolated eigenvalues λl , l ě 1 of finite multiplicities satisfying 0 ă λ1 ă λ2 ă ¨ ¨ ¨ ă λl ă ¨ ¨ ¨ . Let El be the eigenspace of λl , Nl “

l à

Ej ,

j “1

Ml “ NlK .

Then E “ Nl ‘ Ml , u “ v ` w is an orthogonal decomposition with respect to both the inner product in E and the L2 p q-inner product, and }v}2 ď λl }v}2L2 p q }w}2 ě λl`1 }w}2L2 p q

@v P Nl , @w P Ml .

(6.9) (6.10)

Let H px, tq “ 2F px, tq ´ tf px, tq. We have the following theorem. Theorem 6.4.1 If λl t 2 ´ W pxq ď 2F px, tq ď λl`1 t 2 ` W pxq

@px, tq P ˆ R

(6.11)

132

Sandwich pairs

for some l and W P L1 p q, then (6.7) has a solution in the following cases: H px, tq ă 0, |t|τ

(i) H px, tq ď C p|t|τ ` 1q and H pxq :“ lim sup |t|Ñ8

(ii) H px, tq ě ´C p|t|τ ` 1q and H pxq :“ lim inf |t|Ñ8

H px, tq ą 0, |t|τ

for some τ P r1, 2q. Proof By Corollary 6.3.3, Nl , Ml form a sandwich pair, and by (6.11), (6.9), and (6.10), ż Gpvq ď |∇v|2 ´ λl v 2 ` W pxq ď }W }L1 p q @v P Nl , ż

Gpwq ě

|∇w|2 ´ λl`1 w 2 ´ W pxq ě ´ }W }L1 p q

@w P Ml .

It only remains to verify the pPSq condition. By Lemma 3.1.1, it suffices to show that every pPSq sequence puj q is rj :“ uj {ρj we have }r bounded, so suppose ρj :“ }uj } Ñ 8. Setting u uj } “ 1, r weakly in E, strongly so a renamed subsequence of pr uj q converges to some u in L2 p q, and a.e. in . We have ż ż Gpuj q 2F px, uj q W pxq 1´ “ ď λl`1 |r uj |2 ` 2 2 ρj ρj ρj2

r ‰ 0. Since by (6.11), and passing to the limit gives λl`1 }r u}2L2 p q ě 1, so u τ ě 1, ż pG1 puj q, uj q {2 ´ Gpuj q H px, uj q “ Ñ 0. (6.12) ρjτ ρjτ

(i) We have |uj | “ ρj |r uj | Ñ 8 and hence lim

H px, uj q H px, uj q “ lim |r uj |τ “ H pxq |r u|τ τ ρj |uj |τ

a.e. on tr u ‰ 0u, while H px, uj q ďC ρjτ a.e. on tr u “ 0u, so

ż 0 “ lim

˜

1 |r uj | ` τ ρj τ

H px, uj q ď ρjτ

¸ Ñ0

ż r‰0 u

H pxq |r u|τ

6.5 p-Laplacian problems

133

by (6.12) and Fatou’s lemma. This is a contradiction since H ă 0 a.e. and r ‰ 0. u (ii) We have |uj | “ ρj |r uj | Ñ 8 and hence lim

H px, uj q H px, uj q “ lim |r uj |τ “ H pxq |r u|τ ρjτ |uj |τ

a.e. on tr u ‰ 0u, while H px, uj q ě ´C ρjτ a.e. on tr u “ 0u, so

ż 0 “ lim

˜

1 |r uj | ` τ ρj

¸ Ñ0

τ

ż

H px, uj q ě ρjτ

r‰0 u

H pxq |r u|τ

by (6.12) and Fatou’s lemma. This is a contradiction since H ą 0 a.e. and r ‰ 0. u 

6.5 p-Laplacian problems Next we use certain cones as sandwich pairs to extend Theorem 6.4.1 to the p-Laplacian problem $ & ´ p u “ f px, uq in (6.13) % u“0 on B ` ˘ where is a bounded domain in Rn , n ě 1, p u “ div |∇u|p´2 ∇u is the pLaplacian of u, p P p1, 8q, f is a Carath´eodory function on ˆ R satisfying the growth condition (6.8) for some r P p1, p ˚ q, and # np{pn ´ pq, n ą p ˚ p “ 8, n ď p. Weak solutions of (6.13) coincide with critical points of ż 1, p Gpuq “ |∇u|p ´ p F px, uq, u P E “ W0 p q where F px, tq “ the norm

żt 0

1, p

f px, sq ds and W0 p q is the usual Sobolev space with ˙1{p

ˆż |∇u|

}u} “

p

.

134

Sandwich pairs

Eigenvalues of the problem $ & ´ p u “ λ |u|p´2 u %

in

u“0

on B

coincide with critical values of the C 1 -functional  ( 1 puq “ ż , u P S “ u P E : }u} “ 1 . |u|p

Denote by Fl the class of symmetric subsets M of S with cohomological index ipMq ě l. It was shown in Perera et al. [113] (see also Perera [110]) that lě1

λl :“ inf sup puq, MPFl uPM

is a positive, nondecreasing, and unbounded sequence of eigenvalues, and ip λl q “ ipSzλl`1 q “ l

(6.14)

when λl ă λl`1 . Setting H px, tq “ p F px, tq ´ tf px, tq, we prove the following theorem. Theorem 6.5.1 If λl ă λl`1 and λl |t|p ´ W pxq ď p F px, tq ď λl`1 |t|p ` W pxq

@px, tq P ˆ R (6.15)

for some l and W P L1 p q, then (6.13) has a solution in the following cases: H px, tq ă 0, |t|τ |t|Ñ8 H px, tq (ii) H px, tq ě ´C p|t|τ ` 1q and H pxq :“ lim inf ą0 |t|Ñ8 |t|τ (i) H px, tq ď C p|t|τ ` 1q and H pxq :“ lim sup

for some τ P r1, pq. Proof In view of (6.14) we apply Corollary 6.3.2 with A0 “  λl , B0 “ λl`1 . Since }u}Lp p q “ 1{puq1{p , }u ´ v}Lp p q ě }u}Lp p q ´ }v}Lp p q ě

1 1{p λl

´

1 1{p λl`1

@u P  λl , v P λl`1 ,

which together with the Sobolev embedding E ãÑ Lp p q gives distp λl , λl`1 q ą 0.

6.5 p-Laplacian problems

135

So A “ π ´1 p λl q Y t0u , B “ π ´1 pλl`1 q Y t0u form a sandwich pair. By (6.15), ż Gpuq ď |∇u|p ´ λl |u|p ` W pxq

ˆ “ }u} 1 ´ p

ż Gpuq ě

λl pπpuqq

˙ ` }W }L1 p q ď }W }L1 p q

@u P π ´1 p λl q,

|∇u|p ´ λl`1 |u|p ´ W pxq

ˆ “ }u}p 1 ´

˙ λl`1 ´ }W }L1 p q ě ´ }W }L1 p q @u P π ´1 pλl`1 q. pπ puqq

It only remains to verify the pPSq condition, and it suffices to show that every pPSq sequence puj q is bounded by a standard argument (see, e.g., Perera rj :“ uj {ρj we have }r et al. [113]). If ρj :“ }uj } Ñ 8, setting u uj } “ 1, so a r weakly in E, strongly in renamed subsequence of pr uj q converges to some u Lp p q, and a.e. in . We have ż ż Gpuj q p F px, uj q W pxq 1´ “ ď λl`1 |r uj |p ` p p p ρj ρj ρj

p r ‰ 0. Since by (6.15), and passing to the limit gives λl`1 }r u}Lp p q ě 1, so u τ ě 1, ż pG1 puj q, uj q {p ´ Gpuj q H px, uj q “ Ñ 0. (6.16) ρjτ ρjτ

(i) We have |uj | “ ρj |r uj | Ñ 8 and hence lim

H px, uj q H px, uj q “ lim |r uj |τ “ H pxq |r u|τ ρjτ |uj |τ

a.e. on tr u ‰ 0u, while H px, uj q ďC ρjτ a.e. on tr u “ 0u, so

ż 0 “ lim

˜

1 |r uj |τ ` τ ρj

H px, uj q ď ρjτ

¸ Ñ0

ż r‰0 u

H pxq |r u|τ

by (6.16) and Fatou’s lemma. This is a contradiction since H ă 0 a.e. and r ‰ 0. u

136

Sandwich pairs

(ii) We have |uj | “ ρj |r uj | Ñ 8 and hence lim

H px, uj q H px, uj q “ lim |r uj |τ “ H pxq |r u|τ ρjτ |uj |τ

a.e. on tr u ‰ 0u, while H px, uj q ě ´C ρjτ a.e. on tr u “ 0u, so

ż 0 “ lim

˜

1 |r uj | ` τ ρj

H px, uj q ě ρjτ

τ

¸ Ñ0

ż r‰0 u

H pxq |r u|τ

by (6.16) and Fatou’s lemma. This is a contradiction since H ą 0 a.e. and r ‰ 0. u  Remark 6.5.2 Theorem 6.5.1 was proved by Perera and Schechter [121]. Related results can be found in Arcoya and Orsina [12], Bouchala and Dr´abek [20], Dr´abek and Robinson [49], Perera [109], and Perera and Schechter [120].

6.6 Anisotropic systems Finally we use more general curved sandwich pairs made up of orbits of a certain group action on product spaces to extend Theorem 6.5.1 to the anisotropic system $ & ´ p u “ ∇u F px, uq in (6.17) % u“0 on B where is a bounded domain in Rn , n ě 1, p u “ p p1 u1 , . . . , pm um q where u “ pu1 , . . . , um q and p “ pp1 , . . . , pm q with each pi P p1, 8q, and F P C 1 p ˆ Rm q with ∇u F “ pBF {Bu1 , . . . , BF {Bum q satisfying ˜ ¸ ˇ ˇ m ÿ ˇ BF ˇ ˇ ˇ |uj |rij ´1 ` 1 @px, uq P ˆ Rm (6.18) ˇ Bu ˇ ď C i j “1 for some rij P p1, 1 ` pj˚ ppi˚ ´ 1q{pi˚ q and a constant C ą 0. Let (  E “ E1 ˆ ¨ ¨ ¨ ˆ Em “ u “ pu1 , . . . , um q : ui P Ei

6.6 Anisotropic systems

with the norm

˜

¸1{2

m ÿ

}u} “

137

}ui }2i

,

i“1 1, p

where }¨}i denotes the norm in Ei “ W0 i p q. Then weak solutions of (6.17) coincide with critical points of ż Gpuq “ I puq ´ F px, uq, u P E

where I puq “

ż m ÿ 1 |∇ui |pi . p i i“1

Let us recall some recent results on eigenvalue problems for systems proved by Perera et al. [113]. Define a continuous flow on E, as well as on Rm , by pα, uq ÞÑ uα :“ p|α|1{p1 ´1 α u1 , . . . , |α|1{pm ´1 α um q,

α P R.

Noting that I satisfies I puα q “ |α| I puq

@α P R, u P E,

we consider the class of eigenvalue problems $ & ´ p u “ λ ∇u J px, uq in %

u“0

on B

(6.19)

(6.20)

where J P C 1 p ˆ Rm q is positive somewhere and satisfies J px, uα q “ |α| J px, uq

@α P R, px, uq P ˆ Rm

and the growth condition (6.18) (with J in place of F ). Then ż J puq “ J px, uq, u P E

also satisfies J puα q “ |α| J puq

@α P R, u P E.

(6.21)

A typical example is J px, uq “ |u1 |r1 ¨ ¨ ¨ |um |rm where ri P p1, pi q and

m ÿ ri “ 1. pi i“1

(6.22)

138

Let

Sandwich pairs

 ( M “ u P E : I puq “ 1 ,

 ( M` “ u P M : J puq ą 0 .

Then M Ă Ez t0u is a bounded symmetric C 1 -Finsler manifold radially home( ` omorphic to S “ u P E : }u} “ 1 , M is an open submanifold of M, and positive eigenvalues of (6.20) coincide with critical values of the C 1 -functional 1 puq “ , u P M` . J puq Taking α “ ´1 in (6.21) shows that J , and hence also , is even. Denote by Fl the class of symmetric subsets M of M` with ipMq ě l. It was shown by Perera et al. [113] that λl :“ inf sup puq, MPFl uPM

lě1

is a positive, nondecreasing, and unbounded sequence of eigenvalues, and ip λl q “ ipM` zλl`1 q “ l

(6.23)

when λl ă λl`1 . We assume that there are τ “ pτ1 , . . . , τm q with τi P r1, pi q and 0 ă γ ď mini pi {τi such that setting uτ, α “ pα 1{τ1 u1 , . . . , α 1{τm um q, α ě 0 we have α γ J px, uq ď J px, uτ, α q @α ě 1, px, uq P ˆ Rm . In example (6.22) we can take γ “

(6.24)

m ÿ ri when this sum is ď mini pi {τi . Set τ i“1 i

H px, uq “ F px, uq ´

m ÿ ui BF px, uq pi Bui i“1

and T puq “

m ÿ 1 |ui |τi . τ i“1 i

Theorem 6.6.1 Under the above hypotheses, if λl ă λl`1 and λl J px, uq ´ W pxq ď F px, uq ď λl`1 J px, uq ` W pxq

@px, uq P ˆ Rm (6.25)

for some l and W P L1 p q, then (6.17) has a solution in the following cases: H px, uq ă 0, T puq |u|Ñ8 H px, uq (ii) H px, uq ě ´C pT puq ` 1q and H pxq :“ lim inf ą 0. |u|Ñ8 T puq (i) H px, uq ď C pT puq ` 1q and H pxq :“ lim sup

6.6 Anisotropic systems

139

Proof In view of (6.23) we apply Theorem 6.3.1 with A(0 “  λl , B0 “  ` λl`1 Y pMzM q. Identifying E with α u : u P M, α ě 0 , define an odd homeomorphism of E by hpα uq “ uα . To see that (6.4) holds, let p “ maxi pi . Then for r ě 1, distphprA0 q, Bq “

“

inf

uPA0 , vPB0 sě0

}ur ´ vs } ˜

inf

uPA0 , vPB0 sě0

ěr

1{p

m › ›2 ÿ › 1{pi › ui ´ s 1{pi vi › ›r

“ r 1{p

i

i“1

˜

m › ›2 ÿ › › ›ui ´ s 1{pi vi ›

inf

uPA0 , vPB0 sě0

inf

uPA0 , vPB0 sě0

¸1{2

¸1{2

i

i“1

}u ´ vs }

“ r 1{p distpA0 , Bq, so it suffices to show that distpA0 , Bq ą 0. If not,›there are› sequences uj P A0 , j v j P B0 , and sj ě 0 such that, writing vrj “ vsj , ›uj ´ vrj › Ñ 0. Then ˇ› › › › ˇ › › › › ˇ› j› › j› ˇ › j j› ˇ ›ui › ´ ›vri › ˇ ď ›ui ´ vri › ď ›uj ´ vrj › i

i

i

´› › ¯ ´› › ¯ › j› › j› implies, first that ›vri › is bounded since ›ui › is bounded, and then that i i ˇ› ›pi › ›pi ˇ ˇ› j › › j› ˇ ˇ›ui › ´ ›vri › ˇ Ñ 0 via the elementary inequality i

i

|a p ´ bp | ď p max ta, bup´1 |a ´ b|

@a, b ě 0, p ą 1.

Since 1 ´ sj “ I puj q ´ sj I pv j q “ I puj q ´ I pr vj q “

m ÿ 1 ´›› j ››pi ›› j ››pi ¯ ›ui › ´ ›vri › i i pi i“1

by (6.19), then sj Ñ 1. Thus, › › › › › › distpA0 , B0 q ď ›uj ´ v j › ď ›uj ´ vrj › ` ›vrj ´ v j › › › “ ›uj ´ vrj › `

˜

m ´ ÿ i“1

1{p sj i

¯2 › ›2 › j› ´ 1 ›vi › i

¸1{2 Ñ 0.

140

Sandwich pairs

But distpA0 , B0 q ą 0 since for all u P A0 and v P B0 , 1 1 ´ ď J puq ´ J pvq λl λl`1 ż “ J px, uq ´ J px, vq

ż ż1

ı d ” J px, t u ` p1 ´ tq vq dt

0 dt ż ż1ÿ m BJ “ px, t u ` p1 ´ tq vq pui ´ vi q dt

0 i“1 Bui ˜ ¸ ż ż1ÿ m m ÿ rij ´1 ďC |t uj ` p1 ´ tq vj | ` 1 |ui ´ vi | dt “

0 i“1

ďC

m ÿ

˜

i“1

j “1

ˆ ˙rij ´1 }uj } pj˚ ` }vj } pj˚ }ui ´ vi } pi˚ L p q L p q L p q j “1 ¸ m ÿ

` }ui ´ vi }L1 p q ďC

m ÿ

}ui ´ vi }i

since prij ´ 1qpi˚ {ppi˚ ´ 1q ă pj˚ since M is bounded

i“1

ď C }u ´ v} , a contradiction. So  ( r0 q “ uα : u P A0 , α ě 0 , A “ hpA

 ( r0 q “ uα : u P B0 , α ě 0 B “ hpB

form a sandwich pair. Since I puα q “ α, J puα q “ α J puq

@u P M, α ě 0

by (6.19) and (6.21), respectively, (6.25) gives ż Gpuα q ď I puα q ´ λl J px, uα q ´ W pxq

“ α p1 ´ λl J puqq ` }W }L1 p q ď }W }L1 p q ż Gpuα q ě I puα q ´ λl`1 J px, uα q ` W pxq

@u P A0 , α ě 0,

“ α p1 ´ λl`1 J puqq ´ }W }L1 p q ě ´ }W }L1 p q

@u P B0 , α ě 0.

6.6 Anisotropic systems

141

It only remains ` to˘ verify the pPSq condition, and it suffices to show that every pPSq sequence uj is bounded by a standard argument (see, e.g., Perera et al. [113]). Noting that T puτ, α q “ α T puq @α ě 0, u P Rm , › › › › › j› › j› rj :“ ujτ, 1{ρj we have if ρj :“ T p›u1 › , . . . , ›um › q Ñ 8, setting u m › › › 1› ` j˘ › j› › j› r1 › , . . . , ›u rm › q “ 1, so a renamed subsequence of u r converges to T p›u 1

m

r weakly in E, strongly in Lq1 p q ˆ ¨ ¨ ¨ ˆ Lqm p q for all qi ă pi˚ , and some u a.e. in ˆ ¨ ¨ ¨ ˆ . We have γ

ρj ď C I puj q for some constant C ą 0 since τi γ ď pi for each i, and ż ż I puj q ´ Gpuj q F px, uj q λl`1 J px, uj q ` W pxq “ ď γ γ γ ρj ρj ρj

ż rj q ` J px, u

ď λl`1

}W }L1 p q γ

ρj

for all sufficiently large j by (6.25) and (6.24). Combining these inequalities r ‰ 0 since taking α “ 0 in (6.21) and passing to the limit gives J pr uq ą 0, so u shows that J p0q “ 0. Since each τi ě 1, ´ ¯ j j 1 j ż G pu q, pu {p , . . . , u {p q ´ Gpuj q j m 1 m 1 H px, u q “ Ñ 0. (6.26) ρj ρj

(i) We have T puj q “ ρj T pr uj q Ñ 8 and hence lim

H px, uj q H px, uj q “ lim T pr uj q “ H pxq T pr uq ρj T puj q

a.e. on tr u ‰ 0u, while

ˆ ˙ H px, uj q 1 j ď C T pr u q` Ñ0 ρj ρj

a.e. on tr u “ 0u, so

ż

0 “ lim

H px, uj q ď ρj

ż r‰0 u

H pxq T pr uq

by (6.26) and Fatou’s lemma. This is a contradiction since H ă 0 a.e. and r ‰ 0. u

142

Sandwich pairs

(ii) We have T puj q “ ρj T pr uj q Ñ 8 and hence lim

H px, uj q H px, uj q “ lim T pr uj q “ H pxq T pr uq ρj T puj q

a.e. on tr u ‰ 0u, while

ˆ ˙ H px, uj q 1 j ě ´C T pr u q` Ñ0 ρj ρj

a.e. on tr u “ 0u, so

ż

0 “ lim

H px, uj q ě ρj

ż r‰0 u

H pxq T pr uq

by (6.26) and Fatou’s lemma. This is a contradiction since H ą 0 a.e. and r ‰ 0. u  Remark 6.6.2 A result related to Theorem 6.6.1 can be found in Perera and Schechter [122].

Appendix Sobolev spaces

A.1 Sobolev inequality Let Ă Rn be a bounded open set, and let C08 p q denote the set of infinitely differentiable functions on that vanish near the boundary B of . The basic Sobolev inequality is as follows. Theorem A.1.1 For each p ě 1, q ě 1 satisfying 1 1 1 ď ` , p q n there is a constant C “ Cpp, qq such that |u|q ď C p|∇u|p ` |u|p q ,

u P C08 p q,

where ˆż |u|q “

˜

˙1{q |u|q dx

|∇u| “

,

ˇ ¸1{2 n ˇ ÿ ˇ Bu ˇ2 ˇ ˇ . ˇ Bx ˇ

k“1

k

If p ą n, then |u|8 ď C p|∇u|p ` |u|p q ,

u P C08 p q,

where |u|8 “ ess sup |u|.

143

144

Sobolev spaces

A.2 Sobolev spaces For a nonnegative integer m and p ě 1, consider the norm ÿ }u}m, p “ |D τ u|p , u P C08 p q. |τ |ďm

An equivalent norm is ¨ ˝

ÿ ż

˛1{p |D τ u|p dx ‚

.

|τ |ďm

Theorem A.2.1 If p ě 1, q ě 1,

1 1 m ď ` , p q n

then |u|q ď C }u}m, p ,

u P C08 p q.

Proof The theorem is true for m “ 1 in view of Theorem A.1.1. Assume it is true for m ´ 1. Let q1 satisfy 1 1 m´1 “ ´ q1 p n if m ´ 1 ă n{p and q1 ą n otherwise. In either case, 1 1 m´1 ď ` . p q1 n By the induction hypothesis, |u|q1 ď C }u}m´1, p ,

u P C08 p q.

Thus, |Dj u|q1 ď C }Dj u}m´1, p ď C }u}m, p ,

u P C08 p q.

Hence |∇u|q1 ` |u|q1 ď C }u}m, p ,

u P C08 p q.

Moreover, 1 1 1 ď ` q1 q n since 1 m´1 1 1 ´ ď ` . p n q n

A.2 Sobolev spaces

145

So |u|q ď C p|∇u|q1 ` |u|q1 q ď C }u}m, p , 

and the theorem is proved.

Let be the completion of C08 p q with respect to the norm }¨}m, p . m, p m, p What kind of functions are in W0 p q? If u P W0 p q, then there is a 8 sequence puk q Ă C0 p q such that m, p W0 p q

}uk ´ u}m, p Ñ 0. So }uj ´ uk }m, p Ñ 0. This means that

ÿ

|D τ uj ´ D τ uk |p Ñ 0 as j, k Ñ 8.

|τ |ďm

Consequently, for each τ such that |τ | ď m, there is a function uτ P Lp p q such that |D τ uk ´ uτ |p Ñ 0. The function uτ does not depend on the sequence puk q, for if puˆ k q is another m, p sequence converging to u in W0 p q, then }uˆ k ´ uk }m, p Ñ 0. This implies that uˆ τ “ uτ for each τ . We call uτ the generalized strong D τ derivative of u in Lp p q, and denote it by D τ u. We have the following theorem. Theorem A.2.2 Under the hypotheses of Theorem A.2.1, |u|q ď C }u}m, p ,

m, p

u P W0

p q.

(A.1)

Proof For a sequence puk q Ă C08 p q converging to u in W0 by Theorem A.2.1,

m, p

p q, we have

|uj ´ uk |q ď C }uj ´ uk }m,p . So uk Ñ uˆ in Lq p q. Since uk Ñ u in Lp p q, we must have uˆ “ u. Since |uk |q ď C }uk }m, p , we have (A.1). We also have the following theorem.



146

Sobolev spaces

Theorem A.2.3 If m ą n{p, then u P C08 p q.

|u|8 ď C }u}m, p , Proof If m ´ 1 ă n{p, let

1 1 m´1 “ ´ . q p n Otherwise, take q ą n. Then u P C08 p q

|u|q ď C }u}m´1, p , in view of Theorem A.2.1. Hence

|Dj u|q ď C }Dj u}m´1, p ď C }u}m, p , implying |∇u|q ď C }u}m, p . Since q ą n, we have |u|8 ď C p|∇u|q ` |u|q q ď C }u}m, p .  m, p

Theorem A.2.4 If m ą n{p and u P W0

p q, then u P Cp q and

max |u| ď C }u}m, p .

Proof If puk q is a sequence in C08 p q converging to u in W0

m, p

p q, then

|uj ´ uk |8 ď C }uj ´ uk }m, p Ñ 0. ˆ Since uk Ñ u Hence uk converges uniformly on to a continuous function u. p in L p q, we must have uˆ “ u.  m, p

Corollary A.2.5 If m ´  ą n{p, then W0

max max |D τ u| ď C }u}m, p ,

|τ |ď

p q Ă C  p q and m, p

u P W0

p q.

Proof We apply Theorem A.2.4 to the derivatives of u up to order .



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Index

anisotropic systems 136 compactness Cerami 2, 9 Palais–Smale 2, 9 critical groups 3, 18 at a nondegenerate critical point 5 at infinity 5, 20, 110 at zero 50, 116 invariant under homotopies 4, 20 local nature 48 nontrivial 8, 23, 40, 117 of a mountain pass point 7 critical points 1 mountain pass 24 nondegenerate 5 nontrivial 23 pairs 40

example 108 nonresonant 122 resonant 122, 123, 124 solvability 115 linking 30 homological 33 homotopical 31 Schechter–Tintarev 36 local linking 7 alternative 8 generalized 25, 117 minimax principle 30, 31 minimizer 22 Morse inequalities 4 lemma 5 theory 1

deformation lemma 43 first 3, 11 second 3, 15

nonstandard geometries 42 null manifold 94

Fuˇc´ık spectrum 69 examples 69 minimal and maximal curves 78

operators bounded 67 monotone 67 positive homogeneous 67 potential 67

index cohomological 126 Yang 27 jumping nonlinearity 69, 107 crossing the Fuˇc´ık spectrum 122

p-Laplacian 25 problems 133 potential 67 pseudo-gradient 9

156

Index

sandwich pairs 125 semilinear problems 47, 131 asymptotically linear 53, 57 concave nonlinearities 54, 63 nonresonant 53, 57, 58 resonant 53, 57, 59 shifting theorem 6 simple eigenvalue 99, 101

Sobolev inequality 143 spaces 144 splitting lemma 6 three critical points theorem 8, 24 Type II region 100

157