CRITICAL POINT THEORY AND ITS APPLICATIONS
CRITICAL POINT THEORY AND ITS APPLICATIONS
By WENMING ZOU Tsinghua University, Beijing, China MARTIN SCHECHTER University of California, Irvine, California, USA
^
Spri ringer
Library of Congress Control Number: 2006921852 ISBN-10: 0-387-32965-X
e-ISBN: 0-387-32968-4
ISBN-13: 978-0-387-32965-9
Printed on acid-free paper.
AMS Subject Classifications: 35J50, 58E05, 47J30, 49505, 58E30 © 2006 Springer Sciencen-Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 springer.com
W. Zou dedicates this book to his parents: LIANG-SHENG ZOU & GUO-XIU ZENG. M. Schechter dedicates this book to his wife, children and grandchildren (currently 22) and great grandchildren (currently one).
Contents Preface
ix
1
Preliminaries 1.1 Partition of Unity in Metric Spaces 1.2 Sobolev Spaces 1.3 Differentiable Functionals 1.4 Topological Degrees 1.5 An ODE in Banach Space 1.6 The (PS) Conditions 1.7 Weak Solutions
1 1 2 7 13 16 19 20
2
Functionals Bounded Below 2.1 Pseudo-Gradients 2.2 Bounded Minimizing Sequences 2.3 An Application
25 25 26 31
3
Even Functionals 3.1 Abstract Theorems 3.2 High Energy Solutions 3.3 Smah Energy Solutions
37 37 44 49
4
Linking and Homoclinic Type Solutions 4.1 A Weak Linking Theorem 4.2 Homoclinic Orbits of Hamiltonian Systems 4.3 Asymptotically Linear Schrodinger Equations 4.4 Schrodinger Equations with 0 G Spectrum 4.5 The Case of Critical Sobolev Exponents 4.6 Schrodinger Systems 4.6.1 The Superlinear Case 4.6.2 The Asymptotically Linear Case
55 55 64 73 74 89 101 102 112
viii
CONTENTS
5
Double Linking Theorems 5.1 A Double Linking 5.2 Twin Critical Points 5.3 Eigenvalue Problems 5.4 Jumping Nonlinearities
117 117 120 129 131
6
Superlinear Problems 6.1 Introduction 6.2 Proofs 6.3 The Eigenvalue Problem
141 141 145 152
7
Systems with Hamiltonian Potentials 7.1 A Linking Theorem 7.2 Hamiltonian Elliptic Systems
159 159 167
8
Linking and Elliptic Systems 8.1 An Infinite-Dimensional Linking Theorem 8.2 Elliptic Systems
179 179 187
9
Sign-Changing Solutions 9.1 Linking and Sign-Changing Solutions 9.2 Free Jumping Nonlinearities
195 195 202
10 Cohomology Groups 10.1 The Kryszewski-Szulkin Theory 10.2 Morse Inequalities 10.3 The Shifting Theorem 10.4 Critical Groups of Local Linking 10.5 Computations of Cohomology Groups 10.6 Hamiltonian Systems 10.7 Asymptotically Linear Beam Equations
215 215 227 231 242 244 253 269
Bibliography
287
Index
317
Preface Since t h e birth of t h e Calculus of Variations, it has been realized t h a t when they apply, variational methods can obtain better results t h a n most other methods. Moreover, they apply in a very large number of situations. It was realized many years ago t h a t t h e solutions of a great number of problems are in effect critical points of functionals. In this volume we present some of the latest research in the area of critical point theory. Many new results have been recently obtained by researchers using this approach, and in most cases comparable results have not been obtained by other methods. We describe these methods and present t h e newest applications. In a typical application, one first establishes t h a t the solution of a given problem is a critical point of a functional G{u) on an appropriate space, i.e., a "point" in t h e space where G\u) = 0. Finding t h e points where t h e derivatives vanish is t a n t a m o u n t to solving the problem. T h e main difficulty is finding candidates. In this connection, one can use "geometrical" considerations. But geometrical considerations do not involve derivatives, and usually the best they can produce are Palais-Smale sequences, i.e., sequences of t h e form G{uk) -^ a,
G\uk)
-^ 0.
T h e existence of such a sequence is not enough to produce a critical point. It is possible t h a t such a sequence is converging to infinity. However, if one can show t h a t the sequence has a convergent subsequence, then one indeed obtains a critical point. A functional t h a t has the property t h a t every Palais-Smale sequence for it produces a critical point is said to satisfy the Palais-Smale condition. W h a t is one to do if t h e corresponding functional does not satisfy the Palais-Smale condition? In the present volume, one of the purposes is to consider just this situation. T h e trick here is to find bounded Palais-Smale sequences directly from t h e linking geometry. In most cases such sequences
X
PREFACE
produce critical points. One might think that such methods are severely restricted. However, the number of such methods and applications found here should convince anyone otherwise. It is surprising that so much has been accomplished under this handicap resulting in new variational methods. Another purpose of this book is a description of the so-called topological method. We present a new Morse theory which satisfactorily fits strongly indefinite functionals. We include such topics as extrema, even functionals, weak and double linking, sign-changing solutions, Morse inequalities, and cohomology groups. The applications we describe include Hamiltonian systems, Schrodinger equations and systems, jumping nonlinear it ies, elliptic equations and systems, superlinear problems and beam equations. The book is organized as follows. In Chapter 1, we provide some prerequisites for this monograph. We collect some knowledge of degree theory, Sobolev space and so forth. Basically, these theories are essentially known and readily available in many books. Welltrained readers may skip this chapter. In Chapter 2, we present some theorems concerning functionals which are bounded below on Banach spaces or Finsler manifolds. Chapter 3 is devote to critical point theory on even functionals. Some variants of the fountain theorem will be established without (PS) type assumptions. Applications to Schrodinger equations and Dirichlet boundary value problems will be given. We will show readers how to get infinitely many solutions. In Chapter 4, we establish a weak infinite-dimensional linking theorem. It not only unifies the classical results but also gives us more information. This abstract theory works perfectly for some PDEs and ODEs with pure continuous spectrum. Therefore, applications will be considered mainly on homoclinic type solutions of asymptotically linear Hamiltonian systems and Schrodinger equations, superlinear Schrodinger equations with zero as a point of the spectrum or with critical Sobolev exponents. In particular, Schrodinger systems depending on time will be discussed. Chapter 5 concerns twin critical points resulting from double linking. Roughly speaking, if A links 5 , does B link A? Can they yield two different critical points without the (PS) type compactness conditions? We will give positive answers. Applications on eigenvalue problems and Dirichlet elliptic equations with jumping nonlinearities will be studied.
PREFACE
xi
In Chapter 6, we solve elliptic semilinear boundary value problems in which the nonlinear terms are quite weak super-linear. That is, the nonlinearities need not satisfy a superquadracity condition of the Ambrosetti-Rabinowitz type. Because of this, we are able to include more equations than hitherto permitted. Some new tricks will be seen. In Chapter 7, we assume that A links B. Let Bi and B2 be two linear bounded invertible operators. We describe the situation in which the values of the functional H are separated by BiB and B2A. Then BiB and B2A become much more complicated. We prove the existence of a critical point of H without assuming (PS) type conditions. This theory is applied to some special elliptic systems. In Chapter 8 we prove an infinite-dimensional linking theorem and apply it to elliptic systems with gradient type potentials. In Chapter 9, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A linking type theorem is established with the location of the critical point in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities. Under stronger conditions, we show that the existence of sign-changing critical points can be independent of the Fucik spectrum which usually is indispensable for such cases. In Chapter 10, a more advanced Morse theory will be introduced. We first present the W. Kryszewski-A. Szulkin infinite-dimensional cohomology theory and a new Morse theory associated with it. Then we develop some methods of computing the cohomology critical groups precisely. Applications to Hamiltonian systems and beam equations will be considered. The present monograph is based on results obtained by ourselves or through direct cooperation with other mathematicians such as S. Li, A. Szulkin, Z. Q. Wang and M. Willem. It is not intended to be complete. The materials covered in this book are presented at a level suitable for advanced graduates and Ph. D. students following the development of new results, or anyone who wishes to seek an introduction to critical point theory and the study of differential equations by variational and topological methods. The chapters are designed to be as self-contained as possible.
xii
PREFACE
Both Zou and Schechter thank the University of Cahfornia at Irvine for providing a favorable environment during the period 2001-2004 in which the first version of this book was written. Both authors wish to thank the NSF, NSFC (No. 10571096 & No. 10001019) and SRF-ROCS-SEM for supporting much of the work that led to this book.
Wenming Zou Tsinghua University, Beijing Martin Schechter University of California at Irvine
Chapter 1
Preliminaries In this chapter, we present some classical results on nonlinear functional analysis and partial differential equation. Some of them are well known and we shall omit their proofs. For others, although their proofs may be found in many existing books, we make no apology for repeating them.
1.1
Partition of Unity in Metric Spaces
Assume {E,d) is a metric space with a distance function (i(-, •). Let A C E and let 11 be a family of open subsets of ^ . If each point of A belongs to at least one member of 11, then 11 is called an open covering of A. Definition 1.1. Assume that H is an open covering of a subset A of E, then n is called locally finite if for any u ^ A, there is an open neighborhood U such that u e U and that U intersects only finitely many elements of li. The following result is due to A. H. Stone [347]. Proposition 1.2. Any metric space {E,d) is paracompact in the sense that every open covering li of E has an open, locally finite refinement Q, i.e., 6 is a locally finite covering of E and for any Vi of Q, we can find a Ui of li such that Vi C Ui. Proposition 1.3. Assume that {E, d) is a metric space with an open covering n . Then li admits a locally finite partition of unity {\i}i^j subordinate to it satisfying: (1) Xi : E ^ [0,1] is Lipschitz continuous; (2) {Vi}i^j is a locally finite covering of E, where Vi = {u ^ E : \i{u) 0}, J is the index set;
^
2
CHAPTER 1. (3) for each Vi, there is a Ui eli
PRELIMINARIES
such that Vi C Ui]
(4) E i e J ^ i W = l ' V u e ^ . Proof. Since {E, d) is a metric space with an open covering 11, by Proposition 1.2, there is an open, locally finite refinement B, i.e., B is locally finite and for any Vi of B, we can find a [/^ of 11 such that Vi ^Ui. We define pi{u) = d{u,E\Vi),
i G J.
Then pi is locally Lipschitz. Let
Then {Xi}i^j is what we want. This proves the theorem.
1.2
D
Sobolev Spaces
Let O be an open subset of R ^ , AT G N. Define 1/^(0) := {i^ : O ^ R is Lebesgue measurable, ||I^||LP(Q) < oo}, where \\U\\LP{Q)
= y
\u\Pdxj
,
l
If p = +00, \u\\LOOm\ = ess sup \u\ := Q
inf
Acfl,meas{A)=0
sup |i^| Q\j^
where meas denotes the Lebesgue measure. Let ^ L ( ^ ) •={u:n^Il,ue
LP{V) for each V CC O},
where y c c O < ^ y c y c O and V is compact. We will in this book denote \MLP{Q)
by ll^llp or \u\p.
Let C^{fl) denote the space of infinitely differentiable functions (/> : O ^ R with compact support in O. For each (j) G C^(^) and a multiindex a = (o^i,..., Q^AT) with order \a\ = ai -\-... -\- ajy, we denote dx'^'
•••
dx'}" i-jV
1.2. SOBOLEV SPACES
3
Definition 1.4. Suppose u^v e Ll^^{ft). We say thatv is the a^^-weak partial derivative of u, written D^u = v provided uD'^(j)dx = (-1)1^1 / v(j)dx for
aU(l)eC^{n).
It is easy to check that the o^^^-weak partial derivative of u, if it exists, is uniquely defined up to a set of measure zero. Let C^{ft) be the set of functions having derivatives of order < m being continuous in Q {m = integer > 0 or m = oo). Let C"^(0) be the set of functions in C"^(0) all of whose derivatives of order < m have continuous extension to fl. Definition 1.5. Fix p G [1, +oo] and A: G N U {0}. The Sobolev space
consists of all u : Q ^ H which have a^^- weak partial derivatives D^u for each multiindex a with |Q^| < A: and D^u G 1/^(0). If p = 2, we usually write H^{n) = iy^'2(0), Note that H^{n) which agree a.e.
= L'^{n).
A: = 0 , 1 , 2 , . . . .
We henceforth identify functions in ly^'^(O)
Definition 1.6. If u e VF^'^(O), we define its norm to be (
/
\ i/p vjv
Definition 1.7. We denote
as the closure ofC^{Q) in VF^'^(O) with respect to its norm defined in Definition 1.6. It is customary to write
and denote by II~^{Q) the dual space to
IIQ{Q).
CHAPTER 1.
PRELIMINARIES
The following results can be found in L. C. Evans [147]. P r o p o s i t i o n 1.8. For each A: = 1, 2 , . . . and I < p < +oo, the Sobolev space f VF^'^(O), II • ||vi/fc'P(Q)) ^-5 0. Banach space and so is H^{Q),HQ{Q)
WQ'^{Q).
In particular,
are Hilhert spaces.
cY. Definition 1.9. Let (X, || • ||x) CL^id (F, || • \\Y) he two Banach spaces, X We say that X is continuously imbedded in Y (denoted by X ^^ Y) if the identity id : X ^ Y is a linear bounded operator, that is, there is a constant C > 0 such that \\U\\Y < C||i^||x for all u e X. In this case, constant C > 0 is called the embedding constant. If moreover, each bounded sequence in X is precompact inY, we say the embedding is compact, written X ^^^^ Y. Definition 1.10. A function u : Q C R ^ ^ H is Holder continuous with exponent 7 > 0 i/ 1(7) .-
\u(x) — u(y)\ ^ < 00. \x-y\-r
sup
^ ^
Definition 1.11. The Holder space C'^''^(f2) consists of all functions u G C''(f2) for which the norm \\u\\cK.m:=
Y,
P"«llc(n)
\a\
(y.\=k
is finite. It is a Banach space. We set C^'^(O) = C^{Q). We have the following imbedding results, see R. A. Adams [4], L. C. Evans [147] and D. Gilbarg-N.S. Trudinger [174]. P r o p o s i t i o n 1.12. If Q is a bounded domain in H^,
rk,p
<'^(0)
L^(0),
kp
C"^'^(0),
0
then Np/{N - kp), N/p
and
0 < m < k- N/p < m + 1. P r o p o s i t i o n 1.13. If ft is a bounded domain in H^,
<'^(0)
L^(0),
kp
C"^'^(0),
{)
then Np/{N - kp), and
0 < m < k — N/p < m -\-1.
1.2. SOBOLEV SPACES
5
The following proposition can be found in H. Brezis [64] and M. Willem [377]. Proposition 1.14. The following imbeddings are continuous: H\'R^)^LP(R^),
if 2
H^K^)
if
where 2* := 2N/{N-2) exponent.
-^L^(R^), ifN>3
= l,2,
23,
and 2* = +oo if N = 1,2, is called a critical
For AT > 3, let
be the best Sobolev constant. Then, by G. Talenti's results (cf. [362]),
o_l|v^lli
\\u\\l,' where (Ar-2)/4
(iV(iV-2)) U{x) = ^ '
(N-2)/2
Note that if R ^ is replaced by a bounded domain, S is never achieved. Let iVC(a;)e(^-2)/2
^e{x) :--
(£2 + Ixp)
(Af-2)/2'
where A^ = {N{N - 2))^^ ^^^".e > 0 and ^ e Cg°(R^, [0,1]) with i{x) = 1 if |a:| < r/2; ^(x) = 0 if |x| > r, where r can be chosen to meet different requirements. Proposition 1.15. The following estimates are true (see e.g. pp.35 and 52 ofM. Willem [377]):
r
ce2|^n£|+0(e2),
ll^e||i = where c > 0 is a constant.
A^ = 4,
6
CHAPTER
1.
PRELIMINARIES
We shall frequently use the following Gagliardo-Nirenberg Inequality, see L. Nirenberg[262] (see also L. C. Evans [147] and J. Chabrowski [89]). Proposition 1.16. For every u G H^{¥i^)^ \\u\\, 0 and q G [2,2*). For any bounded sequence {u^} of E := HHR^), if ^P / sup
yGR^R ^
where B{y^r) q
\un\^dx ^ 0 ,
n ^ 00,
JBiy^r) JB{y,r)
:= {u e E : \\u — y\\ < r}, then Un ^
0 in I/^(R^) for
Proof. We just consider A^ > 3. Choose Pi,P2,^ > l,t^ > 1 such that Pit = q^p2t' = 2*,l/t + 1/t' = l,pi + p2 = P' By Holder's Inequality and Proposition L13, we have \Un\^dx B{y,r)
<( [
lur^r'dxY^'f [ lur^r^'dxY^'
^JB{y,r)
^
^JB{y,r)
<-^\,
^
'" lB{y,r)
I
<
c
(
•
lB{y,r)
\Unr*dxy"( -
•
/ {ul + ^JB{y,r)
- ' ^ ' ^ ^
^
-
\SJUn\'')dxy''\ - ' ' ' ' '
(In this book, the letter c will be indiscriminately used to denote various constants when the exact values are irrelevant.) Covering R ^ by balls with radius r in such a way that each point of R ^ is contained in at most A^ + 1 balls, then we have /
|u„|^dx<(A^ + l)c sup f /
\un\'^dx)
which impUes the conclusion. This proves the theorem.
1/t
D
1.3. DIFFERENTIABLE
1.3
FUNCTIONALS
7
Differentiable Functionals
Let ^ be a Banach space with norm || • ||. Let U C E he SLU open set of E. The dual (or conjugate) space of E is denoted by E^, i.e., E^ denotes the set of bounded linear functionals on E. Consider a functional I : U ^ H. Definition 1.18. The functional I has a Frechet derivative F ^ E' at u ^ U
heE,h^o
\\h\\
We define r{u) = F or \/I{u) = F and sometimes refer to it as the gradient of / at u. Usually, r{') is a nonlinear operator. We use C^(t/, R) to denote the set of all functionals which have continuous Frechet derivative on U. A point u ^ U is called a critical point of a functional / G C^{U, R), if
r{u) = 0. Definition 1.19. The functional I has a Gateaux derivative G ^ E' atu if, for every h ^ E, t^o
t
^U
^^
The Gateaux derivative ai u ^ U is denoted by DI{u). Obviously, if / has a Frechet derivative F e E^ dit u e U, then / has a Gateaux derivative G e E^ dit u and I\u) = DI{u). But the converse is not true. However, if / has Gateaux derivatives at every point of some neighborhood oi u ^U such that DI{u) is continuous at u^ then / has a Frechet derivative and I'{u) = DI{u). This is a straightforward consequence of the Mean Value Theorem. Let f{x,t) be a function on O x R, where O is either bounded or unbounded. We say that / is a Caratheodory function if / ( x , t) is continuous in t for a.e. x G O and measurable in x for every t G R. Lemma 1.20. Assume p > l^q > I. Let f{x,t) on ft X H and satisfy \f{x,t)\
< a + 6|t|^/^
be a Caratheodory function
V(x,t) G O X R ,
where a, 6 > 0 and O is either bounded or unbounded. Define a Caratheodory operator by Bu:= f{x,u{x)), ueLP{n). Let {i^fcj^o ^ LP{Q). If\\uk -uoWp -^ 0 as k ^ +oo, then \\Buk - Buo\\q -^ 0 as k ^ oo. In particular, if ft is bounded, then B is a continuous and bounded mapping from 1/^(0) to L^{ft) and the same conclusion is true if ft is unbounded and a = 0.
8
CHAPTER 1.
PRELIMINARIES
Proof. Note that there is a renamed subsequence such that (1.1)
Uk{x) -^ uo{x),
a.e. x e ft.
Since / is a Caratheodory function, (1.2)
Buk{x) -^ Buo{x),
a.e. x e ft.
Let (1.3)
Vk{x) := a + 6|^fe(x)|^/^
A: = 0,1, 2 , . . . .
Then by (1.1)-(1.3), (1.4)
\Buk{x)\ < Vk{x) for all x ^ ft;
Vk{x) -^ vo{x) a.e. x ^ ft.
Since \uk\^ -\- \UO\P — \\uk\^ — |i^o|^| ^ 0, by Fatou's Theorem, we have
/ hminf (\ukf + i^or - ii^^r - i^ori)^^ (1.5)
< hminf /
(\ukf^\uof-\\ukf-\uof\)dx.
Combining (1.1)-(1.5), we see that (1.6)
lim
/ ||^fe|^-|^orM^ = 0.
It follows that (1.7)
/ \v^-v^\Ux
\
\\u^\^-\u^\^\dx^^
as A: ^ oo. Since there are constants C > 0, Ci > 0 such that \Buk-Buo\i
<
C{\Buk\'^ +
\Buon
<
C,i\vk-vo\^
+ \von
a.e. X G O, by Fatou's Theorem, / liminf (Ci{\vk - ^ol' + l^ol") - \Buk (1.8)
< liminf /
Buo\'')dx
(Ci{\vk-vo\''^\von-\Buk-Buo\Adx.
1.3. DIFFERENTIABLE
FUNCTIONALS
9
By (1.2), (1.3), (1.7) and (1.8), we have \\Buk-Bu\\q^O. Finally, if O is bounded, then for any u G 1/^(0), evidently we have (1.9)
\\Bu\U
where c > 0 is a constant. Inequality (1.9) remains true if O is unbounded and a = 0. Therefore, 5 is a continuous and bounded mapping from 1/^(0) to 1/^(0) and the same conclusion is true if O is unbounded and a = 0. D
The following lemma comes from M. Willem [377]. Lemma 1.21. Assume pi,p2,qi,q2 tion on O X R and satisfy \f{x,t)\
> 1- Let f{x,t)
< a|t|^i/^i + 6|t|^^/^%
be a Caratheodory func-
\/{x,t) G O X R,
where a,b > 0 and O is either bounded or unbounded. Define a Caratheodory operator by Bu := / ( x , u{x)),
uen:=
L^' (O) H L^' (O).
Define the space <£::=L^i(0) + L^2(0) with a norm \\u\\s = mf^\\v\\L.iin)
+ \\w\\L.2in) :u = v^weS,ve
L^'{n),w
e L^^(O)}.
Then B = Bi -\- B2, where Bi is a bounded and continuous mapping from L^^iQ) to L^^{Q)^i = 1,2. In particular, B is a bounded continuous mapping from Ti to 8. Proof. Let (f : R ^ [0,1] be a smooth function such that £^{t) = 1 for t G (-1,1); ^(t) = 0 for t ^ (-2, 2). Let ^(x,t) = e ( t ) / ( x , t ) ,
h{x,t) = (1 - e ( t ) ) / ( x , t ) .
We may assume that pi/qi < ^2/^2- Then there are two constants d > 0,m > 0 such that \g{x,t)\ < (i|t|^i/^S \h{x,t)\ < m|t|^2/^^
10
CHAPTER 1.
PRELIMINARIES
Define Biu = g{x,u),
u G I/^'(0);
B2U = h{x,u),
ueLP^{n).
Then by Lemma 1.20, Bi is a bounded and continuous mapping from L^^^Q) to L^^ (^), ^ = 1, 2. It is readily seen that B := Bi-\-B2 is a bounded continuous mapping from H to S. D The fohowing theorem and its idea of proof are enough for us to see that the functionals encountered in this book are of C^. Theorem 1.22. Assume a > 0,p > 0. Let f{x,t) on ft xH satisfying
(1.10)
be a Caratheodory function
1/(^,01 < «IC + ^l^r. V(x,t) G O X R,
where a, 6 > 0 and ft is either bounded or unbounded. Define a functional I{u) := / F{x,u)dx,
where F{x,u) = /
JQ
f{x,s)ds.
JO
Assume {E, \\ • ||) is a Sobolev Banach space such that E ^^ L^^^iQ) E ^ L^+i(0). Then I G C^{E,Ii) and I'{u)h = I f{x,u)hdx,
and
\/h G E.
JQ
Moreover, if E ^^^^ L^^'^^E ^^^^ L^^^, then I' : E ^ E' is compact. Proof. Since E ^^ L^^^(Q) and E ^^ 1/^+^(0), we may find a constant Co > 0 such that (1.11)
||u;||<,+i
||u;||p+i
^weE.
We make use of Young's Inequahty and
(\s\ + \t\Y <2--\\s\- + \tY),
T>i,s,teR.
Combining the assumptions on / , for any 7 G [0,1], it is easy to check that \f{x,u^-ih)h\
< Ci(|^P+i + l/iP+i + l^l^+i + |/i|^+i),
where Ci is a constant independent of 7. Therefore, for any u^h G E^ by the Mean Value Theorem and Lebesgue's Theorem,
1.3. DIFFERENTIABLE
FUNCTIONALS
I{u + th) - I{u)
lim
lim / / (1.12)
11
f{x,u-\-Oth)hdx
f{x,u)hdx
: Fo{u,h),
where 0 G [0,1] depends on u, h, t. Obviously, Fo(i^, h) is linear in h. Further, by (1.11), \Fo{u,h)\ < I
\f{x,u)h\dx
a+ll
ll^ll^+lll/^ll.+l)
k+1
DI{v))h\
/ {f{x,u) Jn
I
f{x,v))hdx
{Bu — Bv)hdx
Jn {Biu -\- B2U — Biv — B2v)hdx
< / \Biu - Biv\\h\dx ^ / \B2U - B2v\\h\dx < Co\\B,U
- B,v\\^,^,yjh\\
+ Co\\B2U
-
This implies that (1.13)
\\DI{u) -
DI{V)\\E'
<
Co(||Sl^t-Bl^;||(,+l)/,. where || • \\E' is the norm in E\
\\B2U-B2v\\^p^iyp
B2v\\^p^,y4h\\.
12
CHAPTER 1. Uvk^uinEc
PRELIMINARIES
L^+i(0) n L^+i(0), then
Therefore, DI{vk) -^ DI{u). This means that DI{u) is continuous in u. Hence, r{u) = DI{u), i.e., / G C^(^,R). Furthermore, if E ^ ^ L^+\ E ^^^^ 1/^+^, then any bounded sequence {uk} in ^ has a renamed subsequence denoted by {uk} which converges to i^o in L^^^{Q) and in 1/^+^(0). Hence, 5 i K ) ^ BI{UQ) in L ( ^ + I ) / ^ ( 0 ) ; ^ S K ) ^ ^2(^0) in L ( ^ + I ) / ^ ( 0 ) . Finahy, DI{uk) -^ DI{uo) in ^^, i.e., r is compact in ^ . This proves the theorem. D A relation between pointwise convergence of functions and convergence of functionals can be found in H. Brezis-E. Lieb [67]. Sometimes, we wih use the concepts of the second order Frechet and Gateaux derivatives. Definition 1.23. The functional I G C^{U,Il) has a second order Frechet derivative at u e U if there is an L, which is a linear hounded operator from E to E', such that
r(u^h)-r(u)-Lh
^
We write I"{u) = L. We say that / G C^(t/, R) if the second order Frechet derivative of / exists and is continuous on U. Definition 1.24. The functional I G C^{U^Yi) has a second order Gateaux derivative at u ^U if there is an L, which is a linear hounded operator from, E to E', such that (r{u^th)-r{u)-Lth]v Um -^ t^o t We write D'^I{u) = L.
^— = 0,
V/i,
veE.
Evidently, any second order Frechet derivative of / is a second order Gateaux derivative. Using the Mean Value Theorem, if / has a continuous second order Gateaux derivative on [/, then / G C^(t/, R).
1.4. TOPOLOGICAL
1.4
DEGREES
13
Topological Degrees
Since topological degree is an eternal topic of every book on nonlinear functional analysis, we just outline the main ideas and results without proofs. Readers may consult the books of L. Nirenberg [263], K. Deimling [131] and E. Zeidler [379]. To construct the degree theory, Sard's Theorem plays a key role. For this, we introduce the following definition. Definition 1.25. Let U C X := R^ {N > 1) be an open subset and a mapping I G C^{U,X). A point u e U is called a regular point and I{u) is a regular value if I'{u) : X ^^ X is surjective. Otherwise, u is called a critical point and I{u) is the critical value. Here we state a simplified version of Sard's Theorem. It is from A. Sard [305]. Theorem 1.26. Let U C X := R ^ {N > 1) be an open subset and I G C^{U,X). Then the set of all critical values of I has zero Lebesgue measure in X. Definition 1.27. (Brouwer Degree) LetU C X '.= 11^ {N > 1) be a bounded open subset, I G C^{U,X), p e X\I{dU). (1) If p is a regular value of I, define the Brouwer Degree by deg(/, [/,p) :=
^
signdet/'(i;),
where det denotes the determinant. (2) If p is a critical value of I, choose pi to be a regular value (by Sard^s Theorem) such that \\p — Pi\\ < dist{p,I{dU)) and define the Brouwer Degree by degiI,U,p):=degiI,U,pi). In item (1), I~^{p) is a finite set when p is a regular value. In item (2), the degree is independent of the choice of pi. If / G C(f7,X), we may find by Weierstrass's Theorem an approximation of / via a smooth function. Definition 1.28. (Brouwer Degree) LetU C X := R ^ {N > 1) be a bounded Choose I G C^{Lf,X) such that open subset, I G C{U,X), p G X\I{dU). sup \\I{u) - I{u)\\ < ueu
dist{p,I{dU))
14
CHAPTER 1.
PRELIMINARIES
and define the Brouwer Degree by deg{I,U,p):=deg{i,U,p), which is independent of the choice of I. Proposition 1.29. Let U C X := R^ {N > 1) be a bounded open subset, IeC{U,X),peX\I{dU). (1)
(2) Let Ui, U2 be two disjoint open subsets ofU,p^
I{U\{Ui U U2)). Then
deg(/, U,p) = deg(/, Ui,p) + deg(/, U2,p). (3) LetH eC{[^,l] x f 7 , R ^ ) , P G C ( [ 0 , 1 ] , R ^ ) andp{t) ^H{t,dU). deg{H{t, •), U^p{t)) is independent of t e [0,1].
Then
(4) (Kronecker Theorem) //deg(/, U,p) ^ 0, then there exists a u ^ U such that I{u) = p. Theorem 1.30. (Borsuk-Ulam Theorem) Let U be an open bounded symmetric neighborhood ofO in R ^ . Every continuous odd map f : dU -^ R^"-*^ has a zero. As we know, in treating ODE or PDE, the working space is infinitedimensional. We have to prepare a degree for infinite-dimensional space. Here we introduce a degree for a compact perturbation of the identity. This is the Leray-Schauder degree. Definition 1.31. Let E be a Banach space, M C E. A mapping I : M ^ E is called compact if I{S) is compact for any bounded subset S of M. Further, if I is continuous, we say that I is completely continuous. In this case, id — I is called a completely continuous field. Theorem 1.32. Let E be a Banach space, and let M C E be a bounded closed subset. Let I : M ^ E be a continuous mapping. Then I is completely continuous if and only if, for any £ > 0, there exists a finite-dimensional subspace E^ of E and a bounded continuous mapping I^ : M ^ E^ such that s u p ||/(l^) -In{u)\\ uEM
< S.
1.4. TOPOLOGICAL DEGREES
15
Let ^ be a Banach space, and let U C E he di bounded open subset. Let I : U :^ E he completely continuous and f = id — I. If p e E\f{dU), then by Theorem L32, there exists a finite-dimensional subspace E^ of E and a bounded continuous mapping 1^:1)^ E^ such that sup||/(^)-/^H||
En\fn{dUn).
Definition 1.33. (Leray-Schauder Degree) Let f be the completely continuous field defined as above. Define the Leray-Schauder degree of f at p e E\f{dU) by ^^g{f^U,p) = deg{ fn,Un,p), which is independent of the choice of En^p^I^. Proposition 1.34. Let U C E be a bounded open subset of the Banach space E, f = id — I a completely continuous field, and p G E\f{dU). Then (1)
deg{id,U,p) = ^
J^
P'
(2) Let [/i, U2 be two disjoint open subsets ofU,p^
f{U\{Ui U U2)), then
deg(/, U,p) = deg(/, Uup) + deg(/, U2,p). (3) Let H G C([0,1] X U,E) be completely continuous, p G C([0,1],^) and p{t) i H{t, dU) for each t G [0,1]. Then deg{H{t, •), U,p{t)) is independent oft G [0,1]. (4) (Kronecker Theorem) / / d e g ( / , U,p) ^ 0, then there exists a u ^ U such that f{u) = p. Theorem 1.35. (Borsuk-Ulam Theorem) Let U be an open bounded symmetric neighborhood of 0 in a Banach space E. A completely continuous field f = id — I : U ^ E, where I is odd on dU, p G E\f{dU), then deg(/, U^p) is an odd number. We also refer the readers to N. G. Lloyd [238], L. Nirenberg [264], M. Nagumo [260, 261], D. Guo [177] and C. Zhong-X. Fan-W. Chen [383] for the details of degree theory.
16
CHAPTER 1.
1.5
PRELIMINARIES
An ODE in Banach Space
Let ^ be a Banach space with a norm || • ||. Consider the following Cauchy initial value problem of the ordinary differential equation d(^
-r ^ /
/
NX
(1.14) cr{0,uo) =uo e E, where V is a potential function. We are interested in the existence of solutions to (1.14), which will play an important role in the following chapters. We assume: (V) y : ^ ^ ^ is a locally Lipschitz continuous mapping, i.e., for any u ^ E, there exists a ball B{u,r) := {w ^ W : \\w — u\\ < r} with radius r and a constant p > 0 depending on r and u such that \\V{wi) -V{w2)\\
< p\\wi -W2\\,
y wi,W2 e
B{u,r).
Moreover,
||y(^)|| < a + 6||^||,
yueE,
where a, 6 > 0 are constants. T h e o r e m 1.36. Assume {V). Then for any u ^ E, the Cauchy problem, (1.14) has a unique solution o-{t,u) defined on the interval [0,+00) of t. Moreover, o-{t,u) depends continuously on the initial data u. Hence, a eC\[0,^oo) X E,E). To prove Theorem 1.36, we prepare two auxiliary results. L e m m a 1.37. (Gronwah Inequality) If X > 0, (3 > 0 and f e C([0,r],R+) satisfies
(1.15)
/ ( t ) < A + /3 / f{s)ds,
VtG[0,r],
Jo
then f{t) < Xe^^ for all t G [0,r]. Proof. By (1.15), we observe that
dt
(e~^' / f{s)ds) V
JQ
< Xe-
Integrating both sides on [0,t], we get the conclusion.
D
1.5. AN ODE IN BANACH SPACE
17
Lemma 1.38. (Banach Fixed Point Theorem) Let E be a Banach space, with D C E dosed. Let F : D ^ D satisfy (1.16)
||Fi^ - Fv\\ < k\\u - v\\
for some k G (0,1) and all u,v e D.
Then there exists a unique u^ such that Fu^ = u^. Proof. Choose UQ e D and let i^n+i = Fun. have ll'^^n+m+l - Un\\ < (1 - ky^k'^Wui
Using (1.16) repeatedly, we
- Uo\\ ^
0,
U^
+00.
Therefore, {un} is a Cauchy sequence. The conclusion follows from the continuity of F . n Proof of Theorem 1.36. For any fixed i^o ^ E^ by condition (V), we find a ball B{uo,r) := {w e W : \\w — uo\\ < r} with radius r and a constant p > 0 depending on r and i^o such that \\V{wi) -V{w2)\\
< p\\wi -W2\\,
y wi,W2 e
B{uo,r).
Let A:=
sup
||y||.
B(uo,r)
Then A < +00. Choose £ > 0 such that sp < l^sA < r. Consider the Banach space E := C([0,£],F) := {u : [0,^] ^ F is a continuous function} with the norm ||i^||^ := max^^[o,£] ||'?^(OII for each u e E. Let D := {u e E : \\u — uo\\^ < r}. Define a mapping F : E ^ E hj Fu := 1^0 + / V{u{s))ds, Jo
u e E.
For any u,w ^ D we have \\Fu-uoh<
I Jo
\\V{u{s))\\^ds
and | F ^ - F ^ | | ^ < max /
\\V{u) - V{w)\\^ds < ps\\u - w\\^.
Therefore, F : D ^ D satisfies all conditions of Lemma 1.38. Hence, F has a unique fixed point u^ ^ D, which is a solution of Cauchy problem (1.14).
18
CHAPTER
1.
PRELIMINARIES
On t h e other hand, assume t h a t u{t) and v{t) are solutions of t h e Cauchy problem (1.14) corresponding t o initial d a t a i^o and VQ, respectively. Then
Mt)-v{t)u <\\UO-VO\\E+
I
\\V{u{s))-V{v{s))\\^ds
Jo
\\u{s)-v{s)\\^ds. Jo
By Lemma 1.37, \\uit)-vit)\\^<\\uo-vohe^K This proves t h e continuous dependence on t h e initial d a t a of solution of (1.14). Summing up, (1.14) has a unique solution u{t) on t h e maximal existence interval [0, K.) which is continuously depending on t h e initial data. Next, we just show t h a t n = + o o . Assume t h a t n < + o o . Then u{t) = UQ -\-
V{u{s))ds. Jo
Thus, by ( y ) , ||^(t)|| < WuoW^an^b
[ \\u{s)\\ds. Jo L e m m a 1.37 implies t h a t there is a constant Ci depending on u^^n^a such t h a t \\u{t)\\
and h
It follows t h a t \Ht)-u{s)\\
(1.17) cr(0,l^l) =Ui e E. Similarly, it has a unique solution u{t) on a maximal interval [0,/^i) with initial d a t a ui = u{n — 0). Let
r v{t) = I
u{t),
tG[o,/^),
Then v{t) is also a solution of (1.14) with t h e initial d a t a i^o on t h e maximal interval [0, n -\- ni). This produces a contradiction. D
1.6. THE (PS) CONDITIONS
1.6
19
The (PS) Conditions
Many nonlinear problems can be reduced to the form (1.18)
/'(^)=0,
where / is a C^ functional on a Banach space. Equation (1.18) is called the Euler-Lagrange equation of the functional / . The original idea was to find maxima and minima of / , and the critical point theory was devoted to finding extrema of / . The simplest extrema to find are global maxima and minima if / is semibounded. However, we can derive from the semiboundedness the existence of a sequence {un} C E such that
We now introduce definitions of compactness conditions. Definition 1.39. Any sequence {u^} satisfying (1.19)
sup|/(^^)| < o o ,
l\ur,)^0,
n
is called a Palais-Smale sequence ((PS)-sequence, for short). If any (PS)sequence of I possesses a convergent subsequence, we say that I satisfies the (PS) condition. The original idea of the (PS) condition was introduced by R. Palais [269], S. Smale [343] and R. Palais-S. Smale [272]. One of the weak versions of the (PS) condition was proposed in G. Cerami [86]. Definition 1.40. Any sequence {un} satisfying (1.20)
s u p | / ( ^ ^ ) | < OO, n
(1 + | | ^ n | | ) / ' ( ^ n ) ^
0,
is called a Cerami sequence ((C)-sequence, for short). If any (C)-sequence of I possesses a convergent subsequence, we say that I satisfies the (C) condition. Theorem 1.41. Let E be a Banach space, I G C^(^,R). Assume I'{u) =Lu^J'{u),
ue
E,
where L : E ^^ E' is a bounded linear invertible operator and J' maps bounded sets to relatively compact sets in E'. Then any bounded (PS)-sequence or (C)sequence is relatively compact.
20
CHAPTER 1.
PRELIMINARIES
Proof. Let {un} be a bounded (PS)-sequence or (C)-sequence, then I'{un) -^ 0. The conclusion fohows from the relative compactness of J' and Un = L-^r{Un)-L-^J'{Un). • It is well known, for a given functional / , there may exist critical points of / which are not even local extrema. The existence of this class of critical points was first studied by A. Ambrosotti and P. H. Rabinowitz [19], where a set of sufficient conditions was provided. We also refer the readers to M. Struwe [352]. This abstract critical point theory was based on the (PS) condition. In this book, we will establish a series of theorems without any (PS) type assumption.
1.7
Weak Solutions
In practice, one of the main research projects using critical point theory is the existence of solutions to elliptic equations. For example, consider (1.21)
-/\u
= f{x,u),
xen.
The corresponding functional is defined by I{u) = ^\\\/uf-
J
F{x,u)dx,
where F{x,u)
= / f{x,s)ds. Roughly speaking, if / is in C^ on a suitable Jo space and I\u) = 0 (a critical point), then (1.22)
/ Vu'VwdxJQ
/ f{x,u)wdx
= 0,
\/w e E.
JQ
The critical point u satisfying (1.22) is called a weak solution of (1.21) and obviously u is not necessarily a classical solution. In general, more assumptions on the smoothness of dfl and of / are needed if we want the weak solution to be a classical solution. It is not an easy task. In this section, we just give a simple example to show how the regularity theory of elliptic equations can be applied to obtain a classical solution from a weak solution. Definition 1.42. Consider a bounded domain ft C R ^ with boundary dft. Let k be a nonnegative integer and o^ G [0,1]. O is of class C^'^ if at each point xo G dft there is a ball B = B{XQ) and a one-to-one mapping Lp from B onto a subset D C R ^ such that
1.7. WEAK SOLUTIONS
21
(1) cp{B n O) C R ^ := {x = (xi, X2,..., XN) eK^ :XN > 0}; (2) ip{B n dn) C dK^ := {x = (xi, X2,..., XN) eK^ :XN = 0};
The following proposition is due to M. D. Gilbarg-N. S. Trudinger [174, Theorems 6.6] (see also M. Struwe [352]). Proposition 1.43. Suppose that u G Hi^^{ft) such that —Au = f in ft with f G I/^(^), 1 < p < 00. Then for any O^ CC ft, we have \\U\\H^^P{Q^) < C{\\U\\LP^Q^ + | | / | | L P ( Q ) ) ,
where C depends on ft,ft\N,p. Assume in addition that ft is a C^'^ domain and that there exists a function uo G H'^'^{ft) such that u — uo ^ HQ'-^{ft). Then \\U\\H^^P{Q)
< C{\\U\\LP(Q)
+ ||/||LP(Q) +
||^0||if2,p(Q)),
where C depends on ft^N^p. The following proposition is found in M. D. Gilbarg-N.S. Trudinger [174, Theorems 6.14 and 6.19]. Proposition 1.44. Assume that ft is a Qk+2,a domain, f G C^'^(O). Then the Dirichlet problem —Au = f
in ft,
u=0
in dft
has a unique classical solution u G C^+^'^(0). The following proposition is also in M. D. Gilbarg-N.S. Trudinger [174, Theorem 9.15]. Proposition 1.45. Assume that ft is a C^'^ domain, f G L^{ft),p > 1. Then the Dirichlet problem —Au = f
in ft,
u=0
in dft
has a unique classical solution u G I^o^'^l^) H W'^^P{n). The following result is due to H. Brezis-T. Kato [66] (see also M. Struwe [352]).
22
CHAPTER 1.
PRELIMINARIES
L e m m a 1.46. Assume that ft is a domain of R ^ (N > 3) and that f : ft xH ^H is a Caratheodory function such that \f{x,t)\ where a G LfJ^\n).
< a{x){l + |i^|),
If u e HHifl)
a.e. x e Q,
is a weak solution of
—Au = f{x^u)
in O,
then u G L^^^(Q) for any q < oo. If u ^ HQ' (ft) and a G 1/^/^(0), then u G L^{Q) for any q < oo. Next, we give an example to illustrate when a weak solution becomes a classical solution. As we mentioned, it is not a matter of course. Theorem 1.47. Assume that Q is a bounded domain ofH^ {N > 2), Q is is a Caratheodory function such that (1) there exists a r G (0,1] such that f{x,t)
for any
GC^''^(OX [ - M , M ] , R ) ,
M
>0;
(2) there are C > 0 and 2 < p < 2* such that \f{x,t)\
< C ( 1 + |^|^-^),
(3) there exists a function fo{x) G L^{ft) lim
^— = fo{x)
a.e.xeQ;
such that uniformly for x G O.
Assume u G H^' (Q) is a weak solution of (1.23)
-/\u
= f{x,u)
inn,
u=0
on dn.
Then u must he a classical solution of (1.23). In particular, u G C^+^'^(0), where (3 = ar^~^^. Proof. Let
f{x,t) g{x,t)
t /o(x),
'
if t ^ 0, ift = 0.
1.7. WEAK SOLUTIONS
23
Then there are constants a > 0, 6 > 0 such that (1.24)
\g{x,t)\ < fo{x) + a + b\tf-^ < /o(x) + a + b\tf-^
for ah X G 0 , t G R. By assumption, u G HQ' ( O ) is a weak solution of (1.25)
—Au = g{x,u{x))u
in O,
i^ = 0
on dfl.
If N > 3, by Proposition 1.12, u G L'^*{n). By Lemma 1.20 and (1.24), g{x,u{x)) G 1/^/^(0). Then, Lemma 1.46 implies that u G 1/^(0) for ah 5 > 2. This is naturally true if A^ < 2. Noting the conditions (2)-(3) and using Lemma 1.20 again, we see that f{x,u{x))
GL^(O),
V5>2.
Choose 5 > 2 , 5 > p — 1 . By Proposition 1.45, the problem (1.26)
—Aw = f{x,u{x))
in O,
w=0
on dft
has a unique solution
w G w^'^n) n iy2'^(o), v^ =
^ > i, 5 > 2.
Since u is a weak solution of (1.23), we see from (1.26), that u = w.Uwe choose q = j ^ , then q > 2/{p - 1) if N > 2. By Proposition 1.12, u G I^o'^(^) implies that u G C^'^(O); here 1 — N/q = o^. Then we may find a M > 0 such that |^(x)|<M,
\u{x)-u{y)\
VXGO,
<M|x-7/|^,
x.yen.
Note that O is of class C^+^.Q^ with a G (0,1). Thus / satisfies condition (1) with k replaced by 0,1, 2 , . . . , A: - 1 (see M. D. Gilbarg-N. S. Trudinger [174, Lemma 6.35] and W. Lu [245, Theorem 7.5.4]). Hence, there exists a C > 0 such that + \u-vn \f(x,u)-fiy,v)\
Therefore,
\fix,uix)) - f{y,v{y))\ + \u{x)-v{y)n
24
CHAPTER 1.
PRELIMINARIES
in C^'^^(O). By induction, if we assume that u e C*=+i'"'' (f2), we may prove that f{x,u{x)) G C^'^^ (O) and u = w. By Proposition 1.44 once again, we have that u G C^+2'«^ ^ (O). This proves the theorem. D
We refer the readers to M. D. Gilbarg-N.S. Trudinger [174], L. C. Evans [147], M. Struwe [352] and M. E. Taylor [366] for the regularity theory of elliptic equations. We focus on critical point theory in the present volume.
Chapter 2
Functionals Bounded Below If a functional has the same infimum on two bahs Br C BR, we estabhsh the existence result of a bounded (PS)-sequence. On the other hand, we give a new proof for classical results on functionals bounded below.
2.1
Pseudo-Gradients
The existence of a pseudo-gradient vector field is a foundation stone for some variational problems. Lemma 2.1. Let P be a continuous mapping from a Banach space E to its dual E', and let E := {u e E : P{u) ^ 0}. For any a G (0,1), there exists a locally Lipschitz continuous mapping (i.e., pseudo-gradient vector field) V : E ^ E such that \\V{w)\\ < 1,
a\\P{w)\\ < {P{w),V{w)),
\/weE.
Proof. Take ai G {a, 1). For any u e E, there exists an element (j) = (j){u) G E such that mu)\\
= 1,
ai||P(u)|| < {P{u),cf>{u)),ue
E.
The continuity of P{u) implies that there exists an open neighborhood U{u) of u such that a\\P{w)\\ < {P{w),(l){u)), w e U{u). Then we get an open covering {U{u)} oi E. By Proposition 1.3, there is a locally finite refinement {Vi\i^j and a locally Lipschitz continuous partition
26
CHAPTER 2. FUNCTIONALS
BOUNDED
BELOW
of unity {Xi}iej subordinate to this refinement. For each i e J, Vi C U{ui) for some ui. Define
Then V : E ^ E is locahy Lipschitz continuous. By Proposition 1.3,
\\V{w)\\
Moreover, {Piw),Viw))>aJ2Mw)\\Piw)\\=a\\Piw)\\. ieJ
D Notes and Comments. Many books and papers have addressed the existence of the pseudo-gradient vector field which was apphed directly to prove miscellaneous deformation theorems. For examples, see P. Bartolo-V. BenciD. Fortunato [28], V. Benci-P. H. Rabinowitz [55], K. C. Chang [95, 96] (on a Finsler manifold), Y. Du [143], M. R. Grossinho-S. A. Tersian [176], J. Mawhin-M. Willem [252], L. Nirenberg [265], P. Rabinowitz [293], M. RamosC. Rebelo [298], M. Schechter [310], M. Struwe [352] and M. Willem [376, 377].
2.2
Bounded Minimizing Sequences
Let {E, II • II) be a Banach space and / G C^(^,R). Theorem 2.2. Assume that there exist R > r > 0 such that m := inf / = inf / > —oo, BR
B,
where BR := {U ^ E : \\U\\ < R}. Then there exists {un} C BR such that I{un) -^ m,
I'{un) ^ 0
as n ^ oo.
Proof. Let D{R,e) = {u e BR : I(u) < m + e}. Then
inf
||/'(^)|| = 0
for all £ > 0. Otherwise, there would exist an SQ > 0 such that ||/^(i^)|| > £o/{R-r) wheni^ G D{R, SQ). Let u G D(r,£o/2)(^ 0). By Lemma 2.1, there is a V{u) : E :={ue E : r{u) ^ 0} ^ ^ such that 11^(^)11 < 1 ,
{V{u)J'{u))>\\\I'{u)l
yueE.
2.2. BOUNDED MINIMIZING SEQUENCES
27
Moreover, y is a locally Lipschitz continuous map. Let a{t,u) be the solution of the Cauchy initial value problem
a{0,u) =ue
D{r, So/2).
Then ||cr(t,i^)-1^11 < / \\a\s,u)\\ds
=
Therefore,
-{l'{a),V{a))
<-^\\I'(a{t,um ,
3
for u G D(r, £o/2), and therefore, a{t,u) G D{R, SQ) for t e [0, i? — r] and u e D{r, £o/2). Furthermore, 3 1 3 1 l(cr{R - r, u)) < I{u) - -£o < m + -£o - jSo = m - -£o, a contradiction.
D
Consider a family of C^{E, R)-functionals of the form Ix{u) := -\H{u)
- J{u),
XeA.ueE,
where A C (0, oo) is an open interval; H{u) > 0 for all u e E. Assume that one of the following conditions holds. (A) For any /3 > 0,
sup {H'ull :ue
E with H{u) < (3} < +oo.
(B) For any /3 > 0,
sup {H'ull :ue
E with J{u) < (3} < +oo.
Theorem 2.3. Assume that either (A) or (B) holds and that Ix is bounded below for each A G A. Then for each X e A, there exists a sequence {un} such that s u p ||l^n|| < oo,
Ix{Un)
^
Mx
'•= i n f / A ,
Ixi^n)
^ 0 ,
aS Tl ^
OO.
28
CHAPTER 2. FUNCTIONALS
BOUNDED
BELOW
Proof. We only prove the first case. Note that the mapping X ^ Mx is concave with respect to A G A. Therefore, it is Lipschitz continuous on each closed subinterval of A. For A G A, we choose a closed subinterval AA C A containing A as an interior point. Then, there exists a constant Ai^ > 0 depending on A such that
\Mx-My\<M'^\X-X'\, Choose A„ e (A, 2A) n
AA,A„
VA'eA^.
-^ A as n ^
Then l - ^ - - - ^ - ' <
oo.
An — A A^A f^^ ^11 ^- We claim that there exists a sequence {u^} C E such that ll'^nll < ^o(A) := ko and Mx < Ix{un) < hA^n) < Mx^ + (A, - A) < A^A + {M'x + l){Xn - A). In fact, by the definition of AIA^, there exists a Un such that Ix^{un) Mx^ + (An - A). Evidently, Mx < h{un) < h^i^n) and Hiu^)
= 2 ( ^ A . . K ) - ^ A K ) ) < 2M',
<
+ 4;
An — A J{un) =
^^"^-^
7—^—- < -Mx
+ Mx + 1.
An — A
Therefore, by (A), there exists a /CQ = ko{X) > 0 such that ||i^n|| < ^o- Define I^,(A) :={ueE:
\\u\\
Ix{u) < Mx ^ £}•
Then, for any £ > 0, there exists an n large enough such that Un G ^^^(A). Now we claim that inf{||/A(i^)|| : u G ^^^(A)} = 0 for ah e > 0. If not, there exists an £o > 0 such that ||/^(i^)|| > SQ for u G Ds^{X). By Lemma 2.1, there is a locally Lipschitz continuous map VA • ^ •= {'^ ^ ^ • ^A('^) T^ ^} ^ ^ ^^^^ that \\Vx{u)\\ < 1 and {rAu),Vx{u)) > ^ | | / ; H | | for ah u e E. Therefore, for any u G
DSQ{X),
we have that
A:={u:
||VA('^)||
^ - and (/^(i^), VA('^)) ^ ^^o- Define
\\u\\ >ko^2}U{u:
5 := {^ : ll^ll < A:o + 1,
Ix{u) > At A + ^o/3}, A^A < h{u) <Mx^
£o/4}.
Then, An B = 9. Moreover, if u ^ A then i^ G l^£o(A)- In particular, 5 C D^^X). Define ^(^) := dist(^,A) ^^ ^ dist(^,5) + dist(^,A)'
^^ ^
_ e(^)yAH. '^v ; Av ;
2.2.
BOUNDED
MINIMIZING
SEQUENCES
29
Then it is easy t o check t h a t (/^(i^), ^ A * ( ^ ) ) ^ ^ ^ ^ ^ ||y^*|| < 1 for ah u e E. F u r t h e r m o r e , for u e B C DSQ{X),
C{^) = 1 ^^^
{I',{U),V^{U))=^{U){I'^{U),VX{U))
II^A('^)II — ^o, we have
> 1\\I',{U)\\ > ^£0.
dfi (t u) Now we consider t h e initial value problem — - ^ — = —Vx{r]) with ?^(0, u) = u for each u ^ E (note t h a t ^ vanishes on an open set containing t h e points where / ^ = 0)- It is well known t h a t there exists a unique solution r]{t,u) for t >0. Moreover, Mx
< Ix{v{t.un))
< lx{r]{0,un))
< IxA^n)
< Mx^ + (A - A,) < A^A + ^
for n large enough. Consequently, = \\ [ dr]{s,Un)\\< Jo
\\r]{t,un)-uj
[ Jo
\\V^{r]{s,Unmds
It follows t h a t \\r]{t,Un)\\ < ||t^n||+^ < A:o + l f o r t < 1. Therefore, r]{t,Un) G B for ah t e [0,1]. Moreover, /A(^(l,^n)) '^
-h{Un)
dIx{r]{s,Un))_^^
0
ds [I'xivis^
Un))^ Vxivis,
Un))j ds
10
1
It follows t h a t Mx
<
Ix{r](l,un))
<
h{Un) - -So
<
AiA + (A^l + 2 ) ( A ^ - A ) - i £ o
<
Mx-
-£o-
This is a contradiction. Thus, we know t h a t there exists a sequence such t h a t sup ||i^n(A)|| < oo,
/A(^^n(A)) ^ A^A,
Ix{unW)
{un{X)}
^ 0 as n ^ oo.
30
CHAPTER 2. FUNCTIONALS
BOUNDED
BELOW
If H^ is invertible and J^ is compact, by standard arguments, there exists a ux such that Ix{ux) = Mx and /^(I^A) = 0D We therefore have Theorem 2.4. Assume that Ix is bounded below for each A G A and that any bounded (PS)-sequence oof Ix is precompact. Then for each A G A, Mx = inf/A E
is a critical value of Ix.
Theorem 2.3 can be easily generahzed to a manifold (see R. Palais [269, 270, 271] for the definition of a Finsler manifold). Let M be a complete Finsler manifold with Finsler structure || • || and / G C^(M, R). Let dl{u) denote the differential of / at u. Define \\dI{u)\\M := sup{|a/(^)(/)| : (/) G T^{M), UW < 1}, where Tu{M) denotes the tangent space of M at u. Let T{M) = UueMTu{M) and M = {u ^ M : | | ^ / ( I ^ ) | | M T^ 0}. Then we have the following theorem. Theorem 2.5. Let M ^ 9 be a complete Finsler manifold with Finsler structure II • II and let Ix G C^(M, R) be bounded below for all A > 0. Assume that either (A) or (B) holds with E replaced by M. Then for each A G A, there exists a Palais-Smale sequence {un} C M such that SUpll'Unll < OO, Ix{Un)
^
MX'=^^fI\,
||^^A('?^n) | | M ^
0
aS U ^ OO.
Proof. For any 0 G (0,1), there exists a mapping Y{u) : M -^ T{S) which is locally Lipschitz continuous and satisfies \\Y{u)\\ < 1,
e\\dIx{u)\\M < {dh{u),Y{u)),
ueM.
This is a generalization of Lemma 2.1 (see also K. C. Chang [96, Lemma 3.1]). In fact, choose 0 < 0' < 1. For u G M, there exists an h{u) G Tu{M) such that
\\h{u)\\ = 1, e'\\dix{u)\\M < {dh{u),h{u)). By the continuity of dlx{u), for each u G M, there is a neighborhood N{u) such that 6'||a/A(^)||M < {dlx{v),h{u)) for v G N{u). Then {N{u) : u e M} is an open covering of M. Since M is a metric space, we may find a locally finite refinement {iV^-}. Let {V^r} be a locally Lipschitz continuous partition of unity subordinate to this refinement. For each r, let u^- be an element for
2.3. AN APPLICATION
31
which Nr C N{ur). Let Y{v) = '^iljr{v)h{ur). Then Y{v) is what we want. Similar to the proof of Theorem 2.3, we can prove that ini{\\dh{u)\\M
: \\u\\
h{u) <Mx+s}
=0
for ah £ > 0 smah enough. It is sufficient to know that the flow r]{t, u) stifl exists on the manifold. D Notes and Comments. By I. Ekeland's variational principle (cf. [145]), if / is bounded from below, then there exists a minimizing sequence {un} for / such that I{un) -^ inf/, I'{un) ^ 0 as n ^ oo. E
It is worth noting that if / satisfles the (PS) condition and maps bounded sets into bounded sets, then inf/ > —oo
<^
I{u) -^ +00
as ||i^|| -^ +oo,
E
that is, / is coercive. This was proved in L. Caklovic-S. Li-W. Willem [77] using gradient flow (see also S. Li [211] and M. Willem [377]) and in D. G. Costa-E. A. B. Silva [114] by Ekeland's variational principle. Other properties for functionals bounded below (combining other assumptions) can be seen in H. Brezis-L. Nirenberg [70] (with local linking and without symmetry), in D. C. Clark [102] and in S. Li-Z. Q. Wang [217] (with locations of the critical points) for even functionals bounded below. Theorems 2.2-2.5 were obtained in M. Schechter-W. Zou [326].
2.3
An Application
We just give a simple example. We study the following elliptic eigenvalue problem: (P/3)
—Au = (3g{x,u)
in O;
i^ = 0
on ^O,
where O is a bounded smooth domain in R ^ and g{x, t) is a Caratheodory function such that ^(x,0) G 1/^(0). Let 0 < Ai < A2 < • • • < A^ < • • • be the sequence of eigenvalues of —A with Dirichlet zero boundary condition. Assume that there exists a constant a such that ^^^
^^ ^ g{x,t)-g{x,s)
^ ^ ^ ^^^^
^^^ ^^^ ^ ^ ^ ^^^ ^ ^ ^^
32
CHAPTER 2. FUNCTIONALS
BOUNDED
BELOW
Theorem 2.6. Assume that (C) holds. Then for each (3 G (1, - ^ ) ; problem (PIS) has a solution Uf^. That is, the eigenvalue problem (P/3) has infinitely many solutions. However, we would like to show it by using Theorem 2.4. We need the following lemma. Lemma 2.7. Let Ei fz = 1, 2) be two closed subspaces of a real Hilbert space E with an inner product (•, •) and the corresponding norm \\ • \\ such that E = El® E2. Let I G C^(^,R). / / there exists an increasing function h : R+ -^ R+ such that h{s) -^ +00 as s ^ +00 and that (2.1)
{I\u^v)-I\u^w),v-w)
> \\v - w\\h{\\v - w\\)
for all u G Ei^v^w G ^2, then we have the following results. (1) There exists a continuous function (j) : Ei ^ E2 such that I{u -\- (j){u)) = min I{u + v). veE2
Moreover, (j){u) is the unique member of E2 such that {I'{u^(j){u)),v)
=0,
\JveE2.
(2) The functional J : Ei ^ H defined by J{u) = I{u-\- (j){u)) is of class C^ and {J\u),v) = {I\u^d^{u)),v), yu.veEi. (3) An element u e Ei is a critical point of J if and only if u -\- (j){u) is a critical point of L Proof. (1) For each u G ^ 1 , define H^ : E2 ^ K by Hu{v) = I{u + v). By the assumption (2.1), Hu is of C^ and has at most one critical point. We claim that Hu is coercive. Note that Hu{v) = Hu{0)^
[ Jo
{Hu{sv),v)ds
>H^{0)-\\H:,m\\v\\+
f\\\v\M\sv\\)ds. Jo
By the hypotheses on /i, we may choose R large enough such that h{\\sv\\) > 8||i:f;(0)|| uniformly for \\v\\ >R,se
[1/2,1].
2.3. AN APPLICATION
33
Hence, Hu{v)>Hu{0)^\\vl which imphes that Hu{v) ^ oo as H^;!! ^ oo. Next, we show that Hu is convex. For given v^w G ^ 2 , define C{s) =
Hu{v^s{w-v)).
For 0 < a < f3 < 1, by (2.1), it is easy to show that
r(/3)-e'(«)>0. This means that ^ is convex in 5, and consequently, H^ is convex in v. Combining the above arguments, we see that Hu has a unique minimizer (l){u) G E2 with Hu{(j){u)) = min{/(i^ -^ v) : v ^ E2]. Therefore, we have (2.2)
{I'{u^(l){u)),w)={),
\IweE2.
To show that (j){u) is continuous in u^ we assume on the contrary, that there are £0 > 0 and Uk ^ u SiS k ^ 00 such that \\(l){uk) -(/>(^)|| >£o. Let P be the projection from E to ^2- By (2.2), we see that \\Pr{uk (/>('^))|| ^ h{£o/2) if A: large enough. Therefore, h{eo)muk)
- c^{u)\\
< {I\uk + (/)(^fe)) - I\uk + (/)(^)), (/)(^fe) - 0(^))
< (-r(^fe + H^)), (i){uk) - (i){u)) <\\PI'{uk^cj,{u))\\U{uk)-cj,{u)\\
{J'{u),w) = {I'(u^(t){u)),w),
w G El.
Indeed, for 5 > 0, J{u + sw) — J{u) s I{u -\- sw -\- (j){u -\- sw)) — J{u -\- (j){u)) s I{U + 51(7 + (l){u)) - J{U + (l){u)) < s (2.4)
= /
{r{u^(l){u)^rsw),w)dr.
+
34
CHAPTER 2. FUNCTIONALS
BOUNDED
BELOW
Similarly, we have /^ ^N
J (U -\- SW)
(2.5)
^
f^
— J (U)
^
,-rf,
^
,/
N
N X ,
{r{u^(t){u^sw)^TSw),w)dT.
^
Jo
Summing up (2.4)-(2.5), we get (2.3).
D
Let E := HQ{Q) be with the usual norm induced by the inner product {u, v) =
\/u ' \jvdx. We denote by E{\i) the eigenspace corresponding to Jn Xi. Set E- = E{Xi)QE{X2)Q- • -©^(Afe-i), E^ = E{Xk) and E^ = ^(A^+i)© ^(Afe+2) © • • • . Then ^ = ^ - © ^^ © ^ + . Solutions of (Pp) correspond to the critical points of the C^-functional h{u) = ^\\ufwhere A = —, G{x,u) = /
J
G{x,u)d:\x,
\/ u e E,
g{x,s)ds.
Proof of Theorem 2.6. Choose A = ( v ^ , !)• Then for A G A and u G E^ ®E-,w,v
e ^ + , by (C), we have that (Ixiu^v)
- Ix(u^w),v
-w)
= X\\v — w\\^ — I (g{x,u-\-v)—g{x,u-\-w)]{v
— w)dx
J S2
> X\\v — w\\'^ — / a{v — w)'^dx JQ
>{X-T^)\\v-Wf. By Lemma 2.7, there exists a mapping 6x : E^ ^ E- ^ E+ such that Ix{u^ ^u-^(t)x{u^ Moreover, (t)\{u^ ^u~)
^u-))=
min /A(^^ + ^ " + ^ ^ ) .
is the unique member of E^ such that
for ah V e E^. Define a functional JA : ^^ © ^ ~ ^ R by Jxiu"" + U-) = Ixiu"" + ^ - + (/)A(^^ + u-)).
2.3. AN APPLICATION
35
Then J is of class C^ and
for all u^ -\-u~,z e E^ ^E~. Moreover, u^ -^u~ is a critical point of J if and only if u^ ^ u~ ^ (l)\{u^ -^ u~) is a critical point of I\. Next, we claim that —I\ is bounded below on E~ ® E^. In fact, by condition (C) we see that Afe^^ < t{g{x,t) - ^ ( x , 0 ) ) < at^. Then G{x,t)=
/ 5^(x,5)—>-Afet2 + t^(x,0). Jo
5
^
Therefore, Ix{u) < \{\ - l)\\uf 2
- I ug{x,0)dx ^ - o o Jn
as 111^11 -^ oo. Hence, —Ix and —Jx are bounded below. Evidently, —Jx satisfies the other assumptions of Theorem 2.3. Therefore, for all A G A, there exists a ux such that —J^^{ux) = 0. This completes the proof of the theorem. D Notes and Comments. Lemma 2.7 was established in A. Castro [81]. Some applications of it can be found in A. Castro-J. Cossio [82] and M. Schechter [315]. Theorem 2.6 was given in M. Schechter-W. Zou [326]. Possibly, it can be proved by other methods such as the degree theory or the contraction mapping principle. We believe that Theorem 2.4 has far more extended applications. We would like to leave them to the readers.
Chapter 3
Even Functionals In this chapter we present some abstract theorems which concern the existence of infinitely many critical points for even functionals. The Palais-Smale type compactness condition is not necessary for the new results. By taking advantage of the abstract theorems, we study the existence of infinitely many large energy solutions for nonlinear Schrodinger equations and of infinitely many small energy solutions for semilinear elliptic equations with concave and convex nonlinear it ies.
3.1
Abstract Theorems
Let ^ be a Banach space with the norm || • || and let {Xj} be a sequence of subspaces of E with dimX^ < oo for each j G N. Further, E = ^j^-^Xj, the closure of the direct sum of all Xj. Set
and Bk := {ueWk:
\\u\\ < pk},
Sk := {ue Zk'. \\u\\ = r/e}, for Pk > Tk > 0. Consider a family of C^-functionals ^\ : E ^Yi $A(W) := J(w) - AJ(w),
oi the form:
Ae[l,2].
We make the following assumptions. (Ai) ^\ maps bounded sets into bounded sets uniformly for A G [1,2]. Moreover, ^ A ( - ' ^ ) = ^A('^) for all (A,i^) G [1,2] x E.
38
CHAPTERS.
EVEN
FUNCTIONALS
(A2) J{u) > 0 for all u e E; I{u) ^ 00 or J{u) ^ 00 as ||i^|| -^ 00, or (A3) J{u) < 0 for all u e E; J{u) -^ —00 as ||i^|| -^ 00. Let afe(A) :=
max
^x(u),
bk{X) :=
ueWk,\\u\\=Pk
inf
^x{u).
ueZk,\\u\\=rk
Define Cfc(A) := inf max ^x{j{u)),
where
jeTkueBk
Tk := {7 e C{Bk,E) : 7 is odd,7|a5, = id],
k>2.
Theorem 3.1. Assume that (Ai) and either (A2) or (As) hold. If bk{X) > a/c(A) for all A G [1,2], then Ck{X) > bk{X) for all X G [1,2]. Moreover, for almost every X G [1,2], there exists a sequence {u^{X)}'^^i such that sup ||i^^(A)|| < 00, ^xi^^W)
-^ 0 and ^xi^nW)
^ Cfc(A)
as n ^ 00.
n
Proof. We divide the proof into two cases. Case 1: Assume that conditions (Ai) and (A2) hold. We show that Ck{X) > bk{X) first. For each 7 G r^, let U^ := {u e Bk : ||7('^)|| < 7fe}- Then U^ is an open bounded symmetric neighborhood of 0 in W^. Let P^ : E ^ W^-i be the projection onto Wk-i. Then Pkj : dU^ -^ Wk-i is a continuous odd map. By the Borsuk-Ulam Theorem (cf. Theorem 1.30), there exists a i^ G dU^ such that Pkj{u) = 0, that is, 7(1^) G Z^, ||7(i^)|| = r^. Therefore, j{Bk) nSk j^9. This implies Ck{X) > bk{X). Furthermore, Cfc(A) < max ^A('^) ^ i^ax ^i(i^) := m/c, uEBk
uEBk
where ruk is a constant independent of A. By (A2), Ck{X) is nonincreasing with respect to A. Therefore, c^(A) := ——— exists for a.e. A G [1,2]. dX From now on, we consider those A where the derivative c^(A) exists. Let A^ G [1, 2], A^ < A, A^ -^ A, then there exists an n(A) such that (3.1)
- 4 ( A ) - 1 < " ' - ( y _ - ^ ^ ( ^ ) < -c-(A) + 1
forn>n(A).
Step 1. We show that there exists a sequence 7^ G Tk^m := m(c^(A)) > 0 such that ||7n('^)|| < ^^T. if ^A(7n('^)) ^ ^^(A) — (A — A^) for some u G B^-
3.1. ABSTRACT
THEOREMS
39
Indeed, let 7^ G Tk be such that sup^^^^ ^x^hn{u)) < Ck{Xn) + (A - A^). If ^x{ln{u)) > Ck{X) — (A — An) for some u e Bk, then
.. .NX -^(TnN) = <
^Aj7nH)-^A(7nH) ^^^7^^ -4(A)+ 3
and I{-fn{u))
< <
Cfe(An) + (A-An) + An(-4(A) + 3) Cfe(A) + A(-4(A) + 3).
Recahing (A2), we observe that ||7n('^)|| ^ ^^ for some m > 0. Step 2. Since A^ < A, obviously we have that ^xhn{u))
< ^ A j T n H ) < Cfe(A) + (2 - 4(A))(A - Xn) for ah u e Bk.
Step 3. For e small enough, define ^,(A, k):={ueE:
\\u\\ < m(4(A)) + 4, | ^ A ( ^ ) - Cfe(A)| < e}.
It suffices to prove that inf{||^^^(i^)|| : u G ^^^(A, A:)} = 0. Otherwise, we may assume that there exists an SQ > 0 such that ||^^(i^) || > £o for u G J-SQ (A, A:). Take 7^ G F/^ from step 1 and choose A —A^ < £o,X — Xn < '"'^^^-'"'^^\
(2 - 4(A))(A - A„) < £0. Define
J^{X,k) •.= {ueE:
\\u\\ < m(4(A))+4,Cfe(A)-(A-A„) < $A(W) < Cfe(A)+£o}.
Then J^{\, k) C J^eo{\ k). For any u G J^{\, k), let u;{u) :=
I^IH
12-
Then the mapping v -^ {^^^{v),uu(u)) is continuous for each v G J^{X,k). Moreover, hx{u) := {^'^{u),io{u)) = 2 for i^ G J^{X,k). Hence there exists a neighborhood [/^^ of u such that /IA('^) = ( ^ A ( ' ^ ) ' ^ ( ' ^ ) ) > 1 for 'U G t/t^ H J^{X,k). Therefore, we get an open covering {Uu}ueJ^{x,k) of ^(A, A:). Choose an open set UQ := ^ ^ ^ ( - 0 0 , Cfc(A) - (A - A^)). Then {Uu}ueTix,k) U t/o is an open covering of ^(A, k):={ueE:
\\u\\ < m(4(A)) + 4, ^ A M < Cfe(A) + £0}.
40
CHAPTERS.
EVEN
FUNCTIONALS
Hence, there exists a refinement {Nj}j^j such that Nj C Uu or Sj C UQ and a locahy Lipschitz continuous partition of unity {Pj}jeJi where J is the index set. Define UJJ{U) = uj{u) for Nj C Uu] ^j{u) = 0 for Nj C t/o, and set J\f := U^GJ^j- Then ^(A, k) C A/". Let ^*(A, k):={ueE:
^x{u) < Ck{\) - 2(A - A,)}
and . , ^ dist(^,>F*(A,A:)) "^^^^ • " dist(^,J^*(A,A:)) + dist(^,J^(A,A:))' ^^ Define a vector field l^(i^) := '4^{u)^-^j PJ{U)(JOJ{U) : J\f ^ E and consider djTi
the following Cauchy problem — = —V{r]) and r]{0,u) = u for u ^ E with ll^ll < m(4(A)), Cfe(A) - (A - A,) < ^x{u) < Ck{X) + soThen V is locally Lipschitz continuous and for any u as above, there exists a unique solution ?^(-, u). Noting that for any u G J^£Q{X, A:), we have that either 2 2 cjo = 0 or \\iOn\\ = ||c(;(i^)|| = ,, ^, , ,,, < —. Therefore, V is bounded and ^
^
II^AMII
^0
2 11^(^)11 ^ —• It follows that r](',u) exists as long as it does not approach the £o
boundary of M. Moreover, {^'x{u),V{u)) > 0 , for ^ G AT, {^xi^), V{u)) > 1, for u G J^(A, k). Evidently, ^^^(^(^^^)) < Q. For each u G ^ ^ ^ ( - o o , Cfe(A) - 2(A - A^)), we have V{u) = 0 and ^x{r]{s,u)) < ^x{r]{0,u)) = ^x{u)' It follows that r]{s,u) G ^^^(-oo,Cfc(A) - 2(A - A^)) and V{r]{s,u)) = 0. Therefore, (3.2)
v{s,u) =u
for ue ^ ^ ^ ( - o o , Cfe(A) - 2(A - A^)).
On the other hand, since ^A is even, then we may choose r]{s, u) to be odd in u. Consider ?^(2£o,7n('^))- Then we claim that ?^(2£o,7n('^)) ^ ^k- Iii fact, for u G dBk,jn{u) = u, then r]{2£o,jn{u)) = r]{2£o,u). Since a/c(A) < c/c(A) —2(A —An), we have ^A('^) < «fe(A) < c/c(A) —2(A —A^) for i^ G ^ 5 ^ . By (3.2) r]{2£o,u) = 1^. Hence, 7^(260,7n('^)) = '^- Noting that 7n('?^) and r]{2£o,u) are odd, we see that 7^(2^0,7n('^)) is odd. Consequently, 7^(260,7n('^)) G F^. If ^A(7n('^)) < c/c(A) — (A —A^) for some u ^ B^, then r]{s,jn{u)) is well defined and (3.3)
$A(r/(2eo,7n(w))) <$A(r?(0,7n(w))) = $A(7n(w))
3.1.
ABSTRACT
THEOREMS
41
If ^xi^jn^u)) > Cfc(A) — (A — A^) for some u ^ B^, then by the choice of 7^, we know t h a t ||7n('^)|| < ^(c/c(A)). By step 2, (3.4)
^A(7nH)
Thus, jni"^) ^ -^(A, k). Assume t h a t
(3.5)
^A(^(2£o,7n(^))) > Cfe(A) - (A - A^)
for some u ^ Bk- Note t h a t ll^(^,7n(^))-7n(^)|| < / Jo 2t < —.
\\V{r^{t,^n{u))\\dt
From this we see t h a t \\r]{t,jri{u))\\
<
2t
\- ||7n('^)|| < ^(c/c(A)) + 4 for
t G [0, 2^0]. Now, we have t h a t Cfe(A) - ( A - An) < ^A(^(2£o,7n(^))
for t G [0,2£o]. Since {^'^(u),V(u))
> 1
for i^ G
^A(^(2£o,7n(^)))
= ^A(7n(^)) + j
—^A(^(5,7n(^)))
<^A(7n(^))-2£o < Cfe(A) - ( A - An), which contradicts (3.5). Hence, max^^^^^ ^A(^(2£o,7n('^))) < c/c(A) —(A —A^). Therefore, Cfe(A) = inf m a x ^ A ( 7 ( ' ^ ) ) < m a x ^A(^(2£o,7n('?^)) < Cfe(A) - (A - A^).
42
CHAPTERS.
EVEN
FUNCTIONALS
This contradicts t h e definition of Cfc(A) and completes t h e proof of t h e first case. Case 2: If conditions (Ai) and (A3) hold, t h e proof is quite similar to t h a t of case 1. We omit t h e details. D By Theorem 3.1, we get bounded Palais-Smale sequences for almost all A. In applications, the bounded (PS) sequence readily has a convergent subsequence if t h e working space has some kind of "imbedding compactness." If ^fe(A) ^ 00 as A: ^ 00, we may obtain infinitely many high energy critical points. T h e next theorem is a dual form of Theorem 3.1, by which we may get critical points with small energy.
T h e o r e m 3 . 2 . Assume
that
( B i ) ^x maps bounded sets to bounded sets uniformly ^x{-u) = ^x{u) for all {X,u) G [1,2] x E; (B2) J{u) > 0; J{u) ^
for A G [1,2].
00 as \\u\\ -^ 00 on any finite-dimensional
Further,
subspace
ofE; (B3) there exist pk > rk > 0 such that afe(A) := bk{X)
inf ^x{u) ueZk,\\u\\=pk
:=
max
> 0,
^A('^) < 0
ueWk,\\u\\=rk for all A G [1,2] and dfe(A):=
inf
$A(n)-0
ueZk,\\u\\
for A G [1,2].
Then there exist A^ -^ l,u{Xn)
G Wn such that
^LLJ^(^^)) =^'
^Aj^(An)) ^Cfe
as n ^ 00, where Ck G [c^/c(2), 6/^(1)]. In particular, if {u{\n)} has a convergent subsequence for every A:, then ^ 1 has infinitely many nontrivial critical points {uk} G ^ \ { 0 } satisfying ^i{uk) ^ 0~ as k ^ 00. P r o o f . Choose n > k > ko dnid let Z ^ := ®]=k^j, pk}' Define c^(A) := sup min ^x{l{'^))i where
B^ := {^ G Z ^ : ||^|| <
r ^ := {7 G C{B^, Wr,) : 7 is odd, 7|a5r^ = id}.
3.1. ABSTRACT
THEOREMS
43
If we now apply Theorem 3.1 to —^x on W^, we get that dk{X) < c^(A) < bk{X) and that, for a.e. A G [1,2], there exists a sequence {u'^{X)}'^^i C Wn such that sup ||t^J^(A)|| < oo and that
^'x\wA<W)
- 0 and $ A « ( A ) ) ^ 4(A)
as m ^ OO. Since Wn is a finite-dimensional subspace and bk{X) < 0, there exists a i^^(A) ^ 0 such that
*Ak„(t^"(A)) = 0,
where c^(A) G [(i/c(A), 6fc(A)] of the theorem.
$A(t^"(A)) = 4(A),
for a.e. A G [1,2]. This completes the proof D
Notes and Comments. Theorems 3.1-3.2 are known as fountain theorems since the critical points spout out like a fountain. The earlier form of the fountain theorem and its dual were established by T. Bartsch in [30] and by T. Bartsch-M. Willem in [48] (see also T. Bartsch-M. Willem [49, 50] and M. Willem [377] ) respectively, where the (PS) condition and its variants play an important role for those abstract results and their applications. They are effective tools in studying the existence of infinitely many large or small energy solutions. Another important result on even functionals is the symmetric mountain pass theorem by taking advantage of the index theory and the (PS) condition (cf. A. Ambrosetti-P. Rabinowitz [19] and M. Struwe [352]). Theorems 3.1-3.2 in this section were obtained by W. Zou in [385]. In proving Theorems 3.1-3.2, a " monotonicity method" is applied which was introduced by M. Struwe in [351, 352] for minimization problems and essentially developed by L. Jeanjean in [191, 193, 196] for one positive solution of mountain pass type. The readers may consult the papers of B. Buffoni-L. JeanjeanC.A. Stuart [74], D. G. de Figueiredo-P. L. Lions [163], L. Jeanjean [194], N. Hirano-S. Li-Z. Q. Wang [185], Z. Liu-S. Li-Z. Q. Wang [236], P. L. Lions [227], M. Schechter [316], M. Struwe [349] and other papers for results without the (PS) condition. The ideas adopted here and in the sequel essentially come from L. Jeanjean in [191] (see also the survey of I. Ekeland-N. Ghoussoub [146]). Further developments were made by A. Szulkin, M. Willem, and W. Zou in [361, 378, 386, 395] for homoclinic orbits of Hamiltonian systems and Schrodinger equations which shall be described in later chapters.
44
CHAPTERS.
3.2
EVEN
FUNCTIONALS
High Energy Solutions
Consider the existence of multiple solutions to the nonlinear Schrodinger equations (S)
-Au^b{x)u
= f{x,u),
xeK^.
We make the following assumptions. (Ci) b G C ( R ^ , R ) satisfies inf^-^i^Ar b{x) > c > 0. Moreover, for every M > 0, meas{{x G R ^ : b{x) < M}) < oo, where meas denotes the Lebesgue measure in R ^ . (C2) / G C ( R ^ X R , R ) , | / ( X , ^ ) | < C ( 1 + 1^1^-1) for a.e. x G O and ah ^ G R,
where 2 < p < 2*, / ( x , u)u > 0 for all i^ > 0. fix UiU
(C3) liminf—-^. \u\^oo
\u\^
> c > 0 uniformly for x G R ^ , where /i > 2 is a
constant. (C4) lim li^O
^— = 0 uniformly for x G R ^ ; U
^— is a nondecreasing funcU
tion of u for every x G R ^ . (C5) / ( x , u) is odd in i^ for any u ^ E and x G R ^ . T h e o r e m 3.3. Assume that (Ci)-(C^) hold, then problem (S) has infinitely many solutions {uk} satisfying - I {\\j Uk\^ ^b{x)u\)dx 2 JRN
— I
JRN
F{x,Uk)dx ^ 00
as A: ^ 00,
where F(x, i^) = J^ / ( x , 5)(i5. Let E :={ue
iy^'^(R^, R) : /
(| V ^P + b{x)u^)dx < 00}.
Then ^ is a Hilbert space with the inner product (i^, v) =
{\/u ' \/v -\- b{x)uv)dx
and the norm ||i^|| = {u,uy^'^. Then obviously, E ^^ I/^(R^). By Propositions 1.13 and 1.16 (Gagliardo-Nirenberg Inequality), we see that ^-^L"(R^),
V5G [2,2*].
Without loss of generality, we assume A^ > 3.
3.2. HIGH ENERGY SOLUTIONS L e m m a 3.4. E ^ ^
45
L^(R^) for 2 < s < 2N/{N - 2).
Proof. Let {un} C ^ be a sequence of E such that Un ^ u weakly in E. Then {||i^^||} is bounded and u^ ^ u strongly in Lf^^(R^) for 2 < 5 < 2N/{N-2). We first show that u^ ^ u strongly in I/^(R^). It suffices to prove that ^n •= II'^nII2 -^ II'^112- Assume, up to a subsequence, that 5^ -^ 5. For any bounded domain O in R ^ ,
/ \Unfdx< [
\unfdx -^ 5'^
hence S > \\u\\2. Let A{R,M)
:= {x G n^\BR
: h{x) > M } ,
B{R,M)
:= {x G n^\BR
: 6(x) < M } .
Then
/
'^uldx<\\^.
-Jn-M^
JA(RM)
-
M
Choose t G (1, N/{N - 2)) such that 1/t + 1/t' = L Then, li^^dx
<
( /
\un\'^^dx)
(mesisB{R,M)
^JB{R,M)
B{R,M)
<
lli^^f ('meas5(i?,M)')
Note that {||i^n||} is bounded and that condition (Ci) holds. We may choose
ll'^nlP R^M large enough such that —^-— and meas 5 ( i ? , M ) are small enough. Hence, for an arbitrary £ > 0, we have that u^dx = / II^\BR
JA{R,M)
u^dx -\- /
u^dx < s. JB{R,M)
Therefore, 11^112 = Il^lli2(5^) + ll^lli2(RA^\5^) > Jim^ Il^n||i2(5^) >6^
-s.
It means that 5 = ||i^||2- Finally, by Gagliardo-Nirenberg Inequality (see This completes Proposition LI6): i^^ ^ ^^ in L^(R^) for 2 < s < 2N/{N-2). the proof of the lemma. D Let ^(i^) = - | | i ^ f - /
F{x,u)dx,
ueE.
46
CHAPTERS.
EVEN
FUNCTIONALS
Then by Lemma 1.20, ^ G C^{E, R ) and J^ is compact, where we take J{u) = /
F{x^u)dx.
We shah prove Theorem 3.3 by finding t h e critical points of
^. We choose an orthonormal basis {cj} of E and define X^ : = R e ^ . Then Zk^Wk can be defined as in t h e previous section. Consider ^x : E ^ H defined by ^ A H := - l l ^ f - A
T h e n J{u)
> 0, /(i^) ^
/
F{x,u)dx:=I{u)-XJ{u),
A G [1,2].
oo as ||i^|| -^ oo, ^ A ( - ' ^ ) = ^ A ( ' ^ ) for ah A G [1, 2] and
1^ G ^ .
L e m m a 3 . 5 . Under the assumptions of Theorem 3.3, for each A: > 2, there exist A^ ^ 1 as n ^ oo, Ck >bk > 0 and { ^ n } ^ i C E such that
P r o o f . Evidently, by conditions (C2), (C3) and (C4), for any £ > 0, there exists a C^ such t h a t f{x^u)u > Cs\u\^ —£\u\'^ for any u. Therefore, it is easy to prove, for some Pk > ^ large enough, t h a t afe(A) : =
max ^A(^) < 0 ueWkMu\\=pk
uniformly for A G [1,2]. On t h e other hand, by (C4), for any £ > 0, there exists a Cg > 0 such t h a t \f{x,u)\ < £\U\-\-CS\U\P~'^ for any x G R ^ , i ^ G R . Let ak : =
sup \\u\\p, ueZk,\\u\\=i
then ak ^ 0 diS k ^ 00. Indeed, suppose t h a t this is not t h e case. Then there is an £0 and {uj} C E with Uj ± Wk--i, \\uj\\ = 1, \\uj\\p > SQ, where kj -^ 00 as j ^ 00. For any v e E, we may find a Wj G Wk^-i such t h a t Wj ^ i; as j ^ 00. Therefore, \{Uj,v)\ = \{Uj,Wj -V)\<
\\Wj
-V\\^0
as j -^ 00, i.e., Uj -^ 0 weakly in E. Hence, Uj ^ 0 in I / ^ ( R ^ ) , a contradiction. Therefore, for u ^ Z^ and £ small enough, we have t h e following
3.2. HIGH ENERGY SOLUTIONS
47
estimates:
$AW >
l\\uf-^Ml-^\\u\\;
> \hf-c\\u\\i >
\hf-cal\\ur.
If we choose rk := (4cpQ^^)^/^^~^\ then for u e Zk with ||i^|| = r/c, we get that
^xiu)>iAcpalf/(^-''\\-^):=h. It follows that bkW '=
inf
>bk^oo
ueZk,\\u\\=rk
as A: ^ oo uniformly for A. Therefore, by Theorem 3.1, for a.e. A G [1,2], there exists a sequence {u^{X)}'^^i such that sup||«^(A)||<^,$',(«n(A))-0 n
and ^A(4(A))^Cfe(A)>6fe(A)>6fe, as n ^ oo. Furthermore, since Cfc(A) < sup ^{u) := Cfc and J^ is compact, by Theorem 1.41, {u^{X)}'^^i has a convergent subsequence. Hence, there exists a z^{X) such that ^^(z^(A)) = 0 and ^x{z^{X)) G [bk^Ck]. Evidently, we may find A^ ^ 1 and {zn} desired by the lemma. D L e m m a 3.6. The sequence {^n}^i obtained in Lemma 3.5 is bounded. Proof. If not, we consider Wn '•= ,, ^,|. Then, up to a subsequence, we get
IknII that Wn ^ w in E; Wn ^ w in I/^(R^) for 2 < t < 2* and Wn{x) -^ w{x) a.e. X G R ^ . We first consider that case that w ^ 0 in E. Since f
fix z )z
^A (^^) = ^' ^^ ^^^^ ^^^^ /
—
JR^
^^ ^(ix < c. Further, by Fatou's Lemma
ll^nll
and conditions (C2) and (C3), f{x,Zn)Zn
f
I
2f{^^^n)Zn,
48
CHAPTERS.
EVEN
FUNCTIONALS
a contradiction. For the second case w = 0 in E, we define ^A^(^n^n) •= max ^x^{tZn)- For any c > 0 and Wn := {Acj^^'^Wn, we have, for n large enough,that
=
2c — Xn
>
c;
F{x,Wn)dx
it follows that lim ^A^(^n^n) = oo. Obviously, t^ G (0,1). Hence, (^';,^(t^Z^),t^Z^) = 0 . Thus, /
{-f{x,tnZn)tnZn
- F{x,tnZn))dx
-^ OO.
^
JUN
By condition (C4), h{t) = -t^f{x^s)s
— F{x,ts)
is increasing in t G [0,1];
hence, -f{x,s)s — F{x,s) is increasing in 5 > 0. Combining these with the oddness of / and noting that ^n
/ JUN
{-f{x,Zn)Zn
- F{x,Zn))dx
= ^x^{Zn)
G [6fe,Cfe],
^
thus we see that
/ ^
\'^J\'^i^nZn)^nZn
-t^
yX^tfiZfi)juX
00.
This provides a contradiction.
D
Proof of Theorem 3.3. This is a straightforward consequence of Lemmas 3.5 and 3.6. D Notes and Comments. An important theory for getting high energy solutions is the well known symmetric mountain pass theorem (SMP, for short) based on the (PS)-condition and index theory, see e.g. A. Ambrosetti-P. Rabinowitz [19]). The readers may find some variants of it in M. Struwe [352]. There is an extensive literature concerning the existence of infinitely many
3.3. SMALL ENERGY SOLUTIONS
49
high energy solutions via SMP and the Fountain Theorem (cf. P. Rabinowitz [293], T. Bartsch [30], T. Bartsch-M. Wihem [48], M. Struwe [352], and also M. Willem [377], etc). In particular, in T. Bartsch-Z. Liu-T. Weth [38], S. Li-Z. Q. Wang [217] and W. Zou [391], sign-changing high energy solutions were obtained. We will address this topic later in this book. Several authors considered the existence of high energy solutions with a perturbation from symmetry. For instance, the special case —Ai^ = \u\^~'^u -\-p{x) in O with 1^ = 0 on dQ was first studied by A. Bahri-H. Berestycki [25] and M. Struwe [348] independently (see also A. Bahri [24] and A. Bahri-P. L. Lions [27]). In P. Rabinowitz [293, 296] and K. Tanaka [364, 365] (and also G. C. Dong-S. Li [139]), the authors considered a general case of perturbed elliptic equations. In [367], H. T. Tehrani considered the case of a sign-changing potential. By using the ideas of P. Bolle [59], C. Christine-N. Ghoussoub [101] also obtained some results on perturbed elliptic equations. Applications of the perturbation theory to Hamiltonian systems are given by A. BahriH. Berestycki [26], P. Rabinowitz [296] and Y. Long [239]. Basically, ah the papers mentioned above only concern the existence of the solutions. In M. Schechter-W. Zou [332], infinitely many high energy sign-changing solutions for perturbed elliptic equations with Hardy potentials were initially obtained. However, whether the symmetry can be cancelled completely is even today not adequately solved (see P. Rabinowitz [293, 296], M. Struwe [352] and M. Schechter-W. Zou [332]).
3.3
Small Energy Solutions
We consider the following elliptic equation with concave and convex nonlinearities: (D)
—Au = f{x,u)-\-g{x,u)
in O,
i^ = 0 on dft,
where O is a bounded smooth domain of R ^ , A^ > L (Di) f,g e C{Q X R, R) are odd in u. (D2) There exist cr,5 e (1, 2), ci > 0, C2 > 0, C3 > 0 such that ci|i^|^ < f{x,u)u
< C2\u\^ -\- csli^l
for a.e. x G O and i^ G R.
(D3) There exists p G [2,2*) such that |^(x, 1^)1 < c(l +|i^|^~^) for a.e. x e ft and 1^ G R. Moreover, lim g{x,u)/u = 0 uniformly for x G O. (D4) Suppose that one of the following conditions holds (1) \im.\u\^00 g{x,u)/u
= 0 uniformly for x G O;
50
CHAPTERS.
EVEN
FUNCTIONALS
(2) lim|^|^oo^(x,i^)/i^ = —oo uniformly for x e Q. Furthermore,
fix u) ^—
and
^— are decreasing in u for u large enough; u (3) \im.\u\^oo g{x,u)/u = oo uniformly for x G O; g{x,u)/u is increasing in u for u large enough. Moreover, there exists a > maxjcr, 5} such that _. . ^ g(x,u)u — 2G(x,u) ^ .r i r ^ limmi ^—• > c > 0 umiormly tor x G O.
We let F and G denote the primitive functions of / and g respectively. Consider the example / ( x , u) + g{x, u) = X\u\'^~'^u + fi\u\P~'u. which satisfies (Di)-(D4). For the case of 0 < A < < fi = 1,1 < q < 2 < p < 2*, this problem was solved by A. Ambrosetti-H. Brezis-G.Cerami [15]. They also raised an open problem about the existence of infinitely many solutions for all A > 0. This open problem was studied in T. Bartsch-M. Willem [48]. Theorem 3.7 is an improvement and generalization of the results in [48]. Another example is f{x,u)
=i^|i^|^~^ln(2 + |i^|),
a G (1,2);
g{x,u) =/ii^ln(l + |i^|).
Then (Di), (D2), (D3) and (D4)-(2) hold if /i < 0; (Di), (D2), (D3) and (D4)-(3) hold with o^ = 2 if /i > 0. If we choose g{x,u) = u'^ for |i^| < 1; g{x,u) = c|^|-i/2ln(l + 1^1) for |^| > 1, then (Di), (D2), (D3) and (D4)-(l) hold. Theorem 3.7. Assume that (DiJ-fD^) hold. Then equation (D) has infinitely many solutions {uk] satisfying ^{uk) '•=-\\uk\\^ — I F{x,Uk)dx — I G{x,Uk)dx ^ 0~ 2 JQ JQ
as k ^ 00,
where \\u\\ = (J^ | y i^p(ix)^/^.
We choose an orthonormal basis {cj} of E := HQ{Q). Set Xj = Re^, W^ ®j=i-^j5 ^n = ®^nXj. Consider a family of C^-functionals: ^^[u) := -\\u\\^ 2
/ G{x,u)dx - X / F{x,u)dx JQ
JQ
:= I{u) - XJ{u),
3.3. SMALL ENERGY SOLUTIONS
51
where A G [1,2]. Then J{u) > 0 and J{u) ^ oo as ||i^|| ^ oo on any finitedimensional subspace. Let n > k > 2. Lemma 3.8. There exist Xn -^ l,u{Xn) G Wn such that ^xJwA^iK))
= 0,
^A.(^(An)) ^ Ck
as n ^ oo, where Ck G [(i^(2), 6^(1)]. Proof. We win apply Theorem 3.2. By (D3), for any £ > 0, there exists a Cs such that \G{x,u)\ < £\u\'^ -\- CS\U\P. Therefore, for ||i^|| small enough, ^ A H > ihW^ - c||i^||^ - c||i^|||. Assume a < 5 smd let ak{cr) :=
sup
||^||^, ak{S) :=
ueZk,\\u\\=i
sup
H^H^.
ueZk,\\u\\=i
Then ak{(j) -^ 0, ak{S) ^ 0 as A: ^ 00. For ll^ll := Pfe := (8c<(<7) + 8ca^(^))i/(2-), we have that ^A('^) ^ PI/^ enough, then we have that ^A(^)
<
-\\uf-c 2
< <
^ 0- On the other hand, if i^ G W^, \\u\\ small
I \u\''dx^e Jn
I \u\^^C, JQ
I \u\Pdx JQ
c\\uf+c\\ur-c\\u\r 0.
The above arguments imply that 5fe(A) < 0 < ak{X) for A e [1,2]. Furthermore, a u E Zk with ||M|| < pk, we see that
^x{u)>-caU
/
(^Xnf{x,u{Xn))u{Xn)^g{x,u{Xn))u{Xn)jdx.
If, up to a subsequence, ||i^(An)|| ^ oo as n ^ oo, then by (D2), ||^(A^)f(l + o ( l ) ) = I
g{x,u{Xn)MXn)dx,
52
CHAPTERS.
EVEN
FUNCTIONALS
where o(l) ^ 0 as n ^ oo. Evidently, it is a contradiction if (D4)-(l) holds. ?t) for for 22 < Otherwise, set Wn = .. ..^x,,, then Wn ^ w in E; Wn ^ w in L^(Q) t < 2* and i(;n(^) -^ w{x) for a.e. x G O. If i(; ^ 0 in ^ , and
lim |ii|^oo
U
—CO in (D4)-(2), then, for n large enough, by Fatou's Lemma, we have that 1 + 0(1) -g{x,u{X„))u{X„) a ^
'""' ' "
>c+[ ^
2J
-g(a:^A)MA„)^^^^,^^
oo,
a contradiction. It is similar if
lim |ii|^oo
^— = oo in (D4)-(3). Therefore, U
w = 0. Define ^x (tnu(Xn)) := max ^x (tu(Xn)), then lim ^x (^n'^(^n)) = OO,
{^Xn{tnU{Xn)),tnU{Xn))
OO
=
<
tG[0,l] "^ = 0. I t folloWS t h a t .
lim^ ( ^x^{tnU{Xn))
-
l i m Xn I ( -f{x,tnU{Xn))tnU{Xn) n^oo /o V 2
n^oo
-{^xA^nU{Xn)),tnU{Xn))
- F{x,tnU{Xn))
) dx
In
+
/
I -g{x,tnU{Xn))tnU{Xn)
- G{x,tnU{Xn))
] dx.
If (D4)-(2) holds, we have that
- / ( x , su)su — F(x, su) -\- -g{x, su)su — G{x, su) < c
for all 5 > 0 and i^ G R. This provides a contradiction. If (D4)-(3) holds, then we have that
oo < c / \u{Xn)\'^dx^
[-g{x,u{Xn))u{Xn)-G{x,u{Xn))]
dx,
3.3. SMALL ENERGY SOLUTIONS
53
which imphes that / i -g{x, u{Xn))u{Xn) — G{x, u{Xn)) ) dx ^ oo. However, by the property of i^(A^), we have that c / ( -g{x, u{Xn))u{Xn) - G{x, u{Xn)) ] dx - c Jn v2^ / -
2
\ 2^*^^' ^(^ri))u{Xn)
+ ic /
< An /
\u{XnTdx
f -f{x,
-]-C
u{Xn))u{Xn)
-g{x,u{Xn))u{Xn)
- G{X, u{Xn)) I
\u{Xn)Vdx
- F{x,
j dx - \ c
u{Xn))
f
\u{Xn)\'dx
j dx
- G{x,u{Xn)) ] dx
D
Notes and Comments. Condition (Ci) was first introduced by T. BartschZ. Q. Wang in [41] which implies the compactness of imbedding of the working space E. Lemma 3.4 is due to [41]. Condition (Ci) contains the coercivity condition: b{x) ^ oo as |x| ^ oo, which was used by P. Rabinowitz in [295] where a positive and a negative soluton of (S) were obtained under the Ambrosetti-Rabinowitz superquadraticity condition (cf. [19]): 37 > 2 such that 0 < jG{x,u)
< g{x,u)u,yu
G R\{0} and a.e. x G O,
where g is the nonlinear term and G is the primitive function of g; ft C R is bounded or unbounded. The coercivity condition was also used by W. OmanaM.Willem in [268] for Hamiltonian systems and by D. G. Costa in [106] for elliptic systems. In [268], with the aid of the superquadraticity condition, infinitely many homoclinic solutions were obtained if the system is odd. In [106], the existence of one solution was studied. In Z. Q. Wang [375], the author considered the effect of concave nonlinearities for the solutions of nonlinear boundary value problems such as Dirichlet (and Neumann) boundary value problems of elliptic equations. Infinitely many small energy solutions were obtained by different methods. His theoretical tools are D. C. Clark's theory for functionals bounded below (cf. D. C. Clark [102] and also H. P.
54
CHAPTERS.
EVEN
FUNCTIONALS
Heinz [179]), the Fountain Theorem of T. Bartsch-M. Willem [48] and M. Willem [377], and the trick of modifying the nonlinear term. In [375], Hamiltonian systems and wave equations were studied also. In N. Hirano [184], the author got infinitely many small energy solutions for sublinear equations by using relative homotopy groups. Multiplicity results for some nonlinear elliptic equations can also be found in A. Ambrosetti-J. Azorero-I. Peral [11] and A. Ambrosetti-J. Garcia Azorero-I. P. Alonso [17]. In particular, in S. Li-Z. Q. Wang [217], sign-changing small energy solutions were obtained. Theorems 3.3-3.7 of the present chapter were obtained by W. Zou in [385].
Chapter 4
Linking and Homoclinic Type Solutions In this chapter, we first prove a weak finking tfieorem wfiicfi, to some extent, unifies the classical linking theorems. Moreover, it produces a bounded Palais-Smale sequence for a non-even functional. Applications will be given on the existence of homoclinic orbits for Hamiltonian systems and solutions to Schrodinger equations.
4.1
A Weak Linking Theorem
Let ^ be a Hilbert space with norm || • || and having an orthogonal decomposition E = N Q M, where A^ C ^ is a closed and separable subspace. Since N is separable, we can define a new norm \v\w satisfying \v\w < ||'^||, ^ v G N such that the topology induced by this norm is equivalent to the weak topology of N on bounded subsets of N. For u = v -\- w G E = N ® M with V e N,w e M.v^e define \u\l^ = \v\l^ + ||'"^|P, then \u\w < \\u\\, y u e E. In particular, if u^ = v^ -\- w^ is | • 1^^ - bounded and u^ -^ u, then v^ ^^ v weakly in N, Wn ^ w strongly in M, Un ^ v -\- w weakly in E. Let Q C N be a II • 11-bounded open convex subset, po ^ Q be a fixed point. Let F be a I • I ^-continuous map from E onto N satisfying (i) F\Q = id; F maps bounded sets to bounded sets; (ii) there exists a fixed finite-dimensional subspace EQ of E such that
F{u -v)-
{F{u) - F{v)) cEo,\/v,ue
E;
(iii) F maps finite-dimensional subspaces of E into finite-dimensional subspaces of E;
56
CHAPTER
4. LINKING
AND HOMOCLINIC
TYPE
SOLUTIONS
Set A := dQ, B := F~^(po), where dQ denotes the || • ||-boundary of Q. There are many examples. E x a m p l e 4 . 1 . Let N = E-,M = E+, then E = E' © ^ + and let Q := {u e E~ : ||i^|| < i?},po = 0 G Q. For any u = u~ Qu^ e E, define F : E ^ N by Fu := u~^ then A := dQ^B := F~^{po) = E^ satisfy the above conditions. E x a m p l e 4 . 2 . Let E = E' ® E+, z^ G E+ with \\ZQ\\ = 1 , ^ + = R Z Q © E^. For any u ^ E, we write u = u~ ® sz^ © w^ with u~ G E~,s G Il,w~^ G (E-^Kzo)^ =Ef. LetN := E'^HZQ. For R > 0, let Q := {u :=u-^szo : 5 G R + , i ^ ~ G ^ ~ , ||i^|| < i?},po = 50^0 ^ Q, So > 0. Let F : E ^ N be defined by Fu := i^~ + ||5Zo+t^^||^o, then F , (3,po satisfy the above conditions with B = F~^{soZo)
= {u : = szo^w^
: s> 0,w^
G E^, \\szo ^w^\\
= so}.
In fact, according to the definition, F\Q = id and F maps bounded sets into bounded sets. On the other hand, for any u,v G E, we write u = u~ -\szo -\- w~^,v = v~ -\- tzo -\- w^, then F{u) = u~ ^ \\szo + i(;+||zo, F{v) = Therefore, v~ + \\tzo -^w^\\zo^ F(u — v) = u~ —v~ ^ \\{s — t)zo -\-w~^ —W^WZQ. F{u-v)-{F{u)-F{v)) = (^\\{s-t)zo^w^ C Hzo := Eo
For ^ eC\E,Il),
(4.1) r :=
-wtW
- \\szo ^ w^W ^ \\tzo ^
(a one-dimensional
wt\\)zo
subspace).
define h : [0,1] X Q ^ E is \ ' |^-continuous. For any (50,1^0) ^ [0,1] x Q, there is a | • |^-neighborhood Ui^s^^^^^ such t h a t {u - h{t,u) : {t,u) G U^so.uo) n ([o/l] X Q)} ^ ^ / m . /i(0,i^) =u,^{h{s,u)) < ^{u),\Ju G Q\.
Then F ^ 0 since id ^T. We shall always use Efin to denote various finitedimensional subspaces of E whose exact dimensions are irrelevant and depend on (50,1^0)- A variant weak linking theorem is T h e o r e m 4 . 3 . Let the family
of C^-functionals
^x{u):=I{u)-XJ{u), Assume
the following
conditions
hold.
( ^ A ) have the VAe[l,2].
form
4.1.
A WEAK
LINKING
(a) J{u) > 0 , V ^ G ^ ; ^ i
THEOREM
57
:=^.
(h) I{u) -^ oo or J{u) -^ oo as \\u\\ -^ oo. (c) ^\ is\-\w -upper semicontinuous; ^ ^ is weakly sequentially on E. Moreover, ^\ maps hounded sets into hounded sets.
continuous
(d) s u p ^ A < i n f ^ A , V A G [1,2]. A
B
Then for almost all A G [1,2], there exists a sequence {un} such that sup 11^^II < oo,
^ A K ) -^ 0,
^ A K ) -^ Cx;
n
where C\ := inf sup ^ A ( ^ ( 1 , ' ^ ) ) ^ [inf ^ A , s u p ^ ] .
P r o o f . We shall prove the theorem step by step. Step 1. We show t h a t C\ G [inf ^ A , sup ^ ] . Evidently, by t h e definition of C\,
C\ < sup ^ A ( ' ^ ) < sup ^i(i^) = sup ^{u) uEQ uEQ uEQ
<
OO.
To show t h a t Cx > inf^ ^ A for all A G [1, 2], we have to prove t h a t /i(l, Q) H 5 ^ 0 for ah h eV. By hypothesis, t h e m a p F / i : [0,1] x Q ^ AT is | • |^continuous. Let K := [0,1] x Q. Then K is | • l^^-compact. In fact, since K is bounded with respect to both norms | • |^ and || • ||, for any (tn^Vn) G K , we may assume t h a t Vn -^ VQ weakly in E and t h a t tn ^ to e [0,1]. Then VQ e Q since Q is convex. Since on t h e bounded set Q C N, t h e | • |^-topology is equivalent to t h e weak topology, then Un -^ VQ. SO, K is I • l^-compact. By t h e definition of F, for any (SQ, i^o) ^ K, there is a | • |^-neighborhood Ui^s^^^^^ such t h a t {u-h{t,u) : {t,u) G U^so.uo)^^} ^ ^fin- Note t h a t , K C y^^s,u)^KU^s,u)' Since K is | • |^-compact, K C U]^^[/(5.^^.), (si^Ui) G K. Consequently, {u — h{t,u) : (t^u) G K} C Efin. Hence, by t h e basic assumptions (i)-(iii) on F , F{u-h(t,u) : (t,u) G K } C Efin dind {u-Fh{t,u) : (t,u) G K } C Efin. Then we can choose a finite-dimensional subspace Efi^ such t h a t po ^ ^ / m and t h a t Fh : [0,1] x (Q n Efin) -^ Efin- We claim t h a t Fh{t, u) ^ po for ah u G d{Q n Efin) = dQ n Efin, t ^ [O51]- To see this, assume t h a t there exist to G [0,1] and i^o ^ dQ HEfin such t h a t Fh{to, UQ) = po, i-e., h{to,uo) G 5 . It follows t h a t ^i(i^o) ^ ^i(/^(^o,'^o)) ^ inf^ ^ 1 > sup^g ^ 1 , which contradicts t h e assumption (d). Thus, our claim above is true. By the homotopy
58
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
invariance of the Brouwer degree, we get that deg(F/i(l,.),Qn^/,^,Po) = deg{Fh{0,'),QnEf,r^,Po) = deg{id,QnEfin,Po) = 1. Therefore, there exists UQ ^ Q r\ Efin such that Fh{l,uo)
= po-
Step 2. Similar to step 1 in the proof of Theorem 3.1, we may consider only those A G [1,2] where C^ exists and use the monotonicity method. Let \n G [1, 2] be a strictly increasing sequence of such points satisfying A^ -^ A. Then there exists n(A) large enough such that (4.2)
- C ; - 1 < ^ ^ - ~ ^ ^ < - C ; + 1 for
n>n(A).
A — An.
Step 3. There exists a sequence h^ ^V^k := k{\) > 0 such that ||/i^(l,i^)|| < k if ^x{hn{l,u)) > Cx — {X — An). This is an analogue of step 1 in the proof of Theorem 3.1. In fact, by the definition of CA^, let /i^ G F be such that (4.3)
sup ^xAhn{l,u)) ueQ
< Cx^ + (A - A^).
Therefore, if ^x{hn{^,u)) > CA — (A — A^) for some u ^ Q, then for n > n(A)(large enough), by (4.2) and (4.3), J{hn{l,u)) < - C ^ + 3, I{hn{l,u)) < Cx - A(:7^ + 3A. By assumption (b), \\hn{l,u)\\ < k := k{X). Step 4. By step 2 and (4.3), sup^x{hn{hu)) ueQ
< sup^xAhnihu)) ueQ
< CA + (2 - C'x){X - Xn).
Step 5. For £ > 0, define (4.4)
Te{X) :={ueE:\\u\\
4, | ^ A ( ^ ) - Cx\ < s}.
Then we claim, for e > 0 small enough, that (4.5)
i n f { | | $ ^ ( u ) | | : u e ^ e ( A ) } = 0.
Otherwise, there exists an SQ > 0 such that ||^^(i^)|| > SQ for all u G J^SQ{X). Let /in ^ F be as in Steps 3-4 and n be large enough such that A — A^ < ^o and (2 - C'^){X - A^) < ^o- Define (4.6)
J^:^{X) :={ueE:
\\u\\ < A: + 4, CA - (A - A^) < ^ A ( ^ ) < ^A + ^ o } .
4.1.
A WEAK
LINKING
Clearly, J^^^{X) C J^soW(4.7)
THEOREM
59
Consider
^*(A) :={ueE:
^x{u)
< CA - (A - A^)}
and J^*o(A) U j^*(A). Since 11^^(^)11 ^ ^o for u G J^^oW, there is a ^{u) G E with 11^(^)11 = 1 such t h a t {^'^{u),^{u)) _
p f ^
> -\\^'^{u)\\.
We let hx{u)
:=
o
for u e Tl^(A).
Then ($';,(«), /IA(U)) > 2 for U e ^ , ; ( A ) . Since ^'^
is weakly sequentially continuous, if {i^n} is || • ||-bounded and u^ -^ u, then Un ^ u in E. Hence {^^^{un),hx{u)) -^ {^^^{u),hx{u)) as n ^ oo. It follows t h a t (^^(•), hx{u)) is | • |^y-continuous on sets bounded in E. Therefore, there is an open | • |^y-neighborhood Afu of u such t h a t (^^('u), hx{u)) > 1 for 'u G Afu^u G ^ * Q ( A ) . On t h e other hand, since ^ A is | • |^-upper semi-continuous, j^*(A) is I • l^-open. Consequently, Afx := Wu - u G J^toW) U J^*(A) is an open cover of ^ * Q ( A ) U ^ * ( A ) . NOW we may find a | • | ^-locally finite and \-\w open refinement {Uj)j^j with a corresponding | • li^- Lipschitz continuous partition of unity {f3j)j^j. For each j , we can either find Uj G ^ * Q ( A ) such t h a t Uj C A/'ii^, or if such u does not exist, then Uj C JF*(A). In the first case we set Wj{u) = hx{uj); in t h e second case, we take Wj{u) = 0. Let [/* = Uj^jUj, then [/* is I • 1^ - open, and J^*Q(A) U J ^ * ( A ) C U*. Define
(4.8)
rA(^):=E/^^(^H(^)-
Then FA • ^ * ^ ^ is a vector field which has t h e following properties: (i) Yx is locally Lipschitz continuous in both || • || and | • 1^^ topology; (ii) (iii)
{^'^{u),Yx{u))>0,yueU*; ($^(u),FA(ti))>l,Vue^;„(A);
(iv) | F A ( U ) U < \\Yxiu)\\ < 2/eo for u e U* and all A e [1,2]. dfi (t u) Consider the following initial value problem — - ^ — = —Yx{r]) with ?^(0, u) = u for ah u G J^*(A) U J?='(A,£o), where J^*{X) is given by (4.7) and J^(A,£o) := {^ G ^ : ll^ll < A:, CA - (A - A,) < ^ A ( ^ ) < ^ A + ^0} (4.9)
C^;(A).
Then by Theorem L36, for each u as above, there exists a unique solution r]{t,u) as long as it does not approach the boundary of [/*. Furthermore, t -^ ^x{v{^^^)) is nonincreasing.
60
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Step 6. We prove that r]{t,u) is | • |^y-continuous for t G [0, 2£o] and u G j^(A,£o) U j^*(A). For fixed to G [0,2£o] and i^o ^ ^(A,£o) U J^*(A), we see that (4.10)
r]{t,u)-r]{t,uo)
=u-uo^
/
[Yx{r]{s,uo))
-Yx{r]{s,u))jds.
Since the set A := ?^([0,2£o] x {'^o}) is compact and | • |^y-compact and Yx is I • l^y-locally | • l^^-Lipschitz, there exist ri > 0,r2 > 0 such that {u G E : infeGA \u-e\^ < r i } C t/* and \Yx{u)-Yx{v)\w < r2\u-v\w for any i^,v G A. Suppose that r]{s,u) G t/* for 0 < 5 < t. Then by (4.10), \r]{t,u) -r]{t,uo)\w <\u-uo\w^ <\u-uo\w^r2
/ \Yx{r]{s,uo))
-Yx{r]{s,u))\^ds
/ |??(5,^o)-^(5,^)U^5. Jo
By the Gronwall inequality in Lemma 1.37, \r]{t,u) -r]{t,uo)\^
< \u - uol^e'^^'^ < \u - uol^e"^^.
If \u — uo\w < S, where 0 < S < rie"^^, then \r]{t,u) — r]{t,uo)\w < ^iTherefore, if \t-to\ < 5, \r]{t,u) -r]{to,uo)\w < \r]{t,u) -r]{t,uo)\w + \v{t,uo) <\r]{t,u)-r]{t,uo)\w^\
/
-r]{to,uo)\w
Yx{r]{s,uo))ds\w
^0
as (5 ^ 0. It means that r]{t,u) is | • 1^^-continuous for t G [0, 2£o] and u G ^(A,£o)U^*(A). Step 7. Define ( K(2t,u),
0 < t < 1/2,
^*(^,^):=< [ r]{Asot - 2^0, /^n(l, u)),
1/2 < t < 1.
We show that rj* G F. Obviously, for i^ G Q, we have either hn{l,u) G JF*(A) or C'A — (A —A^) < ^A(^n(l5'^))- For the second case, we see that ||/i^(l,i^)|| < k
4.1. A WEAK LINKING THEOREM
61
by step 3 and ^x{hn{l,u)) < Cx -\-SQ by step 4. Therefore, hn{l,u) G ^(A,£o)- In view of step 6, T^* is | • |^-continuous satisfying ?^*(0,i^) = u and ^{r]*{t,u)) < ^{u). For any (5o,i^o) ^ [0,1] x Q, since hn G F, we may find a I • I ^y-neighborhood U} ^ N such that (4.11)
{u-K{s,u)
: {s,u) e [/(!,„,„„) n ([0,1] X Q)} C Efi
Moreover, it is easily seen that there exists a | • |^-neighborhood U? N of (so,tto) (5o,i^o) such that (4.12)
{K{s,u)
- K{2s,u)
: {s,u) G U^^^^^^ n ([0,1] x Q)} C Ef,^.
Next, we have to estimate hn{t,u) — r]{4:£ot — 2^0, hn{l,u)) ^x{hn{l,uo))
^x{v{t, Kihuo)))
< ^x{hn{huo))
for t G [1/2,1]. If
Xn)
fortG [0, 2£o]. Particularly, ?^(t,/i^(l,i^o)) G J^*(A) (see (4.7)). If ^A(/^n(l,^o)) > Cx — {X — An), then by step 3, ||/in(l,'^o)|| < k and by step 4, (4.14)
/in(l,^o)G^(A,eo)c^,;(A).
Since ||?^(t,/ln(l,^o))
II / Jo < I Jo
-hn{l,Uo)\\
dr]{s,hn{l,uo) \\Yx{r^{s,K{hu^)))\\ds
^0
we have (4.15)
2t ||7?(t,/in(l,^o))|| < \\hn{l.uo)\\ + - < A: + 4
for t G [0,2£o]. Further, by step 4, ^A(^(^,/^n(l,^o)) < ^x{K{l,uo)) £o- It follows that (4.16)
r]{t, /in(l, ^o)) e ^*^(A) U ^*(A),
< CA +
t G [0, 2£o].
Consider Ai := {?^([0, 2£o],/^n(l,'^o))}, which is | • |^-compact and contained in [/* of step 5 because of (4.13) and (4.16). Moreover, there are rs > 0, r4 > 0 such that A2 :={ue E : | i ^ - A i | ^ < rs} C t/*; \Yx{u)-Yx{v)\^ < r4\u-v\^
62
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
for all u^v e A2 and ^^(^2) C ^/m- Evidently, by the | • |^ continuity of and hn, there exists a | • |^-neighborhood [/? ^^ such that (4.17)
YX^T],
r]{t,hn{l,u))cA2
for t G [0,2£o] and i^ G ^^5^^^^)- For t G [1/2,1], note that hn{t,u) -r]{A£ot-
2£o,hn{l,u))
= hn{t, u) - K{1, u)^
i
Yx{r]{s, /i^(l, u)))ds,
from which we conclude by (4.17) that (4.18) {hn{t, u) - r]{4.sot - 2^0, /^n(l, u)) : (t, ^) G t/f^^^^^^ n ([1/2,1] x Q)} C Ef,nBy the definition of TJ* , 1^ — r]*{t^u) u-hn{t,u)^hn{t,u) u-hn{t,u)
-^hn{t,u)
-hn{2t,u),
t G [0,1/2],
-r]{4sot-2so,hn{l,u)),
t G [1/2,1].
Combining (4.11), (4.12) and (4.18), we get that {u - r]%t,u) : (t^u) G %^^^^^ n ([0,1] x Q)} C ^ / . . , which implies that 7?* G T, where %o,uo) = ^lso,uo) ^ ^(so,uo) ^^ %o,uo) =
Step 8. We will get a contradiction in this step. If ^A(^n(l5 '^)) < C\ — {\ — \ra) for some u ^ Q, then by (4.7), hn{l,u) G JF*(A) and (4.19)
$ A ( ' ? * ( 1 , U))
= $A(r/(2eo, ft„(l,«)) < $A(ft„(l,«))) <
CA
- (A - A„).
If ^x{hn{l,u)) > CA — (A — A„) for some u e Q, then by step 3 and step 4, ||/i„(l,u)|| < k and sup„gQ $A(/in(l,M)) < Cx+£o- Thus, /i„(l,u) G J^e*oWAssume that $x{ri*{l,u)) > CA - (A - A„). Then for 0 < t < 2so, we have, - (A - A„) <$A(r/*(l,w)) <$A(r/(i,/i«(l,u)))
CA
(4.20)
<$A(??(0,/I„(1,W)))
4.1. A WEAK LINKING THEOREM
63
Moreover, for any t G [0, 2^0], by the properties of Yx, \\r]{t,hn{l,u))-hn{l,u)\\
'^ 0
dr]{s,hn{l,u)) ds ^^"
t
< / \\Yx{v{s^K{l^u))\\ds Jo < 2t/so. It follows that \\r]{t,hn{l,u))\\ < 2t/so^ \\hn{l,u)\\ < A: + 4 for t G [0,2so]. This with (4.20) imply that r]{t,hn{l,u)) G J^toi^) for t G [0,2£o]. Since {^'^{u),Y\{u)) > 1 on J^*Q(A), we see that ^A(^(2£o,/^n(l,^)))-^A(/^n(l,^))) ^^^° d -^x{r](t,K(l,u)))dt (^l(7?(t,/l,(l, ^))), rA(^(t,/ln(l, ^))))^t 0
< -2£o.
Therefore, by step 4, ^A(^(2£0,/^n(l,^))) <^A(/^n(l,^))-2£o < ^A -
^0
Combining this with (4.19), we observe that ^ A ( ^ * ( 1 , '^)) < C'A — (A — A^) for any {t^u) G [0,1] x Q, which contradicts the definition of C\. D
Notes and Comments. The existence of the topology \ - \w can be found in D. Dunford-J. T. Schwartz [144]. Similar weak linking was developed in M. Schechter [310, 313, 314]. In [310, 313, 314], conditions "Fl^v = id'' and "F('u — w) = V — Fw for all V ^ N,w ^ E"" were stated but not needed. All that was used was F\Q = id and F{v — w) = v — Fw for all v ^ Q,w ^ E. This was noted in [314]. In particular, since the monotonicity method was not used in [310, 313, 314], the boundedness of the Palais-Smale sequence was not a consequence of the theorems. Therefore, some compactness conditions were introduced and played an important role. In W. Kryszewski and A.
64
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Szulkin [201] (see also M. Willem [377]), some theorems were given which contained linking, and the boundedness of the Palais-Smale sequence also remains unknown. In [201, 377], a r—topology is specially constructed to accommodate the splitting of E into subspaces and by this, a new degree of Leray-Schauder type is established. The new degree and W. Kryszewski-A. Szulkin's linking theorem are also applied in T. Bartsch- Y. Ding [34, 35], Y.Ding-M.Willem [136]. Another infinite dimensional linking theorem was due to V. Benci-P. H. Rabinowitz [55] where more stronger assumptions were imposed, which eliminates many applications such as Schrodinger equations with pure continuous spectra. Theorem 4.3 was due to M. Schechter-W. Zou [328]. Note that this theorem contains most linking theorems we met before.
4.2
Homoclinic Orbits of Hamiltonian Systems
In this section we are concerned with the application of Theorem 4.3 on the existence of homoclinic orbits of the Hamiltonian system z = JH,{z,t),
(H)
where z = (p, ^) G R ^ x R ^ = R^^ and
^
0
-/
» /
0
is the standard symplectic matrix. We assume that the Hamiltonian H is 1-periodic in t, H e ^(R^^ x R , R ) , H^ e ^(R^^ x R , R 2 ^ ) and H^ is asymptotically linear as \z\ -^ oo. Recall that a solution z of (H) is said to be homoclinic to 0 if z ^ 0 and z{t) ^ 0 as \t\ -^ oo. Let H{z^t) = ^Az' z^G{z^t)^ where A is a constant symmetric 2A^ x 2A^matrix, and assume without loss of generality that H{0,t) = 0. Suppose a{J'A) n zR = 0 (cr denotes the spectrum). Let /ii be the smallest positive /i, and /i_i the largest negative fi such that a{J'{A -\- jil)) H zR ^ 0, and set /io '-= min{/ii, —/i_i}. We introduce the following assumptions. (Ai) A is a constant symmetric 2A^ x 2A^-matrix such that G{JA) H ZR = 0. (A2) G is 1-periodic in t, G{z^ t) > 0 for all z, t and Gz{z^ 0 / k l ~^ ^ uniformly in t as z ^ 0. (A3) G{z,t) = \A^{t)z ' z + F{z,t), where Fz{z,t)/\z\ -^ 0 uniformly in t as \z\ -^ 00 and Aoo{t)z - z > fiz - z for some /i > /ii. (A4) ^Gz{z,t)'
z-G{z,t)
>Ofor ah z,t.
4.2. HOMOCLINIC ORBITS OF HAMILTONIAN
SYSTEMS
(A5) There exists S G (0,/io) such that if \Gz{z,t)\
65
> (/io — ^)|^|, then
Let Cz := —Jz — Az and denote the inner product in I/^(R, R^^) by (•, •). Note that a{JA)niIl = 0. Then E := V{\/:\i) (V denotes the domain) is a Hilbert space with inner product (Z,'U)D := {z^v) + (|£|2z, |>^|^'^) ^ind E = i!f2(R^R2^). Moreover, to C there corresponds a bounded selfadjoint operator L : E ^ E such that (I/Z, i;)!) = / {—Jz — Az) ' vdt, R
E = ^ + 0 ^ ~ , where ^ ^ are L-invariant and ( Z + , Z ~ ) D = (z+,z~) = 0 whenever z^ G ^ ^ . Also, {Lz,z)j) is positive definite on E~^ and negative definite on E~. We introduce a new inner product in E by setting {z,v) := {Lz^,v^)j)
- {Lz ,v~ IV'
Then {Lz^z)^^ = | | z + p — ||z |p, where || • || is the norm corresponding to (•,•). It is easy to see from the definitions of /io, /i±i that | z + f >;,l(z+,Z+),
(4.21)
\\z-f>-^_,{z-,Z-)
_ and
\\z\\^ > fio{z,z).
Let {Ej^ : fi G R } be the resolution of identity corresponding to C Then EQ is the orthogonal projector of E onto E~ and Ei^{E) D ^ ~ whenever /i > 0. If /i is as in (A3), then /i > /ii and since /ii is in the spectrum of £, it follows that E^{E) ^ ^ ~ and there exists a ZQ G ^ + , ||zo|| = 1, such that (4.22)
/
{-JZQ
-
AZQ
-
IIZQ)
• zodt = 1 - /i(zo, zo) < 0.
By (A2) this implies that Hz{z,t) = Az + c>(|z|) as z ^ 0, where A is independent oft. In general, one can assume that A = A{t); however, as observed in V. Coti Zelati-I. Ekeland-E. Sere [116], in many cases one can get rid of t-dependence of A by a suitable 1-periodic symplectic change of variables. If this is not possible, then the hypothesis a {J A) H iH = 0 in (Ai) should be replaced by the one that 0 lies in the gap (/i_i,/ii) of the spectrum of C = —J-^ — ^(^5 ^^d i^ (^3) the constant ji should be greater than /ii. Of course, (A5) should be changed accordingly. Note that the spectrum of C is completely continuous. That is Lemma 4.4. Let A{t) G C(R, R^^) he a 1-periodic symmetric matrix-valued function and let C = —J-^ — A{t) : L'^ D H^ ^ L'^ be the corresponding selfadjoint operator. Then the spectrum of C is continuous.
66
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Proof. Since C is selfadjoint, it has no residual spectrum and the isolated points of (J{JC) are eigenvalues. Therefore, it suffices to show that C has no eigenvalues. Assume that (3 is an eigenvalue with an eigenfunction u G H^. Then (4.23)
-. =
j^A{t)^(3)u.
Let W{t) be the fundamental matrix of (4.23) with 1^(0) = / . By the Floquet theory (cf. P. Kuchment [202]), W{1) = P{t)e^^, where T = InVF(l) and P{t) is a 1-periodic continuous differentiable matrix valued function with a bounded inverse P~^{t). Let v{t) = P~^{t)u{t). Then v{t) -^ 0 and \t\ -^ oo and
With respect to the eigenspaces of T corresponding to the positive, negative and 0 eigenvalues, we may split R^^ as R^^ = M^^M©M^. Assume P*, * = +, —, 0, are the projections from R^^ to M*. Then
Note that v'^{t) ^ 0 as \t\ -^ oo, we must have v'^{t) = 0. This implies i^ = 0, a contradiction. D Theorem 4.5. Assume {Ai)-{A^). clinic orbit.
Then system (H) has at least one homo-
It follows from (^2) and (^3) that \Gz{z,t)\ < c\z\ for some c > 0 and all z^t. Therefore (4.24)
^{z) :=]- I {-Jz 2 JR
- Az) - zdt - [
G{z,t)dt
JR
is continuously differentiable in the Sobolev space H'^iYi^Yi?^)^ and critical points z ^ 0 of ^ correspond to homoclinic solutions of {H). Let '0(z) := / G{z^t)dt. Clearly, V^ > 0, and it follows from Fatou's lemma that V^ is JR
weakly sequentially lower semicontinuous. Since \Gz{z^t)\ < c\z\ and Zn ^ z implies z^ ^ ^ in I/^^^^(R, R^^), it is easily seen that V^^ is weakly sequentially continuous. Thus, (c) of Theorem 4.3 is satisfied. Set
4.2. HOMOCLINIC ORBITS OF HAMILTONIAN SYSTEMS
* A W := l\\z+f
- A ( ^ | | z - f + ^{z)),
67
1 < A < 2.
Then ^ i = ^ (cf. (4.24)). Choose a ZQ G E~^ as mentioned above and let B:={zeE^:\\z\\=r}; M :={z = z- ^ pzo : \\z\\ 0},
A = dM,
where R > r > 0 are to be determined. Lemma 4.6. There exist r > 0 and b > 0 such that ^X\B ^ b. Proof. Choose p > 2. By (^2) and (A3), for any s > 0 there exists a C^ > 0 such that G{z,t) < s\z\'^ ^ CS\Z\P. Hence ^{z) = [ G{z,t)dt < e\\z\\l + CM\l
< c{e\\zf +
CMH-
It follows that '0(^) = o(||zp) as z -^ 0 and there are r > 0, 6 > 0 such that ^x{z)>b>OfoT
z e B.
D
Lemma 4.7. There exists an R > r such that ^x\dM ^ 0Proof. Since G{z,t) > 0 according to (^2), we have ^z-)
1, = --\\z-f-
/I
G{z-,t)dt G{z-,t)dt<0.
Note that ^ A ( ^ ) < ^(^) for any z e E, we just have to prove that ^\dM < 0 for R large enough. If this is not true, then there exists Si z^ = p^zo -\- z~, \\zn\\ -^ 00, such that
where 6n = TT^ and v~ = TT^' Therefore Sn > \\v~\\. Since ^n + lbn P = I5 I P n 11
11 ^n 11
5n ^ S > 0 and v~ ^- v~ weakly in E after passing to a subsequence. Set V = SZQ -\- v~. Since (^0,^;") = 0, it follows from (^3) and (4.22) that S^-\\v-f<S'^<0.
f A^{t)vvdt Jn \\v~f - /j.S'^{zo,zo) -
ii{v~,v~)
68
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Thus we may find a bounded interval O such that 5 — H'^" || — / A^{t)v'vdt
<
0. On the other hand, by (4.25), we have that
0 < k -i|k - | P - ./ot IUJI2 ^!^dt 2"" 2"^"" ^^I-^IKf-^//oo(iK--nrf*-/^^"''^-
where v^ = ^n^o + '^n • Since v^ -^ ^^o + '^ = U ' in ^ , we have v^ ^ v in 1.2(0, R 2 ^ ) . By (A2) and (A3) it is easy to check that \F{z,t)\ < c\z\^ for ah z G R^^. Since F ( z , t ) / | z p ^ 0 as \z\ -^ 00, it fohows from Lebesgue's dominated convergence theorem that
lim f ^^dt=
lim [ ^^\v^fdt
= 0,
and therefore that .^ - | K f " / A ^ ( t ) . • vdt > 0, a contradiction. Consequently, there exists an i? > 0 such that ^ A ( ^ ) < ^(^) < 0 for z G dM. D
Combining Lemmas 4.6, 4.7 and Theorem 4.3 we obtain Lemma 4.8. For almost every A G [1,2] ^/lere exists a bounded sequence (zn) C E such that ^'^{zn) -^ 0 and ^x{zn) -^ c\. Let {zn) C ^ be a bounded sequence. Then, up to a subsequence, eiry+R
ther (i) Vanishing: Nonvanishing:
lim sup /
|z-^p(it = 0 for all 0 < i? < oo, or (ii)
there exist o^ > 0, i? > 0 and T/^ G R such that ry^+R
lim /
|z^p(it > Q^ > 0.
Lemma 4.9. For any hounded vanishing sequence {z^) C E, we have lim / G{zn,t)dt=
lim / G^{zn,t) - z^dt = {).
Proof. Recall the concentration-compactness Lemma L17 due to P. L. Lions [228]. Although this lemma is stated for z G H^ ^ by a simple modification, the
4.2. HOMOCLINIC ORBITS OF HAMILTONIAN
SYSTEMS
69
conclusion remains valid in H^. Therefore, if {z^} is vanishing, then z^ ^ 0 in L^ for all 2 < 5 < oo. On the other hand, by assumptions (^42) and (A3), for any s > 0 there exists a C^ > 0 such that (4.26)
\G,{z,t)\<s\z\^C,\zr\
where p > 2. Hence, / G(zn, t)(it < c(£||zn|p + C^Hz^H^) and
// R.
| G , ( z „ , t ) | \zt\dt
< C{S\\Z„\\ \\zt\\
+ Cell^nlirl^lW,
and the conclusion follows.
D
L e m m a 4.10. Let A G [1,2] be fixed. If a bounded sequence {vn} C E satisfies 0 < lim^^oo ^A('^n) ^ cx ^^^ l™n^oo ^A('^^) ~ ^' then there exists a yn ^ Z such that, up to a subsequence, Un{t) := Vn{t -\- y^) satisfies Un^ux^O, ^x{ux)
lim ^x{vn) > 0,
and it follows from Lemma 4.9 that {vn} is nonvanishing, that is, there exist a > 0, R > 0 and yn ^ ^ such that lim /
\vnfdt > a > 0. Hence we
may find yn ^ Z such that, setting Un{t) := Vn{t -\-yn), 2R \Un\^dt
/
> Q^ > 0.
-2R
Since G{z,t) is 1-periodic in t, {un} is still bounded, (4.28)
0 < lim ^x{un) < cx
and
lim ^xM
= 0.
Up to a subsequence, we have that Un -^ ux and Un -^ ux a.e. in R for some Ux G E. Note that u^ -^ ux in I/^^^(R, R^^), it follows from (4.27) that Ux ^ 0. Since IIJ' is weakly sequentially continuous, we have ^^(i^^) ^ ^'-^{ux) and by (4.28), ^^;,(I^A) = 0- Finally, by (A4) and Fatou's lemma, cx
>
>
l i m {^x{Un)
-
-(^l(^n),^n))
X f {lG,{ux,t)-ux-G{ux,t))dt R 2
=$ A K ) . D
70
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Lemma 4.11. There exists a sequence {A^} C [1,2] and {zn} C ^ \ { 0 } such that An ^ 1, ^A^ (zn) < cx^ and ^'^^ (zn) = 0. Proof. This is a straightforward consequence of Lemmas 4.8-4.10.
D
Lemma 4.12. The sequence {z^} obtained in Lemma 4-11 ^s bounded. Proof. Assume \\zn\\ -^ oo and set w^ = ^n/||^n||- Then we can assume, up to a subsequence, that Wn ^ w. We shah show that {wn} is neither vanishing nor nonvanishing, thereby obtaining a contradiction. (a) Nonvanishing of {wn} is impossible. If {wn} is nonvanishing, we proceed as in the proof of Lemma 4.10 to find a > 0, R > 0 and y^ ^ Z such that if Wn{t) := Wn{t -\-i/n), then / \wn{t)fdt > a for almost all n. J-2R Moreover, since ^^ {z^) = ^^ {z^) = 0, where Zn{t) = Zn{t -\- y^), for any ( / ) G C ^ ( R , R 2 ^ ) wehave (4.29)
{w:^,(l)) -Xn{w-,(l))
-Xn /
A^{t)wn'(l)dt
Jn
-An / p^ \Wn\dt = {). Jn \Zn\ Since ||w;^|| = \\wn\\ = 1, Wn ^ w in E, w^ ^ w in Lf^^CR^R?^) and '^n(0 ~^ ^if) ^'^' i^ ^ - I^ particular, w; ^ 0. Since \Fz{z^t)\ < c\z\ for all z,t, by {H^) and Lebesgue's dominated convergence theorem and by passing to the limit in (4.29), it gives (w;^, (j)) — {w~,(j)) — / AoQ{t)w - (j)dt = 0, that is, equation z = J{A-\-Aoo{t))z has a nontrivial solution in E, which contradicts Lemma 4.4. Therefore nonvanishing of {wn} is impossible. (b) Vanishing of {wn} is impossible. Suppose that {wn} is vanishing. Since ^A i^ri) = 0, we have
(^1JZ,),4) = | | 4 f - A, / G.(^n,t) . Z^dt = 0, Jn
{^'^Jzn),z-)
= -Xn\\z-f
- Xn / G,{zn,t)'Z-dt Jn
Since ||i(;nll^ = \\w:^P + ll'^n IP = 1^ we have that
n
G^(z^,t) • (A^^+ -w~)dt=
\\zn\\.
= 0.
4.2. HOMOCLINIC ORBITS OF HAMILTONIAN
SYSTEMS
71
\G (z t)\ Let Sji := {t G R : —^ — < /j^o — S}. By using Holder's inequality, the relation {w~^,w~) = 0 and (4.21), we see that
e.
IknII
< (/io -^)An||^n||2 ^ (/JQ - 5)\n
< 1 for almost all n. Hence (4.30)
hm /
^.(^n,t).(A^+-^-)^^^^^
and since \Gz{z,t)\ < c|z|, it follows that
R\e^
IknII
\Wn\^dt
0. Since {i(;n} is vanishing and i^;^ ^ 0 in I/^(R, R^^), we obtain from (4.30) that meas(R\ B^) ^ oo as n ^ oo. So by (^4) and (A5), {-Gz{Zn,t) R ^ > / > /
' Zn
-G{Zn,t))dt
{]-G,{Zn,t)-Zn-G{Zn,t))dt 5dt
-^ 00.
However, recalling that ^A^(^n) ^ c^^ and (^^ {zn),^^) = 0, we obtain 1 {-Gz{Zn,t) R ^
a contradiction.
cx • Zn -G{Zn,t))dt
< —^ < OO, A^
D
72
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Proof of Theorem 4.5. We have proved that there exist A^ ^ 1 and a bounded sequence {zn} such that ^A^(^n) < CA^ and ^^ (zn) = 0. Therefore
= $ ^ j 2 „ ) + ( A „ - l ) ( z - + t/.'(2„)) = (A„-l)(z-+^'(z„))
Since ($^ (^^n), ^^) = 0, by (4.26), we obtain that (4.31)
||z+f = \nj^G,{Zn,t)
• Z+dt < \\\Zn\? + C||z„r,
(4.32)
\\z-f = -j^G,{zn,t)-z-dt<\\\zX
+ C\\znr,
where p > 2. Hence
\Znf c for some c > 0. If {^n} is vanishing, it fohows from Lemma 4.9 that the middle terms in (4.31)-(4.31) tend to 0; therefore z^ -^ 0. Hence {zn} is nonvanishing. Note that if the sequence {v^} in Lemma 4.10 is nonvanishing, then the hypothesis hm ^x{vn) > 0 may be omitted. Therefore, there exist y^ ^'Zisuch that if Zn{t) := Zn{t -\- yn), then z^ ^ z ^ 0 and D ^\z) = 0. This completes the proof. Notes and Comments. Some authors studied homoclinic orbits for Hamiltonian systems via the critical point theory. The second order systems were considered in A. Ambrosetti-M. L. Bertotti [14], A. Ambrosetti-V. Coti Zelati [16], V. Benci-F. Giannoni [54], P. Caldiroli-L. Jeanjean [76], P. C. CarriaoO. H. Miyagaki [79], Y. Ding [132], Y. Ding-M. Girardi [133], F. Giannoni-L. Jeanjean-K. Tanaka [173], W.Omana-M. Willem [268], E. Paturel [274], P. Rabinowitz [294], P.Rabinowitz-K.Tanaka [297] and V. Coti Zelati-P.Rabinowitz [117]; and those of first order in G. Arioli-A. Szulkin [22], V. Coti ZelatiLEkeland-E. Sere [116], Y. Ding-S. Li [134], Y. Ding- M. Willem [136], H.HoferK. Wysocki[188], E. Sere [334, 335], K. Tanaka [363], C. A. Stuart [353] and A. Abbondandolo-J. Molina [3]. Basically, in all these papers the nonlinear term was assumed to be superlinear. Lemma 4.4 is due to Y. Ding-M. Willem [136]. Theorem 4.5 was obtain by A. Szulkin-W. Zou in [361]. Some Poincare-Melnikov type results for Homoclinics can be seen in A. AmbrosettiM. Badiale [12]. See also the survey in T. Bartsch-A. Szulkin [40].
4.3. ASYMPTOTICALLY
4.3
LINEAR SCHRODINGER
EQUATIONS
73
Asymptotically Linear Schrodinger Equations
Consider the Schrodinger equation (SEi)
- A ^ + V{x)u = / ( x , u),
where x e K^, V e C ( R ^ , R ) and / G C(R^ x R , R ) . Suppose that 0 is not in the spectrum of - A + V in ^ ^ ( R ^ ) (denoted by 0 ^ cr(-A + V)). Let /ii be the smallest positive and /i_i the largest negative /i such that 0 G cr(—A -\- V — fi) and set /io •= min{/ii, —/i_i}. It is well-known that if V is periodic in each of the x-variables, then the spectrum of —A + V (in I/^) is bounded below but not above and consists of disjoint closed intervals (see M. Reed-B. Simon [301, Theorem XIII. 100]). Similarly, we introduce the following hypotheses. (Bi) V is 1-periodic in Xj for j = 1 , . . . , N, and 0 ^ cr(—A + V). (B2) / is 1-periodic in Xj for j = 1,...,A^, F{x^u) / ( x , u)/u -^ 0 uniformly in x as i^ ^ 0. (B3) f{x,u) = Voo(x)u -\- g{x,u), where g{x,u)/u |i^| -^ 00 and 1^00(^) ^ /^ for some /i > /ii.
> 0 for all x^u and -^ 0 uniformly in x as
(B4) ^uf{x, u) — F(x, 1^) > 0 for all x, i^. (B5) There exists a (5 G (0,/io) such that if/(x,i^)/i^ >/io—^, then ^i^/(x,i^) — F{x,u) > S. Theorem 4.13. If the hypotheses (Bi) — (B^) are satisfied, then (SEi) has a solution u ^ 0 such that u{x) ^ 0 as \x\ ^ 0 0 . The functional ^{u) :=l
[
{\Vuf + V{x)u^)dx - [
F{x, u)dx
is of class C^ in the Sobolev space E := H^{Rj^)^ and critical points of ^ correspond to solutions u of (SEi) such that u{x) ^ 0 as |x| ^ 00. If a{—A-\V) n (-00,0) ^ 0, then E = E^ QE-, where E^ are infinite-dimensional, and the proof of Theorem 4.13 follows by repeating the arguments of previous sections. Notes and Comments. If cr(—A+ y ) C (0, 00), then E~ = {0}, /i_i = —00 and ^ has the mountain pass geometry. Theorem 4.13 remains valid in this case, and it is in fact already contained in Theorem 1.2 of L. Jeanjean [191],
74
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
where V = constant. For general asymptotically linear cases with periodic potentials, the first result (Theorem 4.13) was due to A.Szulkin-W. Zou in [361] (and later in G. Li-A. Szulkin [209]). For the asymptotically linear cases of non-periodic potential, see L. Jeanjean-K. Tanaka [195] (where V{x) -^ y(oo) > 0 as \x\ -^ oo), C. A. Stuart-H. S. Zhou [354] (where the problem has radial symmetry) and also F. A. van Heerden [370], F. A. van Heerden-Z. Q. Wang [371]. The superlinear case for (SEi) was studied in S. Alama-Y. Y. Li [6], V. Coti Zelati-P.H. Rabinowitz [118], Y. Ding-S. Luan [135], W. Kryszewski-A. Szulkin [201], C. Troestler-M. Willem [372] and the survey by T. Bartsch-Z. Q. Wang-M. Willem [45]. The variational perturbative methods and bifurcation from the essential spectrum can be seen in A. Ambrosetti-M. Badiale [13]. The existence result of a ground state for nonlinear scalar field equations had been obtained in H. Berestycki-P. L. Lions [58]. The readers will be seeing more notes and comments following the next section.
4.4
Schrodinger Equations with 0 G Spectrum
Consider a nonlinear Schrodinger equation with periodic potential: . ^
. ^^
J —Au-\-V{x)u = g{x,u) \ u{x) ^ 0
for X G R ^ , as \x\ -^ oo.
We assume that 0 is an end point of the purely continuous spectrum of the Schrodinger operator — A + V. We introduce the following conditions. (Co) V G C(R^, R) is 1-periodic in x^,
z = 1 , . . . , AT.
(Ci) 0 G c r ( - A + y ) , and there exists a / 3 > 0 such that (0,/3]ncr(-A+y) = 0 . (C2) g G C ( R ^ x R , R) is 1-periodic in x^, z = 1 , . . . , A^. There exist constants ci, C2 and 2 < /i < 2* such that ci|i^|^ < g{x^u)u < C2\u\^ for all X G R^,i^ G R. (C3) g{x, u)u - 2G{x, u)>0
for ah x G R ^ , u G R\{0}. Q(X
U}U
(C4) There exists a 70 > 2 such that liminf ——^—- > 70 uniformly for u^o G[x,u) XGR^.
/ ^ \ rr^i 1 1 1 ^ g(x,u)u — 2G(x,u) (C5) There exists a c > 0 such that limmi :—: > c uni^
^
\u\^oo
formly for x G R ^ . Here, a > a'' := /i — 1
|l^|«
— — if 2* < 00; a > 2*/i — 2* — /i
4.4. SCHRODINGER
EQUATIONS
WITH 0 G SPECTRUM
75
Assumption (CQ) implies that the Schrodinger operator S := —A + V (on I/^(R)) has purely continuous spectrum, written a{—A-\-V), which is bounded below and consists of closed disjoint intervals (cf. M. Reed-B. Simon [301, Theorem XIII. 100]). Assumption (Ci) means that V cannot be constant. It is easy to check that the classical Ambrosetti-Rabinowitz superquadraticity condition (see A. Ambrosetti-P. Rabinowitz [19]): (4.33)
37 > 2 such that 0 < jG{x, u) < g{x, u)u,
\/u G R\{0}, x G R ^ ,
implies (C3) - (C5). But the converse proposition is not true. Here we give an example. E x a m p l e 4.14. Let g{x,t)
=
(4.34)
fi\t\^-H + (/i - 2)(/i -
s)\t\^-^-Hsm\^-^)
+(/i-2)|tr-2tsin2(^),
where 2 * > / i > 2 , 0 < £ < min{/i - 2, /i - /i*}. Hence G{x,t) = | t r + (/i - 2)1^1^-^ s i n 2 ( f f ) . s
Then (C2) - (C5) hold with 70 = /i, o^ = /i — £. However, for any 7 > 2, let tn = (£(n7r + f7r))^^^ Then g{x,tn)tn
-jG{x,tn)
= (/i-7)|tnr + ( / i - 2 ) ( / i - e - 7 ) | t n r - ^ s i n 2 ( ^ ) + (/i-2)|t,rsin2(^)
-^ —00
as n ^
00,
i.e., condition (4-33) can not he satisfied for any 7 > 2. T h e o r e m 4.15. Assume (CQ), (Ci)-(C^). Then solution u e i?ioc(R-^) n L^{R^) for id<s<2*.
(SE2) has a nontrivial
Consider the following cases: (4.35)
g{x,u)=(3o\u\>'-'^u + f{x,u),
/3o > 0,
^£(2,2*).
76
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
(Di) / G C(R^ X R, R) is 1-periodic in x^, z = 1 , . . . , AT. f{x,u)u
= o(|i^|^)
as |i^| -^ 0,
f{x,u)u
= o{\u\^)
as |i^| ^ oo
uniformly for x G R ^ . (D2) 0 < f{x,u)u
< ^ ^ ^ ^ ~ ^ V r for all xeK^.ue
Theorem 4.16. Assume (Co), (Ci), equation r -/\u^V{x)u \
Let X := H^{Ii^).
(Di) and (D2). Then the Schrodinger
= (3o\u\^-^u^ u{x) ^ 0
has a nontrivial solution u G H'^^^{BJ^)
R\{0}.
f{x,u)
forxeK^, as \x\ ^ 00
n I/^(R^) for fi < s <2*.
Then the functional ^ : X ^ R defined by
^{u) = 11
(I V ^ p + V{x)u^)dx - [
G{x, u)dx
is of class C^{H^{¥i^)^ R) and the critical points of ^ are solutions of {SE2). By (Ci), X splits as X = X~ 0 X+ corresponding to the decomposition of cr(-A + y ) n (-00, 0] and cr(-A + V) n [/3, 00). Consider a new norm || • H^; on X^ defined by
h^\\l-=
I
{\Vu^\^^V{x)\u^\^)dx
for
u^eX^.
Then ^ can be rewritten as
Hu) = hu+\\l-h\u-\\l-
f
G{x,u)dx
for u = u'^ -\- u~ G X+ 0 X ~ . Furthermore, since 0 G cr(—A + V) and hence II • 11^; is not equivalent to || • ||i:/i, we let ^ be the completion of H^(Il^) with respect to || • H^;. Then ^ is a Hilbert space with inner product {U,V)E = (16^12 1^, \S\^V)L2, where S := —A + V and (•, ')L2 is the usual inner product of H := I/^(R^). Corresponding to the decomposition of o-{S) we have the orthogonal decomposition E = E~ 0 E~^. Let (PA • H -^ i^)AGR denote the spectral family of S, and set H~ := PQH, H^ := (id—PQ)^- Then
4.4. SCHRODINGER
EQUATIONS
WITH 0 G SPECTRUM
77
H = H~ 0 H^ and i!f~ C E~. Since by (Ci), the norm || • H^; is equivalent to the H^-norm on E+, E+ = H^CR^) n H+. But it is not true on E'. So H^{Il^) n H~ = i!f~ is not complete with respect to || • H^;. We introduce a new norm on E by setting ||i^||^ := (||'^|||; + l^lfiV - Here and in the sequel, I • It denotes the usual norm in I/^(R^). Let E~ be the completion of H~ with respect to || • ||^. Then E^ := E' © E^ is the completion of H^(R^) with respect to || • ||^. Moreover, Ej^ is a reflexive Banach space such that i l ^ ( R ^ ) C E^ C E and afl norms || • ||^, || • H^;, || • H^i^i are equivalent on E^. By the above argument, it is easy to check that E~ C E~ and {U,V)E = 0 for any u G E~, v G E~^. The following Lemma 4.17 can be found in T. Bartsch-Y. Ding [34, Lemma 2.1 and Corollary 2.3]: Lemma 4.17. E' ^ Hl^CR^); E' ^ ^ ^ioc(I^^) for 2 < t < 2\ Furthermore, E- ^ L^(R^) for fi
Proof. For u G E~, there exists a sequence {un}ne'N C H~ such that \\un — u\\fi ^ 0 as n ^ 00. For any bounded domain O C R ^ , choose a function r] G C ^ ( R ^ ) with 7^ = 1 in O. Then, for v e H' C H'^{'R^), we see that —A{r]v)r]v = rf'{—/\v)v + v'^{—Ar])r] — 2r]vVv • V?^. Hence,
\V{vv)\l<{Sv,rj\)
+
^\V{vv)\l+c{n)\v\l,
where c(0) is a constant depending on O. It follows that + \v\, + \v\l),
\^{vv)\l
and hence {un} is a Cauchy sequence in H^{ft) and u G H^{ft). o-{S) > —a > —00,
Note that
\S{Un -Um)\l 0 2
A d\Px{Un <-a
\d\Px{Un
-Um)\2 -Um)\2
J—a = a\\Un
-UmWlj'
Then Sun -^ Su in L^. Now we show that u G Hf^^CR^). By the CalderonZygmund inequality (cf. D. Gilbarg-N.S. Trudinger [174]), given r > 0,£ >
78
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
0,7/ G R ^ , we have \\Un -
Um\\H^{B{y,r))
This means that u e Hl^CR^). Finahy, we prove that u e L^(R^) for t G [/i, 2*]. This is clear for t = fi. For fixed r > 0, £ > 0,7/ G R ^ , we have
+
HLf-{B{y,r+s)))-
Hence, |i^| dx B{y,r) < Cr,J\Su\f-^ ^
I JB{y,r+e)
\Su\^dx^\u\^^-^
I J B{y,r+e)
\u\^dx). ^
We cover R ^ by ah bahs B{y,r),y G F C R ^ , such that for e > 0 smah, at most A^ + 1 bahs B{y,r -\- s) intersect nontriviahy. Therefore,
/
\ufdx
i.e., u G I/^*. By interpolation, we readily get u e L^ for any t G [/i,2*]. Finally, if i^ G ^ ^ solves the equation of {SE2), we have esssup^^^(^^i)|^(x)| < c||^||^2(5(^,2)), and by the Holder inequality, \\u\\Loo(Biy,l))
<
c\\u\\Lf.^Biy,2))
for all y G R ^ , where c is a constant independent ofy. For an arbitrary £ > 0, u G I/^(R^) implies that lim /
\u\^dx = 0. Therefore, we may take R
large enough such that ||i^||L/^(|a;|>i?) < ^- Then for y G R ^ with \y\ > R-\-2 we have ||i^||L«=(s(y,i)) ^ ce. This implies that u{x) ^ 0 as \x\ ^ 0 0 . D L e m m a 4.18. There exists a c > 0 such that |5^-|^
for any u' e
E'.
4.4. SCHRODINGER
EQUATIONS
WITH 0 G SPECTRUM
79
Proof. By Lemma 4.17, Su' G ^^(R^) for u' e E'. On the other hand, there exists a {vm} C H~ such that H'^m —'^~IU -^ 0- By the proof of Lemma 4.17, Svm -^ Su~ in I/^(R^). Moreover, for a G (—m/cr(6'), oo), we have \SU m 1^
-
\ 2 ^ I D , /., M2 /' X^d\Px{Vm)\l J—a
<
Q^
<
^Ibmlll
2
1^1^^.
It fohows that |6'^-|^
D
We need the fohowing concentration-compactness lemma which is similar to Lemma 1.17. Lemma 4.19. Let r > 0 and {un} C (^^, || • ||^) be bounded. If sup yen^
/
\un\ dx ^ 0
as n ^ oo,
JB{y,r)
then Un ^ ^ in L^{Il^) for ji < t < 2*. In particular, if {un} C E^, Un^^ in L\n^) for2
then
Proof. We consider the case A^ > 3. For 5 G (/i, 2*), by Lemma 4.17 and the Holder and Sobolev inequalities, (^•^^)
where A = —
\^n\is^B(y,r))
^ 1'^^ lL2(5(y,r)) 1'^^ U * (5(y,r))'
• —. We write Un = u:^ -\- u~ with u^ G E^, u~ G E~.
Then (4.37)
Wn\L^*{B{y,r)) <
4K\\Hl^(B(y,r))'
On the other hand, by the proof of Lemma 4.17, for any s > 0 there exists Cr,£ depending on r and s such that (4.38)
\Un\L^*{B{y,r)) <
4^n\\H^B{y,r))
< Cr^e {\Su~\L2^B{y,r+s))
+ Wn\L^^{B{y,r+s))) -
80
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Combining Lemma 4.17, (4.36)-(4.38), we have that Wn\L^*(B{y,r))
Choosing sX = /i, we see that \sX 2* \^n\L2*^B{y,r)) ,+ l|M ^ <^\Wn \\H^^^{B{y,r)) + cCr,s\Su^
^cCr^s\Un
\L^{B{y,r+s))
\L^^{B{y,r+e))
< C\\U^ 11^1 \\U^ \\Hl^{B{y,r)) ^cCr^Su^
+ ^^rA^n
\L^-{B{y,r+e))
\l^2(^^N^^\Su^ lL2(5(y,r+£))-
By Lemma 4.17 and the equivalence of norms || • H^;, || • ||^ and || • WH^^ on ^ + , the boundedness of ||i^nlU implies that sup||i^+||^i < oo,
sup||^~||^ < oo,
sup|^~|^ < oo.
By Lemma 4.18, \Su~\2 ^ <^ll'^n 11^ ^ ^' Therefore, \s\ 2* P^lL2*(5(y,r)) ,+ l|2 ^ 4^n \\Hl^^{B{y,r)) + cCr^Su^
\L^{B{y,r+e))
It follows that.
JBiy.r) 3{y,r) J
|^(^~^^
I7/ 1 ^ ^ ,
^ri\L^(B{y,r))\^n\L^*
-^cCr,s
[ JBiy.r+s)
iB{y,r))
{\Su-\^^\u-ndx]
. /
4.4. SCHRODINGER
EQUATIONS
WITH 0 G SPECTRUM
81
Cover R ^ by balls B{y,r), y G R^ such that for e > 0 small enough, at most A^ + 1 balls B(y,r -\- s) intersect nontrivially. Then \Un\ g(l-A)
< (A^ + l)c sup yen^
7
|2.
/
\un\ dx
\JB{y,r)
j
(|V
{\Su-\'^\u-ndx
as n ^ oo, that is, i^^ ^ 0 in L^(R^). Since /i < 5 < 2*, the conclusion follows by Lemma 4.17 , Sobolev, Holder and interpolation inequalities. Since the norms || • \\H^ and || • ||^ are equivalent in ^ + , the second conclusion is obvious by Lemma 1.17 (cf. P.L. Lions [228]) . D
Since ^ := ^ ^ = ^ - © ^ + , we let F = ^ - , Z = ^ + . Consider the functional 1„
^„o
H{u) = -\\u+ l|2_
1, WE •
/
G{x^u)doL
foru = u + 1^+ G Ej^ = E 0 E~^. Then by assumption (Ci) and Lemma 4.17, H eC^E^.K). Set Hx{u) =
l\\u^\\l-xl^l\\u-\\l^J^^G{x,u)dx 1
:= I{u)-XJ{u),
Then Hi = H and J{u) > 0. Since G{x,u) = J^ f{x,su)uds, Cl
by (C2) it is
Co
easily checked that —\u\^ < G{x,u) < —|i^|^. Hence, I{u) ^ 00 or J{u) -^
\u\l^<£.
L e m m a 4.20. Assume (C2). Then Hx is \ • \uj-upper semicontinuous, H'^ is weakly sequentially continuous. Proof. For any c G R, we consider Un ^ {u e E/^ : Hx{u) > c} with Un —^ u. Write Un = u^ ^ u~ with i^+ G E^^ u~ G E~] by the definition of the | • |^topology, we observe that u^ -^ u^ in ^ ^ and hence sup ||i^^||£; < 00. Note
82
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
that Hx{un) > c and G{x, u^) > 0, so we have sup \\u~ H^; < oo. On the other hand,
< I
G{x,Un)da
4111-olKIII
A V2 < OO.
Thus, sup ||i^nlU < OO. Insert Lemma 4.17 and assume that Un ^^ u in E^^
Un ^ u strongly in I/[^^(R ) and
Un{x) -^ u{x) a.e. X G R ^ . By Fatou's Lemma and the weak semicontinuity of the norm, we see that c < Hx{u), i.e., Hx is | • l^^-upper semicontinuous. Let i^^ ^ i^ in Ej^. Then 1^^ ^ 1^ in I / [ Q ^ ( R ^ ) for 2 < t < 2*. Hence, by (C2), g{x,Un) -^ g{x,u) in Li^(R^)and /
g{x^Un)(j)dx ^
I
g{x^u)(j)dx.
Therefore, H'^{un) • (j) -^ H'^{u) • (j) for every (j) G E^^.
D
Lemma 4.21. Assume (C2). Then there exist b > 0, r > 0 such that Hx{u) > 6 > 0 ,
\/ue
E^with \\u\\^ = r , V A G [1,2].
Proof. This is obvious.
D
Lemma 4.22. Assume (C2)- Then there exist R> r > O^d > 0 such that sup Hx = 0
and
dM
sup Hx < d < 00 M
for all A G [1,2], where M:={u
= y^sz^:yeE-,\\u\\^
0}, ZQ G ^ + , ||zo|U = 1-
Proof. For u = y -\- SZQ, by (C2), Hx{u)<-^\\zo\\l-h\y\\l-^
f
ly + szo^dx.
4.4. SCHRODINGER
EQUATIONS
WITH 0 G SPECTRUM
83
Since E/^ is continuously embedded in I/^(R^), for /i < t < 2*, we see that there exists a continuous projection from the closure of E~ 0 R z o in Z/^(R^) to HZQ. Thus, 15^01^ ^ c\y -\- 5Zo|^ for some c > 0. Hence Hx{u) 0 independent of A such dM
that supil^A < d < oo.
D
M
Combining Lemmas 4.21-4.22 and Theorem 4.3, we get L e m m a 4.23. Assume (Co), (Ci)-(C2). exists {un} C Efj such that sup||i^nlU < ^ '
^xi^n)
For almost every A G [1,2] there
-^ 0 and Hx{un) -^ cx e [b,d].
Lemma 4.24. Suppose (Co), (Ci)-(C^), A G [1,2]. For a hounded sequence {un} C En satisfying lim Hx{un) G [b^d] and lim H'^{un) = 0, ^/lere exists a ux ^ 0 such that H^-^{ux) = 0, Hx{ux) < d. Proof. Write Un = u^ ^u^ with i^+ G ^ + , i^^ ^ F^ . Since sup | then sup ||i^^||£; < oo. If {u^} is vanishing, that is lim sup then by Lemma 4.19, u^ and Holder's inequality.
/
\u'^\'^dx = 0,
0 < i? < oo.
0 in L^(R^) for 2 < t < 2*. Therefore, by (C2)
g{x,Un)u:^dx\ IM-1U/+ i^^|(ix
< C R^
Since
< OO,
84
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
and lim H'y^(un) = 0, we see that ||i^^||£; -^ 0. Consequently,
^AK) 0, R > 0 such that lim /
|i^+p(ix > Q^ > 0.
Hence, we may find a ^^ G Z ^ such that lim /
\v:^\'^dx>a>
0,
where '^^(x) := u^{x -\- y^)• By the periodicity, the set {v^ := Un{x -\- yn)} ^^ still bounded and lim Hx{vn) e [b,d],
lim i / ^ K ) = 0.
Since sup ||i;n||^ < oo,
sup \\vn\\E < oo,
we have sup||i;+||£; < oo, sup||i;~||£; < oo. We may assume that v^ ^- u'j^, v~ ^- u^ in E^. Since E^ is compactly embedded in L\^^(BJ^) for 2 < t < 2*, it means that {)
lim / ^ ^ ^
\v^\^dx=
i
JB{0,2R)
\uX\^ dx < \uX\l, JB{0,2R)
i.e., i^J ^ 0, ux := i^J + i^^ ^ 0. To prove H^^{ux) = 0, it suffices to check that H^~^{ux) • (/) = 0 for any (j) G C ^ ( R ^ ) . In fact, by Lemma 4.17, v^ -^ ux strongly in L[^^(R^) for 2 < t < 2*. Thus, {g{x, Vn)(l) - g{x, ux)(l))dx -^ 0, and then H'^{v,,)-ci^-H'^{ux)'cl^
-A / ^0,
{g{x,Vn)
-g{x,ux))(l)dx
4.4. SCHRODINGER
EQUATIONS
WITH 0 G SPECTRUM
85
i.e., that H^^{ux) = 0. Finally, by (C3) and Fatou's Lemma, we have that Hx{ux)
= Hx{ux) - -H'^{ux) • ^A <
lim {Hx{Vn) -
=
lim Hx{vn)
-H'y^{Vn) • Vn)
< d.
n Lemma 4.25. Assume (Co), (Ci)-(C^). Then there exist {\n\ C [1,2], {zn} C ^^\{0} such that Xn -^ l,H^-^^{zn) = 0 and Hx^{zn) < d. Proof. This is a straightforward consequence of Lemma 4.23 and Lemma 4.24. D Lemma 4.26. Assume (CQ), (Ci)-(C^).
Then {z^} is bounded in E^.
Proof. Write Zji ^ z^^ -\- z^ with z+ G ^ + , z~ e E~. Since H(^^{zn) • ^n = 0, (4.39)
| | 4 l l | - A n | k n l l l = An /
g{x,z^)z^dx
> c\z^l^^.
Hence,
(4.40)
k„|^
\\Z-\\E
\Z-\^
We just have to show that {H^-^H^;} is bounded. By (C4), let SQ > 0 he such that 70 — £0 > 2. Hence, there exists an i?o > 0 such that (4.41)
g{x, u)u > (70 - So)G{x, u)
for all X G R ^ and |i^| < R^. By (C5), there exists an Ri > RQ such that g{x^u)u — 2G{x,u) > c\u\^ for ah X e K^ and |i^| > Ri. Noting (Cs), we may choose a c > 0 small enough such that (4.42)
g{x, u)u - 2G{x, u) > c|^|^
for all X G R ^ and |i^| > RQ. Since Hx^{zn) < d and H'^^ {zn) = 0, we have that
(J--^)(ll4ll|-An||^-|||) 1
70-^0 +An
/
(
g{x, Zn)Zn - G{x, Zn) ) dx
86
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Hence, by (4.41) and {C2)-{Cs), we see that
ll4ll|-An||^-|||
+ / ( G{X, Zn) J\z^\>RoJ \
I G{X, Zn) J\z^\>Ro \
g{x, Zn)Zn I dx 70-^0 J
)/ ^
7o-^o
g{x, Zn)Zn ) dx 70-^0 J
g{x, Zn)Zndx
J\Z^\>RQ
\Zn\^dx. J\z^\>Ro
Moreover, H\^{zn) — -H'^^{zn) - Zn < d and (C3) and (4.42) imply that >
/
l-g{x,Zn)Zn-G{x,Zn)jdx
> c/
\zXdx.
J\z^\>Ro
Without loss of generality, we may assume 2* < 00. In particular, since /i < 2*/i(/i-2) ^^ ^ 2*/i-2*c^ < /i, we may assume a < a. Choose t := G 2 , 2*/i — 2* — /i 2*/i — afi (0,1). By Holder's inequality and Lemma 4.17, we have that /
IZn^dx (l-t)/x
<
/
i^nr
\i|2„|>iJo
^
t^x
/
/
1^.
2* \
\J\z„\>Ro
< c\Zn\f,
^1
\^n\' < Il4ll|-An|k-| <
C^ C
\Zn\^dx
J\z^\>Ro
<
c^cWz^C,
4.4. SCHRODINGER
EQUATIONS
WITH 0 G SPECTRUM
87
that is, \zn\,^ < c + c||z+|||;. Further, noting that H'^{zn) • ^^ = 0, we see by {C2) that \z.n
<
WE
c
<
\z.
IM-1U+ z^\dx
c{c^c\\zi\\\Y-'\\zi\\E + l|t(M-l) + l C\\ZZ\\E^C\\Z^\\E
<
Then sup H^^H^; < 00 since t(/i — 1) + 1 < 2.
D
P r o o f of T h e o r e m 4.15. First, we mention that although Theorem 4.3 was built in a Hilbert, it is still true in a reflexive Banach space (cf. M. Willem-W. Zou [378]). We now prove that \\Z^\\E > O {). We flrst note that |z^|^ < ll^nll^, 141^ < 11411^. By (4.39), \\z-\\l < c | | 4 | | | and \z^\^^ < c | | 4 | | | . By (C2) and Holder's inequality. It//,
Zj^
j Zn^
HJILJ
R^
r^\z+\dx
< C /R^
< c(\\z.
^;:iir' + knir')1411^ 2(M-1)
+ lx
It follows that
4111
, /
g{x,Zn)z:^dx
< ci\\z+\r^ + \\z+\\^^ ). Since /i > 2, we have that 0 < c < ||4ll£^ ^ ^Now we show that { 4 } i^ nonvanishing. If { 4 } ^^ vanishing, we see that g{x,Zn)z^dx
-^ 0. Hence ||4ll£^ ~^ ^^ which contradicts the conclusion
JR^
which we have obtained. By standard arguments, there exist a > 0, R > 0,
88
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
i/n G Z ^ such that hm /
\uj^\'^dx > a > 0,
^ ^ ^ J5(0,2i?)
where cj+(x) := z+(x + y^). Set u;-{x) := z-{x + y^), 6J~(x). Then sup p n | U < oo. Assume that (4.43)
uo^ -^ cj^,
uo~ -^ uj~^
Zn -^ ^'^ + ^~ '= z* in ^ ^ .
Note that Ej^ is imbedded compactly in Lf^^(Il^). Z * ,6J, .,+ cj^ strongly in I/J^Q^(R^). Hence, /
\uj~^\'^dx > a > 0,
Zn{x) := cj+(x) +
Thus we have z^ -^
Zn{x) -^ z^{x) a.e. x G R
and it follows that z* ^ 0. Furthermore, since ||i^||^ < 1, we have \\U\\E < 1 for ah u G ^ ^ . Thus, for any h e E* (the dual space of E), h \E^^ ^^, the dual space of Ej^. Hence, 6J+ ^ 6J+, uj~ -^ uj~ in E. By combining H'^^{zn) = 0 and Lebesgue's Theorem, -i7^(z*).(^ = H'^Sz^)^c^-H'{z^)^c^
Xn{g{x,Zn) -
for any
g{x,z*))(i)dx
cf) G C^(R^), i.e., H^z"") = 0 .
Proof of Theorem 4.16. We merely check that ^(x, i^) = (3\u\^~'^u^ satisfies the assumptions of the theorem with ^^ = JJL = a.
D f{x^u) D
Notes and Comments. The equation —Ai^ + V{x)u = / ( x , u) + Ai^, where V is periodic and A lies in the gap of the spectrum (j{S) of S := —A + V^ was first discussed in H. P. Heinz [180, 181] where a nontrivial solution was obtained by using the linking theorem in V. Benci-P. H. Rabinowitz [55]. In particular, he showed that bifurcation may occur. Heinz's approach was subsequently refined in H. P. Heinz-T. Kiipper-C. A. Stuart [182], H. P. HeinzC. A. Stuart [183] and B. Buffoni-L. Jeanjean [73] (and L. Jeanjean [192]). In B. Buffoni-L. Jeanjean-C.A. Stuart [74], they developed an alternative approach which may eliminate the compactness condition. The first result
4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS
89
assuming that zero is an end point of the spectrum is due to T. BartschY. Ding [34] where the Ambrosetti-Rabinowitz superquadratic condition is essential to their arguments. In B. Buffoni-L. Jeanjean-C. A. Stuart [74] it was assumed that min cr(6') > 0. In A. Alama-Y. Li [5, 6], 0 hes in a gap of the spectrum cr(—A + V) and G{x,u) := J^ g{x,s)ds is strictly convex. Without either the convexity or the compactness condition, C. Troestler-M. Willem [372] got a nontrivial solution by a generalized linking theorem due to H. Hofer-K. Wysocki [188]. W. Kryszewski-A. Szulkin [201] obtained one nontrivial solution by establishing a new degree theory and a new linking theorem. By an approximation technique without the new degree, A. A. Pankov-K. Pfliiger [273] also got a similar result for superlinear cases. In H. Zhou [384], V{x) = constant > 0 and g{x,t) = f{x,t)t with f{x,t) = f{\x\,t). For this case, the working space possesses a compactness of imbedding. In T. Bartsch-A. Pankov-Z. Q. Wang [39], Schrodinger equations with a steep potential well which depends on a parameter were studied (see also F. A. van Heerden [370], F. A. van Heerden-Z. Q. Wang [371]). A. Szulkin-W. Zou [361] were the first to consider the asymptotically linear case (including homoclinic orbits of Hamiltonian systems), when 0 lies in a gap of {—A-\-V). The main results of this section are due to M. Willem-W. Zou [378]. When 0 G cr(—A + V), more problems are still open.
4.5
The Case of Critical Sobolev Exponents
Consider the following Schrodinger equation with critical Sobolev exponent and periodic potential: (SEs)
- A ^ + V{x)u = T{x)\uf-^u
+ g{x,u),
u e ly^'^(R^),
where N > 4; 2* := 2N/{N — 2) is the critical Sobolev exponent and g is of subcritical growth. 0;V,T,g
(Di) y , T G C ( R ^ , R ) , ^ G C(R^ X R,R),A:o := inf^^R^ T(x) > are 1-periodic in Xj for j = 1,..., A^.
(D2) 0 ^ cr(-A + V) and cr(-A -^ V) n (-00, 0) ^ 0, where a denotes the spectrum in I/^(R^). (D3) T(xo) := max T(x) and T(x) — T(xo) = o(\x — XQP) as
X
^
XQ
and
V{xo) < 0. (D4) \g{x,u)\ < co(l + \u\P-^) for ah {x,u) G R ^ x R, where CQ > 0 and p G (2,2*). Further, g{x,u)/\u\'^*~'^ ^ 0 as i^ ^ 0 uniformly for x G R ^ .
90
CHAPTER 4. LINKING AND HOMOCLINIC TYPE SOLUTIONS
(D5) g{x, u)u > 0 for all x G R ^ and u j^ 0. Theorem 4.27. Suppose that (DiJ-fD^) hold. If ^ A A A\
^0
(4.44)
N —2
— >
g(x,u)u
,
rrig
where
rria : =
2
"^
max
—•—-7-—,
a;GR^,iiGR\{0}
\u\^
then equation (SE^) has a solution u j^ 0. In particular, if T{x) = ko > 0, (Ds) can be dropped and the same result holds. Does Theorem 4.27 remain true for A^ = 3? This is still open. It is worth noting that the equation (4.45)
-Au^Xu=\uf-'^u,
A^O,
has only the trivial solution i^ = 0 in VF^'^(R^) (cf. V. Benci-G. Cerami [53]). Under the hypotheses on V, the spectrum of —A -\- V in I/^(R^) is purely continuous and bounded below and is the union of disjoint closed intervals (cf. Theorem XIII. 100 of M. Reed-B. Simon [301]). The following example satisfies the conditions of Theorem 4.27. . . _ J c\u\'^*u g[x,u).-<^ f(^+|^|-2/3^)
\u\ < 1, 1^1 > 1 .
Let E := VF^'^(R^). It is well known that there is a one-to-one correspondence between solutions of (SEs) and critical points of the C^{E, R)-functional (4.46)
^{u)
:=
I [ ~ ^ ^
{\Vuf^V{x)u^)dx /
^{x)\u\'^*dx-
JR^
/
G{x,u)dx.
JR^
Let {^(A)}AGR be the spectral family of - A + V in ^^(R^). Let E' ^ ( 0 ) L 2 n ^ a n d ^ + := {id-E{0))L'^nE.
Then the quadratic form /
:=
(|Vi^p+
Vv?)dx is positive definite on E^ and negative definite on E~. We can introduce a new inner product (•, •) in ^ such that the corresponding norm || • || is equivalent to || • ||i^2, the usual norm of VF^'^(R^). Moreover, /
(|Vi^p +
JR^
Vu^)dx = ||i^+f - \\u-f, where u^ G E^. Then the functional (4.46) can be rewritten as (4.47)
^u)
= l\\u+f-l\\u-f-^f I
l
l
T{x)\ufdx-f JYIN
G{x,u)dx. JYIN
4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS
91
In order to use Theorem 4.3, we introduce a family of functionals defined by (4.48) $A(n) = i | | u + f - A ( i | | u - f + l Z
\Z
/
T{x)\ufdx+f
Z J YIN
G{x,u)dx) J YIN
/
for AG [1,2]. Lemma 4.28. ^x is \ - l^-upper semicontinuous. continuous.
^^ is weakly sequentially
Proof. Noting that Un := u~ -\- u^^ -^ u implies that Un ^ u weakly in E and 1^+ -^ u~^ strongly in E, thus we see that the proof is the same as that in the previous section. D Lemma 4.29. Assume that V G I / ^ ( R ^ ) (it need not be periodic). for each /i G R there exists a constant c = c{jii) such that \\u\\g < c{^i)\\u\\2,
Then
\/ueE{fx)L^
where q = 2N/{N — 4) if N > A ( q may he taken arbitrarily large if N = A and q = oo if N < A). Proof. Since {^(A)}AGR is the spectral family of —A -^ V in I/^(R^), we have for a fixed /i G R, that E{jii)L'^ is the subspace of L^ corresponding to A < /i. Note that ( - A + X^)U(^)L2 : E{fi)L^ ^
E{fi)L^
is bounded. Let F be a positively oriented smooth Jordan curve enclosing the spectrum of (—A + V)\E{fx)L^' According to formula (in.6.19) of T. Kato [198] (and J. Chabrowski-A. Szulkin [88, Proposition 2.2]): - — [ {-A^V 27Ti JY
-X)-\dX,
ueE{fi)L'^.
Since V is compact and — A + y — A is invertible for each A G F, we obtain the conclusion by the Sobolev embedding theorem. D
Let S •=
inf
^ ^
be the best Sobolev constant, see Chapter 1. Let ^s{x) :--
(£2 + | x | 2 ) ( ^ - 2 ) / 2 ^
92
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
where N = {N{N - 2))(^-2)/4,£ > 0 and ^ G C ^ ( R ^ , [0,1]) with ^{x) = 1 if \x\ < r/2; ^{x) = 0 if \x\ > r^r sufficiently small. Write 'ds = '^t ~^ '^7 ^^^^ i^t ^ E+,d- e E-. Then by Proposition 1.15,
L e m m a 4.30. Set (4.49)
h{u) := i||^.+ f - i | | « - f " ^ ^ ^ T{x)\uf
dx,
ueE.
Then supii < c :
iV||T||(^-2'/2
/or £ sufficiently small, where Z^ := E~ 0 R'^JProof. We first show that
(4.50)
C, :=
sup
a
/
(|V^p + Vu^)dx < \ 1/2*
T(x)|i^P (ixj
. Note that
R^
l^^{\\/^j\'^V{^jf)dx<0. Then by Proposition 1.15,
/
\v^:\'dx
as £ ^ 0. This implies that
For u ^ Zs with
||I^||2*,T
= 1, we write u = u~ -\- s'd~ -\- si^^.
Then there exists a c > 0 such that ||^"||2*
^
^||(N-2)/iV'
4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS
93
By convexity, lklli:,T
>
(4.51)
\\s^e\\i:,r
>
+r
[
{S^ef-'u-dx
\\s^sf2:,r-c\mi:zi\\u-h.
By Lemma 4.29,
=0{s^^-'y')\\u-\\2.
In (Ds) we may, without loss of generality, choose XQ = 0. Then V{x) < —c < 0 for 5 G supping. Hence, (4.53)
/
-c£2, -c£'^\ln£\,
Vi^ldx < I
AT > 5, AT = 4,
for some c > 0. Furthermore, by the assumption on T in (-D3), we have (4.54)
||^,"2*
^£||2*,T
= ||T||oo/
^fdx+
>/R«
{T{x)-T{0))^fdx JRJV
= ||T||^5^/2 + o(£2). If Af > 5, by (4.50)-(4.54), we have
<
-
/R^(iv^^i^+^^g)^^
(11 di.Mr-^ii.-iL)^/^-
||T||r^)/^5(^-^)/2+o(e2)^'+'"^^"^^-^"" "'^ - c | | n - f + 0(e(^-2)/^)||u-|b
where c* > 0. Then we get (4.50) . If A^ = 4, the proof is similar. If i^ ^ 0, then N/2
(4.55)
(j^N{\^u\^^Vu^)dx max/(ti^) = -^ t>0
-
'
'
/
\ (A^-2)/2
N(j^,T\urdx)
94
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
as long as the denominator is positive. By (4.50) and (4.55), we obtain the conclusion of this lemma. D Lemma 4.31. Suppose that g{x^u)/u ^ 0 as |i^| ^ 0 uniformly for x G R ^ and that g is of subcritical Sobolev exponential growth. If a bounded sequence {wn} C E andXn G [1,2] satisfy Xn -^ A, ^'^^{wn) -^ 0, ^x^{wn) -^ c(A), where 0 < c(X) < cX :=
nrr^^^vTTT, then iwn} is nonvanishinq.
Proof. If {wn} is not nonvanishing, then i(;^ ^ 0 in I/^(R^) for 2 < r < 2* by Lemma LI7. Then, (4.56)
/
g{x,Wn)vndx ^ 0 and /
G{x,Wn)dx ^ 0
whenever {vn} C E is bounded. Hence, (4.57)
^xA^n)
-
-{^xAWn),Wn)
= ^ [
T{x)\w^fdx^o{l)
^ c(A). For any (5 > 0, we choose fi > ||y||oo(l + ^)/^- Write Wn = w:^ -\- w~ G E^ 0 E~, and let w:^ = w;^ + z^, with Wn G E{fi)L'^,Zn G {id — E{fi))L'^, where ( ^ ( A ) ) A G R is the spectral family of —A -\- V in L^. By Lemma 4.29, Wn G E and (4.58)
\\w~\\q < C | | ^ ~ | | 2 < Cll^^ll
and
\\Wn\\q < cWWnh
< c\\Wn\\,
where q = 2N/(N — 4) if A^ > 4 and q may be chosen arbitrarily large if N = 4. Therefore,
= -{^xA^n),W~)
-Xn /
- Xn /
T{x)\Wn\'^*~'^WnW~dx
g{x,Wn)w~dx
<2\\T\Uwnf;-^w-\U^o{l) where r satisfies (2* - l ) / r + 1/q = 1, 2 < r < 2*. Similarly, (4.59)
ll'^nll -^ O5
and hence,
Wn — Zn ^ 0.
4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS
95
Thus,
WW = [
{\Vz^f^Vzl)dx
= Xn
T{x)\Wnf*~'^WnZndx
^ 0{1)
J UN
= Xn
T{x)\Wn\
Since z^ G {id — E{/j.))L'^, we have / (5>0and/i> sf
dx^0{l).
(|Vz^p + Vz'^)dx > /ipn||2- For any
\\V\\^{1^6)/6, [
\Wzn\^dx>6{fi-\\V\U\\zJl>-
Vzldx.
It fohows that, (4.60)
(1-^)/
\Vzn\^dx<j
{\Vzn\^^Vzl)dx.
By (4.59) and (4.60), we have the fohowing estimates: (A /
T{x)\wnfdxf^^*
<{X\\T\U'/''\\wJl = {X\\T\W^''\\zJl^o{l) <{X\\T\U'/''\\VzJl/S^o{l)
Let n ^ oo and use (4.57), it follows that (iVc(A))2/2* <
i m ^ N c i X ) ,
which implies that either c(A) = 0 or c(A) > (1 — 6)^^'^c\. Either way, we get a contradiction since 5 is chosen arbitrarily. D For i? > 0, set Q := {i^ = 1^ + SZQ : ||i^|| < R,u e E ,s e R ^ } , where zo := '^^/ll'^^ll G E^. Let po = SQZQ e Q.SQ > 0. For any u e E, we write
96
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
u = u~ -\- SZQ -\- w with u~ G E~,w G {E~ 0 R Z Q ) ^ , 5 G R. Consider a map F : E ^ E~ 0 Rzo defined by F{u~ + 5Zo + i(;) = i^~ + \\szo + i(;||^o- Let 5 := F-^(po). Then 5 = {i^ = 5Zo +1(; : i(; G (^~ 0 R^o)^, ||^^|| = ^o}. It is easy to check that F,PQ, B satisfy the basic assumptions of Theorem 4.3. Lemma 4.32. For almost all A G [1,2], there exists a {un] G E such that sup \\un\\ < oo,
^'x{un) ^ 0
and
^\{un) -^ Cx,
n
where C\ G [inf ^A? sup^]. Furthermore, there exists a (^o > 0 small enough ^
Q
such that, for almost all X G [1,1 + SQ], there exists a ux j^ 0 such that ^ A ( ^ A ) = 0, ^ A ( ^ A ) < supg ^ . Proof. By hypotheses {D4) and (-D5), there exist i? > 0, 5o > 0, such that inf^A>0,
sup^A<0,
for ah A G [1,2].
dQ
^
The first conclusion follows immediately from Lemmas 4.28, 4.31 and Theorem 4.3. We give the proof for the second conclusion. Since g{x^u)u > 0 and Q C Z^, we get that 0 < CA < s u p ^ < sup/i < c*,
(4.61)
where / i , c* and Z^ come from Lemma 4.30. Therefore, there exists SL 60 > 0 such that 0 < CA < c^ for almost all A G [1,1 + So], where c^ comes from Lemma 4.31. For those A, by Lemma 4.31, {un} is nonvanishing; that is, there exist i/n G R ^ , hi > O^Ri > 0 such that limsup /
\un\'^dx > hi > 0. 1)
We may find a T/^ G Z
such that
limsup/ n^oo
Kfd.>.>0,
JB{0,2RI)
where Vn{x) := Un{x-\-yn)- Since V, T and g are periodic, {vn} is still bounded and therefore, lim ^ A K ) e [inf ^A,sup^],
lim ^ A K ) = 0.
4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS
97
We may assume that Vn ^ ux. Since E is embedded compactly in I/[^^(R^) for 2 < t < 2*, we have 0 < K. < lim / ^ ^ ^
\vn\'^dx = I
JB{0,2RI)
\ux\'^dx < \ux\ 2JB{0,2RI)
Hence, ux ^ 0. Since ^^ is weakly sequentially continuous, then ^'^{ux) = 0. By Fatou's Lemma,
=
^A(^A)-2(^'A(^A),^A)
= A /
(^-(T(x)I^Ar + ^ ( ^ , ^ A ) ^ A ) - — T ( x ) | ^ A r
= A /
lim (-{T{x)\vn\'^*
<
lim
UxiVn)
< lim
-
-G{x,ux)jdx
^ g{x,Vn)vn) - —-T{x)\vn\'^* -
G{x,Vn))dx
-(^'xiVn),Vn))
^AK)
< sup^. Q
D By Lemma 4.32, we have the following immediate consequence. L e m m a 4.33. There exist X^ G [1,1 + ^o] with A^ ^ 1, and z^ G ^ \ { 0 } such that ^A^ (Zn) = 0,
^A^ (Zn) < SUp ^ .
L e m m a 4.34. T/ie sequence {z^} obtained in Lemma 4-33 is bounded. Proof. Let gi{x,u) := and Gi{x,u) /Tf-1 I O^
lim ^ \ '
:= J^ gi{x,s)ds.
T{x)\u\'^*~'^u^g{x,u) Then by the assumption (-D4), we see that
7/17/
\
= 2*
uniformly for x G R ^ . Let £1 > 0 be such that 2* - £ i >
2. Hence, there exists an i?i > 0 such that (4.62)
gi{x,u)u>
(2* - £i)G'i(x,^),
for x G R ^ , |^| < Ri.
On the other hand, since g{x^ u) is of subcritical growth, we must have that
98
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
gi{x,u)u — 2Gi{x,u)
^^
SOLUTIONS
2 2*^
uniformly for x G R . Furthermore, condition (4.44) implies that 0 < g{x, u)u < 2 * for all x G R^,i^ ^ 0. Hence N -2-k{)\u\^ (4.64)
^i(x,i^)i^-2G'i(x,i^) > 0
for ah x G R ^ , i ^ ^ O .
Therefore, (4.63) and (4.64) imply that there exists a c > 0 small enough, such that (4.65)
gi{x,u)u-2Gi{x,u)
Note that ^\^{zn)
> c\uf
for allx G R ^ , |i^| > Ri.
< s u p ^ and ^^^^{zn) = 0. Then
(^-^73^)(ll4f-An|k-f)
+^^ /
(^
g{x,Zn)Zn-G{x,Zn))dx
J^N V2* -£i <sup^.
(4.66)
J
Q
By (4.62), (4.64) and (4.66), we have that
(^-^7^)(ll4f-An|k-f)
+ /
Ri
(Gi{x,Zn) ^
J\z^\>Ri
(-gi{x,Zn)Zn^^
c^c
)(Gi(x,z^)-— - — ^ -^1
gi{x,Zn)Zndx. '\z^\>Ri
— ^ -Si
^i(x,z^)z^)(ix
gi{x,Zn)Zn)dx ^ gi{x,
Zn)Zn)dx ^
4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS Note by (D^) that \g{x,z)z\ < c\zf that
99
for all {x,z) G R ^ x R. Thus we see
ll4f-An|k-f C^
L
C
L L
c^ c c^ c
gi{x,Zn)z
'f2, UjtJ-^
.\>Ri
[i:{x)\Zn .\>Ri
r^
\Zn\^* dx. .\>Ri
However, (4.64)-(4.65) imply that SUp^
>
= >
>
^Aj^n) -
-{^xA^n),Zr,
/
(^-gi{x,Zn)Zn-Gi{x,Zn)jdx
/ J\z^\>Ri
(-gi{x,Zn)Zn-Gi{x,Zn))dx ^
^^
CI
\Zn?* dx.
'\z^\>Ri
It implies that (4.67)
\\z+f-K\\z-f
Note that (^^ (^^), ^^) = 0. Thus we observe that
ll4f-An||^-f
>c
A^ /
[i:{x)\Znf
/
\Znfdx.
^g{x,Zn)Znjdx
By (4.67), J^j^ \znfdx < c. Note that {^'y^^{zn), z^) = 0. By {D4) and Holder's inequality, we obtain that
Il4f = A„ /
T{x)\Znf'~'^ZnZ:^dx
Kf-'\z+\
+ Xn
g{x,Zn)z'^dx
100
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS D.
Therefore, ||z+|| < c, and hence, ||^~|| < c. L e m m a 4.35. The sequence {zn} is nonvanishing. Proof. Since (zn) is bounded, by (4.61) we may suppose that (4.68)
^Xr^{zn) ^ ci < s u p ^ < c*. Q
If {zn} is not nonvanishing, then it follows from Lemma 1.17 that z^ ^ 0 in L^ as long as 2 < r < 2*. The assumption (1^4) implies that (4.69)
/
g{x, Zn)Zndx ^ 0 ,
/
G{x, Zn)dx -^ 0,
and consequently, (4.70)
^^^{zn) -\{^'xSZn)^Zn)
= ^
f
T{x)\Znf
dx ^ o{l) ^
C^.
Since T(x) > 0, then ci > 0. We distinguish two cases. If ci > 0, then by (4.68) and Lemma (4.31), z^ is nonvanishing. If ci = 0, then (4.70) implies that /
(4.71)
\znfdx^O.
J UN
Since ^^ {z^) = 0, by (-D4), for any £ > 0, we have that v+l|2
JR^
Since ||z-|| < ||z+|| and £ is arbitrary, then c||z+f < c | | z + f + c||z+f*, which implies that ||z+|| > c > 0. But, ^'^^{zn) = 0 and {D4) imply that
Il4f = A„ /
(T{x)\zn\'^''~'^Znz:^
+g{x,Zn)z:^jdx
4.6. SCHRODINGER
SYSTEMS
101
Proof of Theorem 4.27. Since {z^} is nonvanishing, we may find r > 0, Q^ > 0 and i/n G R ^ such that (4.72)
limsup /
z^dx > a.
We may assume that i/n G Z ^ by taking r large if necessary. Now set Zn{x) := Zn{x -\- Vn)' Since ^\ is invariant with respect to the translation of x by elements of Z ^ , i.e., ^\{u{')) = ^A('^(* + y)) whenever y G Z ^ , we see that ll^nll = \\Znl
^Aj^n) =
^Aj^n).
Without loss of generality, we may suppose that z^ -^ ^* for a renamed subsequence. Then (4.72) implies that z* ^ 0 and ^'{z*) = 0. D Notes and Comments. Note that equation (SEs) was studied in J.Chabrowski-A.Szulkin [88], which also generalized the early results obtained in J.Chabrowski-J.Yang [90, 91]. Lemma 4.29 was obtained in J. ChabrowskiA. Szulkin [88]. Lemma 4.30 was due to J. Chabrowski-A. Szulkin [88] based on M. Willem [377]. Equation (SE^) with T(x) = 0, i.e., the nonlinear term is of subcritical growth, has been studied by several authors, for example, S. Alama-Y. Li [5, 6], T. Bartsch-Y. Ding [34], B. Buffoni-L. Jeanjean [74], V. C. Zelati-P. Rabinowitz [117], L. Jeanjean [191, 192], W.Kryszewski-A. Szulkin [201], C. Troestler-M. Willem [372], M. Willem [377]. In Y. Y. Li [224], the author studied a special case —Ai^ = Ti^^ in R^ (see also Y. Y. Li [225] for a higher dimension case on S^). Theorem 4.27 was established by M. Schechter-W. Zou [328]. Without periodicity, the first result on the Dirichlet boundary value with critical Sobolev exponents was due to H. Brezis-L. Nirenberg [71]. See also J. G. Azorero-L P. Alonso [23], H. Berestycki-P. L. Lions [58], D. Cao-S. Peng [78], G. Cerami-S. Solimini-M. Struwe [87], D. G. Costa-E. A. B. Silva [115], A. Ferrero-F. Gazzola [155], J. P. Garcia AzoreroL Peral Alonso [169], N. Ghoussoub-C. Yuan [172], S. Li-W. Zou [221], P. L. Lions [228], D. Ruiz-M. Willem [304] and E. A. B. Silva-M. S. Xavier [342].
4.6
Schrodinger Systems
Consider the Schrodinger system dfU — AxU -\- V{x)u = Wy{t, X, u, v), (SE4)
; -dfV - AxV + V{x)v = Wu{t, X, u, v),
102
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
where {t,x) G R x R ^ , V : R ^ ^ R and l y : R x R ^ x R ^ ^ ^ R are periodic in t and x. We are interested in finding the non-stationary solution z = {u,v) iKxIl^ ^ R 2 ^ of (6'^4) satisfying z{t, x) ^ 0 as \t\ + \x\ -^ oo. Throughout this section, we need the fohowing basic conditions on V and W. (V) V G C(R^, R) and V is T^-periodic in Xj for j = 1 , . . . , AT; 0 ^ cr(-A^ + V), where a denotes the purely continuous spectrum of — A^^ + V. (W) W G C^(R X R ^ X R 2 ^ , R ) is To-periodic in t and T^-periodic in Xj, j = 1 , . . . , AT. W{t, x,z)>0 for ah (t, x, z), z = {u, v) G R ^ ^ .
4.6.1
The Superlinear Case
We need the following assumptions. (El) Wz{t,x,z)
= o{\z\) SiS z ^ 0 uniformly in t and x.
(E2) \Wz{t,x,z)\ < c\z\^ for ah {t,x) and \z\ > Ro, where Ro > O^fi > 0 are constants, 1 < /i < (A^ + 4)/A^. (E3) ^ i y ^ ( t , x , z ) z - i y ( t , x , z ) >c\z\f^ for ah (t,x,z), where
T h e o r e m 4.36. Assume that {Ei)-{Es) hold. Then {SE4) has at least one nontrivial solution. In the next case, the potential satisfies local conditions both at zero and at infinity. (Fi) There exist z/ > p > 2, z/ < {2N + 4)/Ar, ci, C2, C3 > 0 such that c i k r < W.(^,^,^)^ < \W,{t,x,z)\\z\
< C2\z\^ ^ cs\z\P
for ah (t, X, z) G R X R ^ X R ^ ^ . (F2) W^{t,x,z)z-2W{t,x,z)
>0
for ah {t,x,z)
^ (0,0,0).
(F3) There exists a 70 > 2 such that liminf —TTT^—^—^ > 7o \z\^oo
W{t,X,z)
-
^
uniformly for (t, x) G R x R ^ . ^
^ ' ^
4.6. SCHRODINGER
SYSTEMS
103
(F4) There exists an o^ > p such that ^Wz(t,x,z)z — 2W(t,x,z) ^ , . / X -r^ -r^/\r hminf ^ \ , ^ ' ' ^ > c > 0 uniformly for (t,x) G R x R ^ .
Theorem 4.37. Assume that {Fi)-{F4) hold. Then {SE4) has at least one nontrivial solution.
Let
and A := Jo(—A^^ + y ) . Then {SE4) can be rewritten as JdfZ = —Az -\Wz{t^x^z) for z = (u^v). In this way, {SE4) can be regarded as an unbounded infinite dimensional Hamiltonian system in I/^(R^, R ^ ^ ) . Let HQ := L 2 ( R ^ , R 2 ^ ) with the inner product {\J\'^^w, \J\^l^v), Then D{A) = V{JA)
here \J\ =
{-J^f^.
= W^^^ n iyo'^(R^, R^^) and H := L^(R,Ho)
L 2 ( R X R ^ , R 2 ^ ) . Let S = - A ^
=
+ V.
Lemma 4.38. //O ^ cr(-A^ + V), ^/len 0 ^ cr(^) U cr(J'^). Proof. Assume that 0 ^ cr(jr.4). Then there exists z^ = {un^Vn) G ^(^^^4) such that Iknll^ = ll^nll^ + Ibnll^ = 1,
\\JAZr,\\l
= \\Sujl
+ \\Svjl
^
0.
Then, we may assume that ||i^n||2 > c > 0. Let Un := -r.—^. Then ||6'iZn||2 -^ O5 ||'^n||2 = 1- It follows that 0 G cr(6'). This is a contradiction. Similarly, we have 0 ^ cr(^). D Let L = J^t + v4 be the selfadjoint operator on H with domain
V{L) = {ze and norm
iy^'2(R,^o) : z{t) G P(A), / ||^z(t)|||^(it < 00}
104
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
By Lemma 4.38, 0 ^ cr{JA). Then there are r < 0, p > 0 such that (r, p) H crisJA) = 0. Sphtting according to the positive and negative spectrum, we have Let P^ : HQ -^ HQ be the orthogonal projections and {^(A) : A G R } be the spectral family of J A. Then dE(\),
P+ = /
dE{\).
P- = / J — oc
Jp
Set
OO
J{JA) _ I C/(t)=e*^^-^^ = /
^tA, e'^'dEiX). -oo
Then \\U{t)p-U{s)-^\\n
< e-''(*-'*)
for t > s;
\\Uit)P+Uis)-'\\n
< e-''^^-*)
for t < s;
where 0 = min{—r, p}. Lemma 4.39. 0 ^ cr{L). Proof. If it were not true, then there would exist a i^^ G T){L) with ||i^n||2 = 1 and ||I/i^^||2 ^ 0 as n ^ oo. Hence, dtUn = JAun — JLu^ and Un{t)
=
-
I J — oo oo
/
U{t)P-U{s)-^JLUn{s)ds
U(t)P"'U{s)-^JLun{s)ds.
Let (f ^ : R ^ R be the characteristic function of RQ and oo
U{t)P^U{s)-^i'^{t / Then
-
s)JLun{s)ds.
-OO
oo J — OO
If we let C^{t) = e=F^*C=^(t), we have
where * denotes convolution. Note that / ^ C^dt = 1/0, by the convolution inequality. Thus
\\ut\\2<\\Luj2/e^o
4.6. SCHRODINGER
SYSTEMS
105
as n ^ oo. This is impossible since Un = u~ ^u^.
D
By Lemma 4.39, there is an associated orthogonal decomposition H = 7Y~ 0 Ti'^^z = z~ + z+, where z^ G 7Y^, such that L is positive in 7Y^ and negative in 7Y~. Furthermore, by Lemma 4.39, there exists a 7 > 0 such that [—7,7] n CF{L) = 0. Let {G^(A)}AGR be the spectral family of L and U = 1 — 2G{0) be the unitary isomorphism of H and L = U\L\ = \L\U. From A2d(G(A)2,z)i2 + / -00
A2d(G(A)2,z)i2 >
j^zg,
«/7
we see that \\Lz\\l<\\z\\l
+
\\Lz\\l<{l+^-^)\\Lz\\l
Hence, V{L) is a Hilbert space with inner product {U,V)L
=
{LU,LV)L2.
For r > 1, we introduce the anisotropic space B^: Bj,
= =
Bj^i^ri. X R , R j iy^'^(R, L^(R, R 2 ^ ) ) n L^(R, ly^'^ n i^o'"^(R, R ^ ^ ) )
with the norm
([
N
{\u\'^\dtu\'^y\diundxdt
1/
L e m m a 4.40. V{L) ^ L^(R x R ^ , R 2 ^ ) for any r > 2 if N = 1,2 and for r G [2, 2{N + 2)/{N - 2)] if N > 2. In particular, V{L) ^ ^ LlocC^ x R ^ , R 2 ^ ) /or any r > 2 if N = 1,2 and for r G [2, 2(Ar + 2)/(Ar - 2)) if N >2. Proof. Let u,v e C^(R) and z,i(; G If we integrate by parts, we obtain
C^(R,R2^).
{{dtu)Jz,
V • ^i(;) + ((i^ • ^ ^ ,
{dtu)vdt\ ( I
{Jz,Aw)dx\
RxR^
/R
(4.73)
= 0.
Note that J^A
u{dtv)dt\ ( I ^ ^Jn^
{Jw,Az)dx\
=
{^tv)Jw)^dtdx
A^J.
106
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Let Co^(R)(g)Co^(R,R2^) n
= [^u,v,
: n e N,u, G C^{Il),v, G C O ^ ( R , R 2 ^ ) } .
i=l
Then for any z in C^(R) (g)C^(R, R ^ ^ ) , by (4.73), n
RxR^
=
Il^t^ll2+M^ll2.
Since C^(R) (g) C^(R, R ^ ^ ) is dense in V{L) = B2, the above identity holds for all z G 1){L). Hence, we may find ci, C2 > 0 such that
Clll^b, <|k|U
D
Let ^ := P(|I/|^/^) be equipped with the inner product
and norm \\z\\ = {z^z)^!^. We have the decomposition E = E^ 0 E~^ where E^ = E n TC^ are orthogonal with respect to both (•, •)L2 and (•,•). Lemma 4.41. E ^ L^(R x R ^ , R 2 ^ ) for any r > 2 if N = 1 and for r G [2, 2(Ar + 2)/N] if N >2. In particular, E ^ ^ L^oci^ x R ^ , R ^ ^ ) /^^ a n ^ r > 2 i/A^ = 1 and for r G [2, 2 (AT + 2)/AT) i/AT > 2. Proof. Note that
E:=V{\L\'/^)^[V{L),L\i2, where [•^•]s represents complex interpolation. By Lemma 4.40,
E^[U,L\i2^L^ for r = 00 if AT = 2; r = 2{N + 2)/{N - 2) if AT > 3 and ^ = 2{N + 2)/N. For 2 < r < g', we have, using Holder's inequality, that \\z\\r < ||^||2~^||^||Q with
4.6. SCHRODINGER
SYSTEMS
107
s = q{r- 2)/r{q - 2). That is, E ^ U ioi r e [2, 2{N + 2)/N]. By Lemma 4.40 again, E ^ ^ U for r G [2, 2{N + 2)/Ar). The case of AT = 1 is trivial. D
Let H{z) := i ( | | z + f - \\z-f)
- f
W{t,x,z)dtdx.
Then by standard arguments, H G C^{E,IV) and the critical points of H are weak solutions of (6'^4). To study {SE4), we consider
Hx{z):=h\z+f-x(hz-f+
[
W{t,x,z)dtdx
for a h ^ G ^ , A G [1,2]. L e m m a 4.42. For almost all A G [1, 2], there exist 6 > 0, {z^} C E such that sup\\zn\\
H'^{zn)^^,
Hx{zn) ^ Cx ^ [b, d],
as u ^ oo,
n
where d := supg H, Q := {z = z~ -\- SZQ : 5 G R+, z~ G E~, \\z\\ < RQ}. Proof. By (^1) and (^2), for any £ > 0, there exists a C^ > 0 such that W{t,x,z) < s\z\^ ^ Cs\z\^+K Therefore, A^ll^+lli - ^.ll^+li;::i > ^ > 0, VA G [1, 2] Hx{z^) >l\\z^ffor 6 > 0 and z e B := {z : z e E^^ \\z\\ = r^}. The constants h and TQ are independent of A. On the other hand, by (^1) and (^3), W{t,x,z) > c\z\^ for all (t^x^z). Then for a fixed ZQ G E~^ with ||zo|| = 1 and z = z~ -\- SZQ, we have Hx{z) < -s^ - - | | ^ " f - c\\z\f^ < 0 for z G A := dQ, where Q := {z = z~ +5^0 : 5 G R+, z~ G E~^ \\z\\ < R^} and R^ large enough. Moreover, it is easy to check that Hx is | • 1^^-upper semicontinuous and H'^^ is weakly sequentially continuous. By Theorem 4.3, we get this lemma. D L e m m a 4.43. For almost all A G [1,2], there exists an wx such that H'-^{wx) = 0,Hx{wx) 0 and a sequence {i/n} G R^+^ such that lim / ^^^
Izi'l'^dtdx > a > 0, lB{y^,l)
108
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
where B{y, r) denotes the bah centered at y with radius r. Similar to the proof of Lemma 1.17, we have that z+ ^ 0 in L^(Ri+^) for 2 < t < {2N + A)/N. By (^i) and (^2), for any £ > 0, there exists a C^ > 0 such that Wz{t^ X, Zn)z^dtdx <£
: /
\zn\\z^\dtdx^Cs
j R l + Af
[
\znr\z^\dtdx,
j R l + Af
and the left-hand side converges to 0. Therefore, 2H,{Zr,)
< =
||4f H'^{Zr^)zl^\
W,{t,X,Zn)z^tdx^^O,
which is a contradiction and our claim is true. We may assume that there exist an r > 0, independent of n, and a y* := {yo,yi,...,yN)
G TQZ x . . . x TNZ
such that /
\z:^\'^dtdx>a/2,
where By periodicity, {zn} is also bounded and moreover lim Hx{zn) e [b,di lim H'^izn) = 0. Without loss of generality, we may suppose that z+ -^ w^, z~ -^ w^. The compactness of the embedding of ^ + into L[^^(R^+^) for 2 < t < 2{N^2)/N implies that w1^ ^ 0, and it follows that wx := w^ -\- w^ ^ 0. Evidently, i 7 ; ( ^ A ) = 0 . Finally, Hx{wx) = X
lim
= lim Hx{zn)
(-Wz{t,x,Zn)'Zn-W{t,x,Zn))dtdx
4.6. SCHRODINGER
SYSTEMS
109
L e m m a 4.44. There exist Xn ^ l^Wn ^ 0 such that Hx^ (wn) < d, H'^ (wn) = 0. Moreover, {wn} is bounded. Proof. By Lemma 4.43, we have only to prove the boundedness of {wn}Since Hx^{Wn)
- -{Hx^{Wn),Wn)
< d,
by condition (^3) we see that \wn\^dtdx < c. R l + AT
By (^2)-(^3), we can suppose /3 < /i + L Thus, /
\Wn\^''^dtd', 2X ( l - t o ) ( l + /x)
'
\wnfdtdx)
^
A^to(/x+l)
(
\wn\^^^^
dtdx)
< cWWr, |(M+l)to
11'^^iP
<
cs
\wn\\w^\dtdx -\- c /
<
\wn\^\w^\dtdx
ce\\wtf+\\wtr+\
Note that tofi -\-1 < 2. We can therefore conclude that both {w^} and {wn} are bounded. D Proof of Theorem 4.36. Since H(^ {w^) = 0, we see that ||i(;+|p
=
K
<
c /
<
ce\\wtf
Wz{t,x,Wn)w^dtdx {e\Wn\^Ce\Wn\^)\w^\dtdx
+
c\\w+r+\
Therefore, \\w^\\ > c > 0. By the Concentration Compactness Lemma L17, we know that there exist an £0 > 0 and a sequence {yn} C R^+^ such that lim /
\w'^\dtdx > £0 > 0-
no
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
By standard arguments, there exists a z* = z+ + z~ such that z+ ^ 0 and H\z*) = 0 . D
Proof of Theorem 4.37. Under the hypotheses (Fi)-(F4), the conclusions of Lemmas 4.42-4.44 are stih true. It suffices to show that {wn} in Lemma 4.44 is bounded. Note that (4.74)
l l ^ + f - A ^ l l ^ - f = A^ /
W,{t,x,Wn)wndtdx
>
c\\wnt.
By (F3), there exist RQ > 0, SQ > 0 such that ro — £0 > 2 and (4.75)
Wz{t,x,Wn)wn
> (ro - £o)W{t,x,Wn)
for \wn\ > Ro-
By (F2) and (F4), there exists a c > 0 such that (4.76)
W,{t,X,Wr^)Wr^-2W{t,X,Wr^)>c\wX
for
\Wr^\ < RQ.
Note that H\^{wn) < d, H'^ (wn) = 0, we see that
il--^)i\\y^if-^n\\w-f) +A„ /
(
JRI+N
)(Wz{t,X,Wn)Wn-W{t,X,Wn))dtdx ro - £
V
By ( F i ) ,
+[
^+ C/
)(Ty(t,x,z.„)-^-(;'"'"")"")citd.
(iy(t,X,^^)
W^{t,X,Wn)Wn)c
Wz{t,X,Wn)Wndtdx J\w^\
{{Wnl""
^\Wnf)dtdx.
J\w^\
Recah that H\^{wn) — \H'^ (wn)wn < d. Therefore by (4.75), (4.76) and (Fi), we see that (4.77)
cj
\wX<
f
(lw,{t,x,Wn)wn-W{t,x,Wn))
4.6. SCHRODINGER
SYSTEMS
111
and that
>
C / J\w„.\>Rn '\w^\>Ro
>
C
W{t,X,Wn)
•j'\w^\>Ro
\Wr,
J\w
It follows that / \wn\-^ < c. By (Fi) and (F4), it is easily seen that J\w^\>Ro or a>iy> p. either iy>a>p (i) If j9 < Q^ < z/, then / \wn\^ < c and for t small enough, l\w^\
\w^\
<(f
(Kr)""*""°(/
i»..i'"^"'
(4.78)
/
k„r
J\w^\
/
\wnr
J\iWn\
Combining (4.76)-(4.78), we have the following estimates:
< c||«;+ii( / _^ /
Kr+/ f
^J\Wr^\>Ro
Kr) f
J\w^\
\(P-1)/P ^
Since t is arbitrarily small, both {||'W^^||} and {||'W^n||} are bounded.
D
112
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
4.6.2
SOLUTIONS
T h e Asymptotically Linear Case
We assume the following conditions hold. (Gi) W{t,x,z) = ^/3o|^P + n ( t , x , z ) , where Iiz{t,x,z) = o{\z\) as \z\ -^ oo uniformly for all (t,x); /3o > /ii, where /ii is the smallest positive point in the spectrum of — A^^ + V. (G2) There are m G (2, (2Ar + 4)/N),Ro c | ^ r < W,{t,x,z)z
> 0 such that < \W,{t,x,z)\\z\
for ah {t,x) G R X R ^ and \z\ < RQ. (G3) U^{t,x,z)z-2U{t,x,z)
> 0 for ah {t,x,z)
^ (0,0,0).
(G4) There exists a /i > 2 such that liminf —TTTT-^—^—^ = /^, uniformly for (t, x) G R x R ^ . z^o W{t,x,z) ^' J V . y (G5) There exists an o^ G (0, 2) such that ,. . ^Iiz(t,x,z)z — 2Ii(t,x,z) ^ ., n . / N -r^ -r^/v liminf ^^ ' ' \ . ^ ' ' ^ > c > 0, uniformly for (t, x) G R x R ^ . \z\^oo
|Z|^
T h e o r e m 4.45. Assume {Gi)-{G^). solution.
Then {SE4) has at least one nontrivial
In order to prove Theorem 4.45, we first check the conditions of Theorem 4.3. L e m m a 4.46. There exist ro > 0, 6 > 0 such that H\\B > h for all A G [1,2], where B = {z : z e ^ + , \\z\\ = ro}. Proof.
Trivial.
D
L e m m a 4.47. There exist ZQ G E~^ with \\zo\\ = 1 and R > r^ such that HX\A < 0, where A = d{z = z' ^ SZQ : z' G E-,\\z\\ 0}. Proof. Since /3o > /ii, we can find a ZQ G ^ ^ \ { 0 } such that the quadratic form corresponding to — A^^ -\-V — (3o is negative on HZQ 0 E~. Hence,
\\zor-f3o
/
^<0.
4.6. SCHRODINGER
SYSTEMS
113
Choose ZQ = zo/||^o||- Now we have only to prove that H\A < 0 for large R, since W is positive. If this were not true, we would find diWn = SnZo-\-w~ with \\wn\\ -^ oo such that H{wn) > 0. Setting tn = Sn/\\wn\\,u~ = w~/\\wn\\, we have tn > \\u~ \\. Since t^ + \\u~ p = 1, we may assume that t^ ^ txj > 0 and u~ -^ u~ weakly in E. Write u = wz^ + u~. Since {ZO,U~)L2 = 0, we have w'^ — \\u~ W'^ — Po /
u • udtdx
= w'^ — \\u~ W^ — (3^ I <w'-\\u-f<w^{l-
• /3otx7^ / (3o /
{wzo-\-u~){wzo-\-u~)dtdx z^dtdx — Po
{u
Ydtdx
z^dtdx)
<0. Therefore, there exists a bounded set O such that txj^ - ||i^~ f - /3o / u'^dtdx < 0. On the other hand, since H{wn) > 0, we see that n
^
^^2
1|| - | | 2
/
W{t,X,Wn),,,
^
1.2
1|| - | | 2
/ ^ ( ^ ^ ^ ^ ^ n ) ,, ,
1,2
1|| - | | 2
f ^Po\Wn\'^
^Il{t,X,Wn)
By the Lebesgue Dominated Convergence Theorem and (G^i), lim / 5 f e ^ d i r f ^ = o. n^oo
n
IhnP
Hence, txj^ — ||i^ P — /3o / ^ t^^ > 0, and we get a contradiction.
D
Lemma 4.48. There exist Xn G [1,2], Wn G ^ \ { 0 } such that Xn -^ I ^ ^ A ('^^) 0 anti H\^{wn) < d. In particular, the sequence {wn} is bounded. Proof. The proofs of the existence of Wn^Xn are similar to those of the previous section. We now prove that {wn} is bounded. Since H'^ (wn) = 0
114
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
and Hx^{wn) < d, we have that (4.79)
(i_i)(||^+||2_AJ|^-f) +An /
(-Wz{t,X,Wn)Wn-W{t,X,Wn))dtdx
On the other hand, by {G2) — (G^s), we may assume that (4.80)
W^{t,X, z)z > fiW{t,X, z)
(4.81)
n^(t, X, z)z - 2n(t, X, z) > c\z\'^
for \z\ < Ro; for |z| > R^.
Therefore, by (4.79) and (4.80),
< C-\- C
(W{t,X,Wn)
Wz{t,X,Wn)Wn
/ + + // '\w^\
C^C(
) dtdx
))(W(t,X,Wn) [W{t,x,Wn)
Wz(t,X,Wn)Wn)dtdx W^{t,x,Wn)wn)c
Therefore, E
/
(W{t,X,Wn)
/ (J\w^\>Ro ^
< c-\- c
Wz{t,X,Wn)Wn)dtdx
)Wz{t,X,Wn)Wndtdx ^
\wn\'^dtdx. J\w^\>Ro
Since C
i^A^(^n) - - ( i ^ A ^ ( ^ n ) , ^ n ) < ^ , >
(Wz{t,X,Wn)Wn—2W{t,X,Wn))dtdx
) ( ^ z ( ^ 5 ^ 5 ' ^ n ) ' ^ n — 2W{t,X,Wn) \Wr^\
>
c
J\Wr^\>Ro
\wn\^dtdx -\- c /
J\w^\
J\w^\>Ro
Choose i :-
{2-a){N^2) {2{N^2)-aN)'
\wn\^dtdx.
j dtdx
4.6. SCHRODINGER
SYSTEMS
115
Then L G (0,1) and /
\wnfdtdx
J\w^\>Ro
= [
\Wr,\^'-'^^\Wr,\^'dtdx
J\w^\>Ro r
(1-02
< ( /
\wXdtdx)
^J\w^\>Ro
^
r
ILN
(2N + 4)
" ( /
\wn\ ^
^J ^\w^\>Ro
\ (2Ar + 4)
dtdxj ^
\\
Wz{t,x,Wn)w^dtdx JRXR^
[
+/
^J\w^\>Ro
)\W,{t,x,Wn)\\wi\dtdx
J\w^\
\wn\\w:^\dtdx ^ c / J\w^\>Ro
\wn\^~'^\w:^\dtdx.
J\w^\
Thus, /
r
ik+ip < c /
'\Wn\
-\-c( /
<
\ (m-l)/m /
iwr^rdtdx]
(/
^
\wn\'^dtdx]
r
\ I/TI
^J\Wn\
( /
iwtrdtdx]
|i(;^p(it(ix)
c||^+ii(i+c||^jr),
which imply that both {w:^} and {wn} are bounded. Proof of Theorem 4.45. £ > 0, we have that ^nlP ^n / Ikn f = K
^nll = -• //
D
By the hypotheses of Theorem 4.45, for any
Wz{t,X,Wn)w:^dtdx W,{t,X,Wn)w:^dtdx < s\\Wnf ^ c\\WnV,
Wz{t,x,Wn)w^dtdx <£\\wn\\ + c | | ^ ^ | | ^ . Wz{t,x,Wn)w^dtdx
JRI+^
where p > 2 is a constant. Hence, ||t^n|| > c > 0. Similarly, there exists a z* ^ 0 such that H{z*) = 0. D.
116
CHAPTER 4. LINKING AND HOMOCLINIC TYPE
SOLUTIONS
Notes and Comments. By Schauder's fixed point theorem, H. Brezis-L. Nirenberg [72] considered the following system dfU - Aj^u = -v^ + / ,
-dtv - A^v = u^ ^ g,
for (t,x) G ( 0 , r ) X 0, for {t,x) e ( 0 , r ) X 0,
satisfying u = v = 0 on {0,T) x dfl and i^(0, x) = v{T, x) = 0 in O. Here O is a bounded domain of R ^ and / , ^ G 1/^(0) . They obtained a solution {u, v) with u e l/^((0,r) X n) and v G L^{{0,T) x Q). In P. Clement-P. Felmer-E. Mitidieir [103], the authors studied the following problem dtu - A^u = \v\'^-'^v, - ^ t i ; - A^i; = \u\P-'^u,
for (t, x) G ( - T , T) x O, for (t,x) G ( - T , r ) x O,
where O is a smooth bounded domain in R ^ and N/{N-\-2) < l/p-\-l/q < 1. By the usual Mountain Pass Theorem, they obtained at least one positive solution. The system {SE4) was explored in T. Bartsch-Y. Ding [35] by a new linking theorem due to W. Kryszewski-A. Szulkin[201], where the AmbrosettiRabinowitz superquadratic condition was imposed and the asymptotically linear case was not discussed. Lemmas 4.38-4.41 were established in T. BartschY. Ding [35]. In particular, a regularity result was found and infinitely many solutions were obtained if there is a symmetry of W in (u,v). Theorems of this section are due to M. Schechter-W. Zou [325].
Chapter 5
Double Linking Theorems If A links B, does B link A7 Can they yield two different critical points without the (PS) type compactness conditions? To find out, read on...
5.1
A D o u b l e Linking
Let ^ be a refiexive Banach space with norm || • ||. Define a class of contractions of E as follows: (5.1)
T
Ti;-)eCi[0,l]xE,E); r(0, •) = id, for each t G [0,1), r(t, •) is a homeomorphism of E onto itself and T~^{t, •) is continuous on [0,1) x E; there exists a XQ G ^ such that r ( l , x) = XQ for each x e E and that r(t, x) ^ XQ as t ^ 1 uniformly on bounded subsets of E.
Definition 5.1. A subset A of E is linked to a subset B of E if An B and, for every F G T, there is at ^ [0,1] such that r(t, A) f^ B ^ ^. Proposition 5.2. Let M G C{E, R^) and Q ^ E be such that MQ = M\Q is a homeomorphism of Q onto the closure of a bounded open subset ft of R ^ . Let pen, then M~^{dQ) links M-^{p). Proof. Assume that M^^{dQ) does not link M~^{p). Then there is a F G T such that (5.2)
T(t,M^^(dn)\nM-^(p) = $, te[o,i].
118
CHAPTER 5. DOUBLE LINKING THEOREMS
That is, (5.3)
M(r{t,M^\dn))^
n {p} = 0,
te [o, i].
Set 0{t) :=MoT{t,')oM^\
(5.4)
Then we see that 0{t) G C ( 0 , R ^ ) for each t G [0, l]_and (9(0) = id on O. If r ( l , ^ ) = xo, then 0{l)x = MXQ ^ p for dl\ x e Q since by (5.2)-(5.4), M(T{l,M^^{dn)))
n {p} = 0. By the Brouwer degree,
deg(^(t),0,p) = deg(^(l),0,p) = l,V t G[0,l]. This is a contradiction.
D
P r o p o s i t i o n 5.3. Let A, B he two closed, hounded suhsets of E such that E\A is path connected. If A links B, then they link each other. We call them, douhle linking sets. Proof. Assume that B does not link A. Then we may find a F G T such that (5.5)
r ( t , 5 ) n A = 0 , v t G [0,1].
By the definition of T, we assume that r ( l , ^ ) = XQ, hence XQ ^ A. Let O be a closed ball such that A C O. Note that E\A is path connected, then there is a path 7 connecting XQ ^ A to a point xi ^Vt. Let to G [0,1) be such that the diameter of r(to, B) —XQ is less than min{dist(7, A), dist(xi, O)}. Parametrize 7 in such a way that it is given by j{t),to < t < 1,7(^0) = xo,7(l) = xi. Then (5.6)
[T{to,B) + 7(t) - xoj n A = 0,
to < t < 1
and (5.7)
( r ( t o , 5 ) + x i - x o ) n O = 0.
Define r i ( t , x ) = T{t,x) for t G [0,to]; and = r ( t o , x ) - I ^ Q + 7 ( t ) for t G [to,l]. By (5.5), r i ( t , 5 ) n A = 0for ah t G [0,1]. Hence, (5.8)
Bf^V^^{t,A)
= 0 , V t G [0,1].
By (5.7), (5.9)
r i ( i , 5 ) n o = 0.
5.1. A DOUBLE LINKING
119
Let T2 be any map in T such that T2{t,Q) C Q for ah t. Take Ts{t,-) = ri(2t,-)~^ for ah t G [0,1/2] and Ts{tr) = r i ( l , - ) ~ ^ o r2(2t - 1,-) for ah (1/2,1]. It is easy to check that Ts G T. But (5.8)-(5.9) imply that BnTs{t,A)
= 9,
t G [0,1].
This contradicts the fact that A links B.
D
P r o p o s i t i o n 5.4. Let E = M ® N^ where M^N are closed subspaces with one of them finite dimensional. If yo G M\{0} and 0 < p < R, then the sets A B
:= {u = v^syo := Mn dBp
:v e N,s>
0,\\u\\ = R} U
[NHBR],
link each other in the sense of Definition 5.1, where Br := {u e E : \\u\\ < r}. Proof. We first consider the case of dim N < 00 and identify N with some R ^ . We may assume that \\yo\\ = 1- Let Q = {syo ^v:veN,s>0,\\syo^v\\<
R}.
Then A = dQ in R ^ + ^ Let u = v ^ w with v e N,w e M. We define Fu = V -\- ||t^||^o- Then F\Q = id and B = F~^{pyo). We can apply Proposition 5.2 to conclude that A links B. Since A and B are bounded and E\A is path connected, B links A as well. D P r o p o s i t i o n 5.5. Let E = M Q N, where M,N are closed subspaces with dim AT < 00. Let BR = {u e E : \\u\\ < R} and let A = OBR nN,B = M. Then A links B. Proof. We identify N with some R ^ and take Q = BRH N,Q = Q. For u = V -\- w^v e N^w e M, define the projection Fu = v. Since F\Q = id and M = F~^(0), we observe by Proposition 5.2 that A links B. D P r o p o s i t i o n 5.6. Let B be an open set in E and A = {a, b} such that a e B^b ^ B. Then A links dB. Moreover, dB links A if B is bounded. Proof. Let F G T. If F ( l , ^ ) = ^0, then F(t,_a) (F(t,6)) is a curve in E connecting a ( 6, respectively) with i^o- If '^0 ^ B, then F(t,a) intersects dB. If uo G B, then F(t, 6) intersects dB. Hence A links dB. Obviously, E\A is path connected, therefore, dB links A if 5 is bounded. Notes and Comments. Definition 5.1 was introduced by M. SchechterK. Tintarev [323] and Propositions 5.2-5.6 were proved there (see also M. Schechter [309, 310, 312, 317, 319]). More examples can be found in M.
120
CHAPTER 5. DOUBLE LINKING
THEOREMS
Schechter [310] and M. Schechter-W. Zou [327]. Note that this kind of new linking includes almost all classical linkings as special cases. It is more general and realistic. Actually, many works of the authors are based on this concept. We refer the readers to the papers of M. Schechter and W. Zou in the Bibliography.
5.2
Twin Critical Points
Suppose that ^ G C^(^,R) is of the form: ^{u) := I{u) - J{u),u G E, where I^Je C^{E, R) map bounded sets to bounded sets. Define
^x{u)=XI{u)-J{u),
AG A,
where A is an open subinterval of (0,+oo). Assume one of the following alternatives holds. (Ai) I{u) > 0 for all 1^ G ^ and either I{u) ^ oo or \J{u)\ ^ oo as ||i^|| -^ oo. (A2) I{u) < 0 for all 1^ G ^ and either I{u) -^ —00 or \J{u)\ ^
00 as
ll^ll ^ 0 0 . Further, we need (A3) ao(A) := sup^A < ^o(A) := inf ^A, A
VA G A.
^
Theorem 5.7. Suppose that (Ai) {or {A2)) and (As) hold. (1) If A links B and A is bounded, then for almost all X ^ A there exists an Uk{X) G E such that sup ||i^fc(A)|| < 00, ^^(i^fc(A)) -^ 0 and that k
^\{uk{\))
-^ a{\) as k ^ 00, where a{X) := inf sup ^xC^is^u)). ^^^ se[o,i],ueA
In particular, if a{\) = 60(A); then d\st{uk{X)^ B) ^ 0 ,
k ^ 00.
(2) / / B links A and B is bounded, then for almost all X e A there exists a Vk{X) G E such that sup ||'U/c(A)|| < 00, ^^i^kW) -^ 0 and that k
^A('^fe(A)) -^ b{X) as k ^ 00, where 6(A) := sup inf ^xC^is^v)). TGT se[o,i],veB In particular, if ao{X) = b{X), then dist{vk{X),A) ^ 0 ,
k ^ 00.
5.2. TWIN CRITICAL POINTS
121
(3) If A^B are bounded subsets which link each other and, for any X e A, every bounded (PS)-sequence of ^\ possesses a convergent subsequence, then for almost all A G A, ^A has two different critical points u\ and v\ satisfying
$AK)=a(A),
$';,K)=0;
$;,(«;,)= 5(A),
$ 1 ^ ) = 0.
In particular, if a{X) = b{X), then v\ ^ A^u\ G B. Proof. (1) We first prove conclusion (1) with (Ai) holding. Obviously, a (A) > 60(A) since A links B. By (Ai), the map A -^ a{X) is non-decreasing. Hence, a\X) := —-—— exists for almost every A G A. We consider those A where a^(A) dX exists. For a fixed A G A, let A^ G (A, 2A) H A, A^ ^ A as n ^ 00, then there exists n(A) such that (5.10)
a\X) - 1 < ^(^^) - ^(^) < a\X) + 1, A„ — A
for n > n(A).
Step 1. We show that there exist r „ G T,w := w{\) > 0 such that ||r„(s,u)|| < w whenever ^x{rn{s,u)) > a{X) — (A„ — A). Indeed, by the definition of a(A„), there exists a r „ e T such that (5.11)
sup $A(r„(s,M))< sup $A„(r„(s,M))
If <^x{T„{s,u)) > a{X) - (A„ - A) for some u e A,s e [0,1], then by (5.10) and (5.11), we have that (5.12)
/(r„(s,u))
and that (5.13)
J{Tn{s,u))
However, by (Ai), (5.10) and (5.11), we have that (5.14)
J(r„(s,u)) >-$A„(r„(s,u)) > - ( a ( A ) + (A„-A)(a'(A) + 2)) > - o ( A ) - A | o ' ( A ) + 2|.
122
CHAPTER 5. DOUBLE LINKING
THEOREMS
Combining (5.12)-(5.15) with (Ai), we see that there exists a w{X) := w such that ||r^(5,1^)11 < w. Step 2. By the choice of F^ and (5.10), we observe for all {s^u) G [0,1] x A that (5.15)
^x{Tn{s,u)) <
sup
^xA^n{s,u))
sG[0,l],iiGA
< a{\n) + (A^ - A) 6o(A). For e G (O, (a(A) — 6o(A))/3), we define Q,(A) := {^ G ^ : ll^ll < 117 + 1, | ^ A ( ^ ) - a{\)\ < e}.
(5.16)
Let n be so large that (a^(A)+2)(A^-A) < e, A^-A < s. Then, ^x{Vn{s,u)) < a{X) -\- £ for all {s,u) G [0,1] x A. If there exists a {so,uo) G [0,1] x A such that ^A(rn(5o, ^o)) > a(A) - (A^ - A) > a(A) - s, then by Step 1, ||F^(5o,i^o)|| < tx7(A). It follows that F^(5o,i^o) ^ QsW- By the definition of a(A), we see that the case of ^A(rn(5, u)) < a{X) — (A^ — A) for all {s,u) G [0,1] x A cannot occur. Therefore, Q£{X) ^ 0. Next, we show that (5.17)
ini{\\^'xiu)\\:ueQei\)}=0
for £ G (0,
) sufficiently small. If not, then there exists an SQ G
, , ,^ (a(A)-60(A)), (0, ) such that ||*'A(W)||
>3£o
forallueQeo(A).
Take n so large that (a'(A) + 2)(A„ - A) < £0, A„ - A < SQ. Define (5.18)
Q:„(A)
•.= {ueE:
\\u\\ < t u + l , a ( A ) - ( A „ - A ) < <S>x{u) < a(A)+eo}.
Then Q:O(A) C Qe„(A).
Similarly, Q*Q(A) ^ 0. By Lemma 2.1, we may construct a locally Lipschitz continuous map Hx of E such that
5.2. TWIN CRITICAL POINTS (a)
\\nx{u)\\
(b) {'^'^{u),Hx{u)) >2eo,yu (c)
123
e
QliX);
{^\{u),nx{u))>OyueE.
Consider the Cauchy initial boundary value problem:
dt
-nxm,u)),
^{0,u)=u.
By Theorem 1.36, there exists a unique continuous solution S,{t,u) such that ^\{^{t,u)) is nonincreasing in t. Define f C(2s,w)
0 < s < 1/2,
tis,u):=\ [ C(l,^„(2s-l,^t))
l/2<s
Then it is easy to check that F G T. We shall prove that (5.19)
$A(f(s,u))
W{s,u) e [0,1] x A,
which yields the desired contradiction. Choose any u e A. If 0 < s < 1/2, by (A3), we get that (5.20)
$A(f(s,^t))
< < < <
^x{u) 60(A) a(A) — 3£o a(A)-(A„-A).
If 1/2 < s < 1, then f(s,u) = ^ ( l , r „ ( 2 s - l,u)). «(A) - (A„ - A) for s e [1/2,1], then *A(f(s,^t)) (5.21)
If $A(r„(2s - l,u))
<
$A(C(0,r„(2s-l,w)))
<
a(A)-(A„-A).
<
If there exists an SQ G [1/2,1] such that $A(r„(2so - 1, u)) > a{\) - (A„ - A), then by Step 1, ||r„(2so — 1, w) || < w. Recalling step 2 and (5.18), we see that r „ ( 2 s o - l , w ) e 2*0(A). Since | | C ( t , r „ ( 2 s o - l , ^ * ) ) - r „ ( 2 s o - l , ^ * ) | | = || / dC{a, r„(2so-i,w))||
124
CHAPTER 5. DOUBLE LINKING
THEOREMS
we have (5.22)
U{t,rn{2so-l,u))\\
< <
\\r„{2so-l,u)\\+t w+1,
for a l H e [0,1].
Therefore, if $A(^(io,r„(2so-l,u))) < a ( A ) - ( A „ - A ) we must have (5.23)
for some to G [0,1],
$A(f(so,w)) = $A(e(l,r„(2so - l,u)) < a{X) - (A„ - A).
If it is not the case, by step 2, we must have (5.24)
a(A) - (A„ - A) <$A(C(l,r„(2so-l,w))) <$A(^„(2so-l,^t)) < a(A) + £o
for all t e [0,1]. Thus, (5.22)-(5.24) impUes that e(t,r„(2so - l,u)) e QliX) Note that
($^(U),7^A(U))
for all t e [0,1].
> 2eo on Q*,(A). Thus we have
$A(C(i,r„(2so - l,u))) - $A(r„(2so 0 < J
l,u))
da -($A(?('^,r„(2so - l,u))),WA(?(a,r„(2so - l,u))))d(7
< -2teoIt follows that, $A(f(so,u))
< <
$A(r„(2so-l,M))-2£o a(A)-(A„-A).
Combining this with (5.20), (5.21) (5.23) and (5.25), we get $A(f (s,u)) < a{\) - (A„ - A),V(s,u) e [0,1] x A, which contradicts the definition of a (A). This implies that (5.17) holds in the case of a(A) > ho{\). By (5.17) we get the conclusion (1) of this theorem. Step 4' We show that the conclusion (1) of Theorem 5.7 holds true in case of a(A) = 6o(A). For £ > 0 , r > 0, we define (5.25)
Q(£,r,A):= {ueE : \\u\\ <w{X)^4^dA,\^x{u)
- a ( A ) | < 3s,d{u,B)
<4T},
5.2. TWIN CRITICAL POINTS
125
where dA '-= max{||i^|| : u e A}, which is finite since A is bounded. We claim that Q(£, T, A) ^ 0. We choose n so large that (5.26)
sup
^xi^nis^u))
<
sG[0,l],iiGA
sup
^xA^n{s,u))
sG[0,l],iiGA
<
a(A) + 3s.
Since A links 5 , there exists {so,uo) G [0,1] x A such that r^(5o,i^o) ^ ^• Hence, dist(r^(5o,i^o)5 ^ ) = 0 and (5.27)
^A(rn(5o,^o))
> bo{X) inf ^A B
a(A) a(A) - (An --A) a(A) — 3s. By step 1, ||rn(5o,i^o)|| < ^ ; hence, rn(5o,i^o) ^ Q(£,T, A). /S^ep 5. For s, T small enough, we show that (5.28)
inf{||$^(u)||:ueQ(£,T,A)}=0.
If this were not true, there would exist (5 > 0, £i > 0, Ti G (0,1) such that (5.29)
II^A(^)II
> 3(^ for u G Q(£i,Ti, A).
Define (5.30)
{
1^ G ^ :
||i^|| < tx7 + 4 + (iA,
a(A) - (A, - A) < dist(^,5) < 4 r i .
^A(^)
< a{\) + S^i,
Similarly, by (5.26), Q*(£i,ri,A)^0,
Q*(ei,Ti,A)cQ(£i,Ti,A).
Let n be so large that (A^—A) < £i, (a^(A)+2)(An—A) < Si and (A^—A) < 6Ti. We may construct a locally Lipschitz continuous map Hx of ^ such that (d)
\\nx{u)\\
(e) ($^(u),7iA(w)) > 2<5,Vu e Q(ei,Ti,A);
(f)
{^'M,nx{u))>0,'iueE.
126
CHAPTER 5. DOUBLE LINKING
THEOREMS
Define
(5.31)
\\u\\ < tx7 + 2 + (iA, I ^ A H - a{X)\ < 2si,
( ueE: Qi := <^ [
dist{u,B)
<3Ti.
It is easily seen that Qi ^ 0 and Qi C Q(£i,Ti,A). Choose a Lipschitz continuous map a from E into [0,1] which equals 1 on Qi and vanishes outside Q(£i,Ti, A). Consider the initial boundary value problem
Let ^i{t,u) be the unique continuous solution. Then we have that '-^^^fl^<-2Sam,u))<0.
(5.32) For u ^ A, by (5.32), we have that (5.33)
^x{Ci{t,u))
< ^x{u) < ao(A) < bo{X) = a(A),
Vt G [0,ri]
and
^A(a(^,^)) = ^ A ( ^ ) + / <
^x{u)-
/
^^^^
^^^^
2fo(^i(cr,l^))(icr
for ah t G [0,ri]. We shah show that Ci{t,u) ^ B for ah 5 G [0,ri] and u ^ A. If this were not true, then there would exist a to G [0, Ti] such that ^i(to,^) G 5 . Then ^xmh.u))
> a{X) = bo{X) = inf ^A. B
By (5.32)-(5.33), we see that
Jo /O Hence, c^(ei(a,^)) = 0,V(7G [0,to], i e . , e i ( ^ , ^ ) ^ Q i V ( 7 G [0,to].
5.2. TWIN CRITICAL POINTS
127
Therefore, one of the fohowing three cases occurs: (5.34) (5.35)
\M<^,n)\\>07 + 2 I'^xiCMu)) -a{X)\
+dA; >2ei; dist(^i(a,u), B) > 3Ti.
(5.36) If (5.34) holds, since
Il?i(a,u)-?i(a', u)\\<\a-a'\,
(5.37) \M'7,u)\\
< \MO,u)\\+T,
V(7 e [0,io],
we get a contradiction. If (5.35) holds, then ^x{^i{a, u)) < a(A) — 2ei. Hence, ^i{a,u) ^ B. Evidently, (5.36) implies that ^i(a,u) ^ B. Therefore, that ^i(t, u) ^ B for all s G [0, Ti] and u E A is true. We are going to show that (5.38)
Ci(Ti,r„(2s - l,u)) ^ B,yu eA,se
[1/2,1].
For any fixed u e A and s G [1/2,1], we distinguish two cases. (I) If Ci((T, r„(2s - 1, u)) G Qi for all a G [0, Ti], by (5.32), we have that $A(?i(Ti,r„(2s-l,u))) C5 (r (0. ^ ,\\M r^ = *A(r„(2s - 1, u)) + / <$A(r„(2s-l,u))- I Jo = $A(r„(2s-l,u))-25Ti
rf^A(gi(a,r„(2s-l,n)))^ da '2Sa{Cii(T,T„{2s-l,u)))da
< a{\) - 2STi + (a'(A) + 2)(A„ - A), then it follows that .^i(ri,r„(2s — l,u)) ^ B since a(A) = bo{\). (II) If there exists a to G [0,Ti] such that ^i(to,r„(2s — l,u)) ^ Qi, then one of the following alternatives holds: (5.39) (5.40) (5.41)
| | e i ( t o , r „ ( 2 s - l , u ) ) | | >w + 2 + dA; |$A(a(to,r„(2s-l,u)))-a(A)| >2ei; d i s t ( 6 ( i o , r „ ( 2 s - l , u ) ) , B ) > 3Ti.
Assume that (5.39) holds. If Ci(Ti,r„(2s - l,u)) G B, then 5o(A)
= < <
a(A) $A(6(ri,r„(2s-l,w))) $A(r„(2s-i,u)).
128
CHAPTER 5. DOUBLE LINKING
THEOREMS
Recalling the proof in step 1, thus we see that ||r^(25 — l,i^)|| < ru. Since
||a(to,r,(25-1,^)) -ei(o,r,(25-1,^))|| < to, we get that \\^i{to,Tn{2s — l,i^))|| < tx7 + to < tx7 + l, which contradicts (5.39). Hence, ^ i ( r i , r ^ ( 2 5 - l,i^)) ^ B. Assume that (5.40) holds. Note that
<^A(ei(0,r,(25-l,^))) =
^x{rn{2s-l,u))
< a(A)+£i. Therefore, (5.40) implies that ^x{UTi,Tn{2s-l,u))) <$A(?i(to,r„(2s-l,u)))
>||ei(to,r,(25-l,^))-^||-|t-to|,
[0,ri]. Thus,
dist(a(t,r^(25 - i,u)),B) > 2ri, yt e [o,ri]. In particular, ^i{Ti,Tn{2s
— l,u)) ^ B. This completes the proof of (5.38).
Step 6. Define
r a(25Ti,^), [ Cl{Tl,Tr^{2s-l,u)),
o<5
Then T^ G T. However, by the proofs of step 5, Tl{s,A) 5 G [0,1]. We get the final contradiction.
n 5 = 0 for ah
As for the conclusion (1) with the second alternative (A2), it is quite similar to the first case under the assumption (Ai). We omit the details.
5.3. EIGENVALUE
PROBLEMS
129
Finally, conclusion (2) can be proved immediately by interchanging A and B, replacing ^x by — ^A and using conclusion (1). Conclusion (3) is an immediate consequence of conclusions (l)-(2). D Notes and Comments. Theorem 5.7 was obtained by M. Schechter-W. Zou [324].
5.3
Eigenvalue Problems
Let O C R ^ be a bounded smooth domain and let £ > Ai > 0 be a self adjoint operator on 1/^(0) with compact resolvent and eigenvalues: 0 < Ai < • • • < A^ < • • • . We assume that C^{n)
CD:=
D{C^/^) C H'^{Q)
for some m > 0 {m need not be an integer). Let q he di number satisfying 2
2m/{N - 2m),N < 2m; 2 < q < oo,2m < N,
and let / ( x , t) be a Caratheodory function on O x R. Suppose that (Bi) \f{x,t)\ < Vo{xy\t\^-'^Vo{x)Vi{x), where Vo G L%n),Vi G L^'(Q) and multiplication by VQ is a compact operator from D to 1/^(0). (B2) Xi-it^ - Wo{x) < 2F{x,t) < ue + \V{x)\P\t\P + Wi{x); Xit^-W2{x)<2F{x,t); for all {x,t) G O X R, where u < Xi,p > 2;F{x,t)
= / Jo
f{x,s)ds;
Cj := / Wj{x)dx < 00, j = 0,1 JQ
and ||y^||^ < Co\\C^/^u\\P for ah u e D. (B3) C* + C* < (1 - - ) ( 1 - ii)^/(^-2)( J_)2/(p-2)^ P M P^O Theorem 5.8. Assume that {Bi)-{Bs) hold. Then there exists an SQ > 0 such that, for almost all (3 e [1,1+ ^o], the equation (5.42)
Cu = (3f{x,u)
has two solutions. In particular, the eigenvalue problem (5.4-2) has infinitely many solutions.
130
CHAPTER 5. DOUBLE LINKING
Proof of Theorem 5.8.
THEOREMS
Under hypothesis (5i), the functional defined by
^x{u) '= Xa{u) - 2 /
F{x,u)dx
JQ
is in C^{D, R), and the Frechet derivative on D is given by {^'x{u),v) =2Xh{u,v)
-2{f{-,u),v),
where a{u,v) = {jCu,v),h{u) = h{u,u) := \\u\\'^ for all {u,v) ^ D X D. Let N be the subspace spanned by the eigenfunctions corresponding to the eigenvalues: Ai,...,A/_i and M := N-^ n D he the orthogonal complement of N on D. Choose £i > 0 so small that C* + d
< (1 - ^)(1 - £i - |l)P/(P-2)( A.)2/(p-2) .^ ^^
For w e M with ||u;|| = 5, where 5^''^ = 2(1 - Si - f^)/(j)Co),
we have
$AH > A/.H-Hhlli-r«^ll^-cr > >
(A-f)||u;f-Co||u;r-Ci* Az /^ - Ci*
for A > l-si. Obviously, ^ A ( ^ ) < ^o for ^Wv eN and A < 1. Let A^i := N® {swo : 5 G R } , where w^ is an eigenfunction corresponding to the eigenvalue A/, \\wo\\ = L For any £ G (1 — £i, 1) and v G A^i, ^A(^)
= < = <
\{Cv,v)-\i\\v\\l^B2 \\i\\v\\l-\i\\v\\l^B2 {s-l)Xi\\v\\l^B2 Bo
uniformly for A G (l — Si^s) for H^;!! large enough. By Theorem 5.7, for almost all A G [1 — £i,£), there exist ux ^ vx such that ^^(I^A) = O^^AI'^A) = 0, which implies the conclusions of the theorem. D Notes and Comments. Theorem 5.8 is a simple application of Theorem 5.7. From this we observe that Theorem 5.7 is a powerful tool for studying the eigenvalue problem. It is easy to get infinitely many eigenvalues and each of them has at least two-dimensional eigenspace. Several authors had discussed the eigenvalue problems. See for example, in H. Amann [7], P.
5.4.
JUMPING
NONLINEARITIES
131
Rabinowitz [293], A. Szulkin [356] and E. Zeidler [381], where the nonhnear t e r m is odd. In Y. Li [222], Y. Li [223] (based on K. C. Chang [93]), M. Schechter [306, 310, 314, 316, 318, 321], H. T. Tehrani [368] and E. Zeidler [380] (also E. Zeidler [379, Vol. Ill]), finite solutions were obtained without oddness. Eigenvalue problems with indefinite weight were studied in A. Szulkin-M. Willem [359]. For eigenvalue problems of non-smooth functionals, see M. Struwe [350] and G. Arioli [21]. About the nonlinear eigenvalue problems in ordered Banach spaces, see H. A m a n n [8]. T h e estimates of t h e growth of t h e eigenvalues for linear eigenvalue problems can be found in P. Li-S. T. Yau [210]. Theorem 5.8 was obtained in M. Schechter-W. Zou [324] which extended a result of M. Schechter-K. Tintarev [322, 323].
5.4
Jumping Nonlinearities
Consider the elliptic boundary value problem of t h e form
(5.43) ^ ^
(-An [
= f{x,u), 1^ = 0,
in fi, on ail,
where O C R ^ is a bounded domain with smooth boundary dft, and is a Caratheodory function on O x R such t h a t .
.
J f{x,t)/t^a
a.e. X G O as t ^ —OO,
^ '
'
I f(^x,t)/t^b
a.e. X G O as t ^ OO.
f{x,t)
Recall the Fucik spectrum of —A: (5.45)
E := {(a, b) G R^ : —Ai^ = bu'^ — au~ has nontrivial solutions},
(cf. e.g. S. Fucik [167]). Although t h e existence of solutions of (5.43) is closely related to the equation (5.46)
— Ai^ = bu^ — au~,
where u
= max{±i^, 0},
no complete description of E has been found. So far only some partial results have been obtained. Let 0 < Ai < • • • < A^ < • • • denote t h e distinct Dirichlet eigenvalues of —A on O with zero boundary value conditions. It was proved in M. Schechter [307] t h a t in the square (A/_i, A/+i)^ there exist two decreasing curves Ca,(^Z2, which may or may not coincide, passing through the point (A/, A/) such t h a t all points above or below b o t h curves in the square are not in E, while points on the curves are in E. This is t h e so-called type-(I) region.
132
CHAPTER 5. DOUBLE LINKING
THEOREMS
Usually, the status of points between the curves which is referred to as the type-(II) region if the curves do not coincide is unknown. It was shown in T.Gallouet-0. Kavian [168] that when A/ is a simple eigenvalue, points of the type-(II) region are not in E. On the other hand, C. A. Margulies-W. Margulies [250] have shown that there are boundary value problems for which many curves in E emanate from a point (A/, A/) when A/ is a multiple eigenvalue. These curves are contained in the region-(II). In this section, (a, 6) is allowed to be a point outside the square (A/_i,A/+i)^ or inside the region-(II) between the curves Cn and C12 when they do not coincide. Such points may or may not belong to E. We prove an existence result of multiple solutions. Let i^o(^) be the usual Sobolev space with norm || • || and let Ek denote the subspace spanned by the eigenfunctions corresponding to A^, which is contained in 1/^(0) since all of the eigenfunctions corresponding to A^ are bounded. Set Nk = Ei® • • • ® Ek- Then there exists a constant C^ > 0 such that
(5.47)
ll^lloo = m a x | ^ ( x ) |
\/u e
Nk-i.
Define <^k := I inf{ / {u^fdx JQ ^
: u e Ek, \\u\\ = 1}.
Then, 0 < <^k < l/(4Afe) and <;k = l/2M{[{u-f:ueEkA\u\\ Jn
= l},
where u"^ = max{±i^,0}. Set 1/2
^ ' -^ V2(<^fe + C2|0| + l/Ai)J and Efc = E^i U Efc2, where Efei :={(a,6) G R^ : A^ < 6 < a}, Sfe2 := {(a, 6) G R 2 : A^-i < 6 < A^ < a, 7
^k-l^k
XfeAfe + (1 - xD^fe-
I
b> Afe(l - ^<^k)
and
5.4. JUMPING NONLINEARITIES
133
Define (5.48)
n:=
sup xen, |t|>4
^l^, ^
For /3o G (Afc_i,Afc), we choose / G N such that (5.49)
\i > 10Afe(2ri+Afe) , 32A| Afe-i Afc — /3o
Then by (5.47), there exists a Q > 0 such that (5.50)
||^||oo
for ah u G Ni-i. We make the fohowing assumptions. (Di) (a,6)GEfe. (D2) 2F{x,t) > Afe-it^, (D3) 2F{x,t)
V(x,t) G O X R.
< pot'^ + M*(x)
for |t| < 4,x G O, where M*(x) > 0 in O, M* G Li(0), and / M%x)dx < A/c-i(Afc — po)
^>icf
The points of E^ may be outside the square [A/c_i, A^+i]^, may be inside of the type-(II) region of the square [A^, A/c+2]^ or may be on the curves Ck+i^i and Cfc+1^2 which pass through the point (A^+i, A^+i). We have Theorem 5.9. Assume that (Di) — (Ds) hold; /(x,0) ^ 0. Then for almost all A G [1, 3^^A° ]' ^^^ following equation has two nontrivial solutions: (5.51)
—Au = Xf{x,u)
in ft; u = 0 on dft.
If in addition, (D4) 2F(x, t) — / ( x , t)t ^ ±00,
uniformly for x e ft as \t\ ^ 00,
^/len ^/le equation (5.52)
—Au = f{x,u)
in ft; u = 0 on dft
has two nontrivial solutions.
As usual, we seek solutions of equations (5.51) and (5.52) as critical points of the C^ functional ^x defined by
134
CHAPTER 5. DOUBLE LINKING
^A(^)=A||^f-a||^-||^-6||^+||2_2 / Jn for u e
HQ{Q),X>
THEOREMS
H{x,u)dx,
0, where
r
H{x,u) : = h{x,t):
/ Jo
=
h{x,t)dt,
f{x,t)-{bt^
-at-),
and t^ = max{±t, 0}. L e m m a 5.10. Let u = U- -\- UQ e N^ with U- G N^-I^UQ 1, 11'^-II ^ Xfe, ^^e^ 11'^^ 111 ^ ^fe/2, w;/iere i^^ = max{±i^, 0}.
G E^.
If \\u\\
Proof. Since ||i^|| = 1 and ||i^_|| < ^ , ||i^o|| > 0. It follows that, 2<^k{l \\u-f)
< [
{u-^fdx^nd
JQ
({utfdx Jn
=
f {uof Jun>o
<
2 /
u'^dx^2
Juo>0
/
u'idx
JQ
< 2/
u^dx^^Wu-f
< 2/(^+)2^x + 2C,2||^_f|0| + f ||^_f < 2 We have that / {u-^fdx > —, JQ 2
f{u^fdx^2ClxlM^^xl / {u-fdx JQ
>—. 2
D
L e m m a 5.11. Let {a,b) G E^. Then ^x{u) -^ —oo uniformly for A G (0,1] as ||i^|| ^ oo, 1^ G NkProof. We first consider the case of (a, 6) G E/c2- For i^ = i^_ + i^o ^ ^k^
5.4. JUMPING NONLINEARITIES
135
U- G Nk-ijUo G ^fc, we have $AN
=
\\\uf-a\\u-g-b\\u+g-2[Hix,u)dx Jn b)\\u-\\l - b\\u\\l - 2 / H{x,u)dx
<
\\uf -{a-
<
( l - - ^ ) | | ^ , _ f + (l_A)||^„||2 -(a-6)||i^~||2-2 /
H{x,u)dx
JQ
=
(i--^)lluf+
6(-^-J-)||uof
-(a-6)||^-||^-2 / <
H{x,u)dx
|H|2((i__^) + 5 ( ^ _ _ L ) ^k-1
^
^k-1
^k
-(a-6)^1^) -2 I \M ^ Jn If ^ 1 ^ < Xfe, then by Lemma 5.10, \\U\\
||T|^||2
<
b
Wufil
<
> —• It fohows that Z
rU
$A(U)
H(x,u)dx.
(a-b) — )-2
Afe
8A k
I
H{x,u)da
JQ
-Pi\\uf-2 / H(x, u) dx, JQ
where i^i := - ( 1 - ^Afe - ^ V2 ^ * ) > 0' ^^ W > ^fc' *hen ||^*of < (1 ||u|| x D l k f , and ^x{u)
<
{l--^)\\u.f M-1
+
<
(1 - ^)xl\\uf ^k-1
=
\\uf(l-xl-
=
-Jy2\\uf - 2 /
^k
{l-^)\\uof-2[H{x,u)dx Jn
+ (1 - f )(1 - X ^ ) l k f - 2 f H{x,u)dx ^k Jn (l-x|)^)-2/
H{x,u)dx
H{x,u)dx,
JQ
where 1^2 '•= Xk\ Afe-i
^ (1 ~ Xk)\ Xk
1 > 0. We see that there exists an e > 0
136
CHAPTER 5. DOUBLE LINKING
THEOREMS
such that (5.53)
^ A ( ^ ) < - ^ l l ^ f - 2 / H{x,u)dx,
It is also true if (a, b) G E^i. Note that lim |t|^oo
\/u G Nk.
h(x,t) ^— = 0. By (5.53), we obtain t
hmsup 11 ..^ < -£
\\u\\^oo Mr uniformly for A G (0,1), which implies the conclusion of Lemma 5.11.
D
As an immediate consequence of condition (-D2), we have L e m m a 5.12.
^A(^^)
< 0 for all u G A^^-i, A G (0,1].
L e m m a 5.13. Under the assumptions of Theorem, 5.9, there exist go > 0,eo > 0 such that ^x{u) > CQ for ||i^|| = po, u e Nj^_-^, where A G 3po + Afc ^2(A, + po)' ^' Proof. By ( 5.48), we see that 2F{x,t) < rit^for \t\ > 4,x G O. Moreover, condition (Di) implies that ri > A^. Choose r2 := 2ri + A^. Then (5.54)
2F(x, t) < r2t^ - 8(r2 - po) for \t\ > 4, x G O.
For any u G N^_-^^ write u = v ^ w with v ^ E^ ® ^fe+i 0 • • • 0 ^ z - i and w G A^^-ii. Let (Az+r2)A-2r2 8
2 , A(Afe - po) 2 4
x
If ^(^Y^"Vl - Apohl > 0, then (5.55)
Vi >
^
w' + (
^
|v|-Apo|w|)|f| > 0 .
If
(5.56)
M^Y^°^H-VoH<0,
by (5.49), we have that (5.57)
(A;+r2)A-2r2 ^
4X^pl -PO)'
A(AA;
5.4. JUMPING NONLINEARITIES
137
It follows by (5.56)-(5.57) that (5.58)
Vi > i^^^+'f-^'^ - ^ # ^ ) « ; ^ + hi^l^v^ 8 X{Xk - po) 4
> 0.
Let ( A z + r 2 ) A - 2 r 2 2 ^2 - Apo 2 (5.59) '02 := ^^^' ' '^^^^—^^w'^ ^'^""v^ - (r2 - \po)vw + 4(r2 - po). Then
It is easy to verify that 0^2 — ^ whenever II II
.
. r-, 4 ( r 2 - p o )
i; 00 < m i n { l , —
(Az+r2)A-2r2
- — , ——-
2|r2 - Apol
- — — } = 1.
8|r2 - Apo|
Let (r2 + Az)A^ 2 , (Afe +Po)A 2 If \v^w\> that
4, then by (5.54), (5.55), (5.58) and (5.59), for ||i;||oo < 1, we get
; . / ( A z + r 2 ) A - 2 r 2 2 , (A^ + po)A - 2r2 2 , /,/ N\ 0^3 > ( T w '^ H T '^ - r2'yt(; + 4(r2 - Po)) = V^l + 0^2 > 0. If IU ' +1(;| < 4, then by condition (-D3), we have that
V^3 >
(r2 + A/)A 2 , (Afe + Po)A 2 1 / , N2 1 ./T*/ N ^ ^ + ^ ^ -2/^o(^ + ^ ) --M [x)
> ^'' + Y - '^%^ + ^^' + ^ f - '^%^ - p,\vw\ - 1M*(X) >
(((^2A + AA - 2po)(A.A + poA - 2po))^^ _ ^^^ |^^| _ 1 ^ . ^ ^ ^ --M%x).
> 2
^ ^
138
CHAPTER 5. DOUBLE LINKING
Choose ll^ll := l/Q ^x{u)
:= ^o, then \\v\\^ < Ci\\v\\ < Ci\\u\\ = 1. By (Ds),
=
X\\vf^X\\wf-2
>
2[h\\vf
>
J A ( 1 - ^)\\vf ^
>
THEOREMS
f
+ h\\wf
:=
eo.
-
Z
Ak -t- PO
^l)\\wf
+
hxi\\w\\l-fF{x,u)dx)
+ I 2i^sdx
Ai
+ h{l
^k
Afc-l(Afc - po)
-
Z
J A ( 1 - ^)\\vf
.
+ hx,\\v\\l
+ h{l
^k
^
F{x,v^w)dx
^l)\\wf
JQ
- I
Ai
Ak
Al
M*{x)dx
JQ
JQ
D P r o o f of T h e o r e m 5.9. By Lemmas 5.11-5.13, we may find Ro > go > 0. Let A = {u = v^syo:ve
N^-us
> 0, ||^|| = Ro} U [N^-i H 5 ^ J ,
B = {ue N^_, : ll^ll = ^o}, where yo G Ek with \\yo\\ = 1. Then (5.60)
" - ^ := ao(A)
. . . ^^2g2 (Afe-po) SUP$A(U)
^ ^o(A) := inf
$A(U),
for A e ( ^^""^'^^,,1]. By Theorem 5.7, for almost all A G [ ^^""^'^'^ 1], 2(Afe+po) 2(Afe+poj there are two different critical points Ux,vx satisfying (5.61)
$1(UA) = 0,
$,(«,) = a ( A ) > 6 o ( A ) > ^ ^ ^ i ^ ^ ^
and (5.62)
^ I K ) = 0,
^x{vx) = b{X) < ao(A) < 0.
This is the first part of Theorem 5.9. For the second conclusion of the theorem, by (5.61), we choose A^ ^ 1 and u^ such that ^x^{un) = a(A^), ^^ (un) = 0, Therefore, «(^n) =
{f{x,Un)Un-2F{x,Un))dx.
Jn
5.4. JUMPING NONLINEARITIES
139
Note that T{s,u) := (1 — s)u G T and that A^B are bounded; by the definition of a(An), a{Xn) is bounded from below and above by two positive constants which are independent of A^. Recah condition {D4). By standard arguments, it is easily seen that {||i^n||} is bounded. This yields a critical point u^ satisfying
Similarly, by (5.62), we get another nontrivial critical point v* of ^1 satisfying ^i(i;*) <0,^;(i;*) = 0. D Notes and Comments. The study of the Fucik spectrum began with A. Ambrosetti-G. Prodi [18], E. N. Dancer [120] and S. Fucik[167]. They first realized that the set in (5.45) is an important factor in the study of semilinear elliptic boundary value problems with jumping nonlinearities. When (a, b) G (A/_i, A/+i)^, some cases have been studied. If (a, 6) G (A/_i, A/+i)^, (a, 6) G region of type-(I) (that is the nonresonant case), see M. Schechter [307], E. N. Dancer [122, 123] and K. Perera-M. Schechter [284, 285]). If (a, 6) G (A/_i, A/+i)^, (a, 6) G region of type-(II), (a, 6) ^ E (also nonresonant case), see M. Schechter [315], E. N. Dancer [122, 123] and K. Perera-M. Schechter [282, 284]. See also N. Hirano-T. Nishimura [186] and A. Marino-C. Saccon [251] for results of multiple solutions for nonresonant case. If (a, 6) G (A/_i, A/+i)^, (a,6) G {Cn U C12) C E (that is resonant case), see K. Perera-M. Schechter [283] and K. Perera-M. Schechter [286]. If (a, 6) G (A/_i, A/+i)^, (a, b) ^ TiH (region of type (II)) (also resonant case), see PereraSchechter [286]. One solution was obtained there. For the Dirichlet problem involving the Laplacian, it was shown in N. P. Cac [75], E. N. Dancer [120], P. Drabek [141], P. Krejci [199] A. C. Lazer[203], A. C. Lazer-P. J. Mckenna [204, 205], T. Gallouet-0. Kavian [168], C. A. Magalhaes [248, 249], A. M. Micheletti [253], A. M. Micheletti-A. A. Pistoia [254], B. Ruf [303] and others that the curves in ^ of (5.45) exist locally in the neighborhood of (A/, A/). They also showed that in the square A/_i < a < A/, A/ < a < A/+i, ^ has one or more curves emanating from (A/, A/), while the square A/_i < a, 6 < A/ and Xi < a^b < A/+1 are free of the Fucik spectrum. It was conjectured that there are at most 2m such curves, where m is the multiplicity of the eigenvalue, see M. Schechter [310, page 267] ( and also E. N. Dancer [123] and C. A. Margulies-W. Margulies [250]). Note that the critical groups for jumping nonlinearities were studied in E. N. Dancer [123, 124], E. N. Dancer-Z. Zhang [128], S. Li-K. Perera-J. Su [214], J. Liu-S. Wu [235], K. Perera-M. Schechter [283, 284]. Theorem 5.9 was obtained by M. Schechter-W. Zou [324].
140
CHAPTER 5. DOUBLE LINKING
THEOREMS
Before closing the chapter, we mention some papers on p-Laplacian problems with jumping nonlinearities and its Fucik spectrum: S. Carl-K Perera [80], M. Cuesta-D. G. de Figueiredo-J. P. Gossez [119], E. N. Dancer-K. Perera [125], P. Drabek-S. Robinson [142], A. M. Micheletti-A. A. Pistoia [255], K. Perera [280, 281] and K. Perera-A. Szulkin
Chapter 6
Superlinear Problems We solve elliptic semilinear boundary value problems in which the nonlinear term is superlinear, but the nonlinearities need not satisfy the superquadracity condition. Because of this, we are able to include more equations than hitherto permitted. The region is not assumed to be bounded.
6.1
Introduction
Let O be a domain (which need not be bounded) in R ^ and .4 be a selfadjoint operator on 1/^(0). Some semilinear elliptic problems can be described in the following way. We assume that ^ > AQ > 0 and that (6.1)
C^{n) CV:=
V{A^I^) C R'^^^iSl)
for some TTI > 0, where Co^(O) denotes the set of test functions in Vi (i.e., infinitely differentiable functions with compact supports in O) and H'^''^{Q) denotes the Sobolev space. If m is an integer, the norm in H^'^^iQ) is given by
(6.2)
||«|U,2 := I E
ll^''"ll^ '
\|//|<m
Here D^ represents the generic derivative of order |/i| and the norm on the right-hand side of (6.2) is that of 1/^(0). If m is not an integer, there are several ways of defining the space H^''^(Q), all of which are equivalent. We shall not assume that m is an integer. A typical example of an operator A satisfying these hypotheses is a second order elliptic operator with smooth coefficients applied to functions satisfying zero Dirichlet boundary conditions
CHAPTER 6. SUPERLINEAR
142
PROBLEMS
on a smooth bounded domain in R ^ . Only the abstract properties listed above are relevant to our analysis. Let q he di number satisfying (6.3)
2
2m < N, N < 2m.
Assume that O is bounded and has a smooth boundary. Then • if 2m < n,l < q < oo and (n — 2m)q < 2n, then ||^L
ueH^^\n);
• if n < 2m, then sup|^(x)| < C | | ^ | U , 2 ,
ueH'^^^n)
and U{x) - U{y)\ ^ ,, ,, sup ^ ^ < C U m,2.
TTm2/r^\ U G i7^'2 O ,
where ^ < a < 2m — n. By the first item, H^''^ ^^ L^(Q) for the given values of q. In particular, ^m,2 ^^^^ L^{Q) if (n — 2m)q < 2n. The second item implies that every function in H^''^{Q) is a.e. equal to a continuous function bounded in O and satisfies a Holder condition with suitable exponent. Proofs for these statements can be found in R. A. Adams [4] and M. Schechter [308]. Let f{x,t) (6.4)
be a Caratheodory function on O x R. We consider the problem Au = f{x,u),
ueV.
By a solution of (6.4) we shall mean a function u eV such that (6.5)
{u,v)v = {f{',u),v),
\/veV.
The problem (6.4) has been studied by many people. In A. Ambrosetti-P. Rabinowitz [19] the basic assumption was (6.6)
0 < /iF(x, t) < tf{x, t),
\t\ > r
for some ji > 2 and r > 0. This is a very convenient hypothesis since it readily achieves mountain pass geometry as well as satisfaction of the PalaisSmale condition. However it is also a severe restriction; it strictly controls the growth of f{x,t) as \t\ -^ oo. Most authors discussing superlinear problems
6.1.
INTRODUCTION
143
have made this assumption. In this section, we have been able to weaken this assumption considerably by a method different from the previous chapters. We assume either that fiF{x, t) - tf{x, t) < C{t^ + 1),
\t\ > r
for some /i > 2 and r > 0 or that (6.7)
L(x, t) := tf{x, t) - 2F(x, t)
is convex in t. These allow much more freedom for the function f{x,t). make the following assumptions. (Ai) \f{x,t)\
< M{xy{\t\^-^
We
+ 1) for ah (x,t) G O x R and
f{x,t)/M{xY
= o(|t|^-^) as \t\ -^ oo,
where M{x) > 0 is a function in 1/^(0) such that
(6.8)
IIM^II, < C | | ^ | | D ,
(6.9)
ueV,
\\uy:=\\A'/\l
If O and M{x) are bounded, then (6.8) will hold automatically by the Sobolev inequality. However, there are functions M{x) which are unbounded and such that (6.8) holds even on unbounded regions O (cf., e.g., M. Schechter [308]). With the norm (6.9), V becomes a Hilbert space. (A2) The point AQ is an isolated simple eigenvalue with a bounded eigenfunction eo{x) ^ 0 a.e. in O. (A3) There is a (5 > 0 such that 2F{x,t) F{x,t) := jlf{x,s)ds.
< Aot^,
\t\ < 8, x e Q, where
(A4) There is a function W{x) G L^{Q) such that either ,,,, , F(x,t) Wix) <
,,,, , F(x,t) > 00 as t ^ 00; or Wix) <
> 00 as t ^ —00,
for X e ft. The function W{x) need not be positive. (A5) There are constants /i > 2, C > 0 such that
fiF{x,t)-tf{x,t)
ten,
xen.
CHAPTER 6. SUPERLINEAR
144
PROBLEMS
Theorem 6.1. Under the above hypotheses, the problem (6.10)
Au = f{x,u),
ueV
has at least one nontrivial solution. We also consider the following condition. (AQ)
The function L{x,t) := tf{x,t)
— 2F{x,t) is convex in t.
Theorem 6.2. / / we replace hypothesis (A^) with (AQ), then the problem (6.10) has at least one nontrivial solution.
Problem (6.10) is called sublinear if f{x,t) satisfies | / ( x , t ) | < C{\t\ -\l ) , x G 0 , t G R. Otherwise it is called superlinear. Hypothesis (^4) requires (6.10) to be superlinear. (A7) There are a (5 > 0 and a A > AQ such that 2F{x,t) > At^,
\t\ < S, x e
n. (Ag) There is a function W{x) G L^{Q) such that W{x) > Q{x,t) -^ —00 as \t\ ^ 0 0 , where Q{x,t) := F{x,t)
X G O,
Aot^.
If we drop hypothesis (^5) completely, then we are able to prove the following theorems. Theorem 6.3. If we replace hypotheses (As), {A4) with {A7), (Ag) and drop hypothesis (A^), then problem (6.10) has at least one nontrivial solution.
Notes and Comments. There have been many results studying superlinear elliptic equations under the assumption (6.6) since the celebrated paper by A. Ambrosetti-P. Rabinowitz [19]. In particular, a 3-solution theorem was obtained in Z. Q. Wang [373]. We make no attempt here to give an exhaustive account of this aspect or a complete survey of the literature. Our main aim is to weaken the condition (6.6) via a different way. Notes and comments on this line will be seen at the end of this chapter.
6.2. PROOFS
6.2
145
Proofs
Define (6.11)
^{u) := ll^lll, -2
f
F{x,u)dx.
Under hypothesis ( ^ i ) , we show that ^ is a continuously differentiable functional on the whole of V. Lemma 6.4. Under hypothesis (Ai) , F{x,u{x)) 1/^(0) whenever u, v eV.
and v{x)f{x,u{x))
are in
Proof. By (Ai) we have \F{x,u)\ < c(|M^|^ + |M^||M|^-^). Since M, Mu G L^, |M|^-^ G L^', where q' = q/{q - 1), the right-hand side is in 1/^(0). Similarly, \v{x)f{x,u)\
< \Mv\{\Mu\'^-^ + 1).
Since Mu, Mv are in L^, we obtain the conclusion.
D
Lemma 6.5. ^{u) has a Frechet derivative ^'{u) on V given by (6.12)
{^\u),v)v
= 2{u,v)v - 2(/(-,u),v),
where (•, •) is the usual inner product of L'^. Furthermore, the derivative ^^{u) given by (6.12) is continuous in u.. Proof. We write f{u) := f{x,u{x)). (6.13)
^{u -^v)-
Then we observe that
^{u) - 2{u, v)v + 2(/(^), i;)
= \\v\\% -2
(F{X,U^V)
= o{\\v\\v) — 2 I I
-F{X,U)
lf{x,u-\-rv)
-vf{x,u)\dx,
—
f{x,u)]vdrdx
^
\M-' (/(x, u^rv)-
^
^ 1/q'
fix, u)^ l^'drdx^
\\Mv\\,
m Jo where q' = q/{q — I)- We want to show that
( J J |M-1 (/(X, U + TV)- fix, U)) \l'dTdx) '^' (6.14)
=o{\\vy)
as||w|b^O.
146
CHAPTER 6. SUPERLINEAR
Otherwise, we may find a sequence {vk} C V such that
(6.15)
/ /
PROBLEMS -^ 0 while
H'^/CUD
\M-^(f{x,u^rvk)-f{x,u))\'^'drdx>s>0.
By (^i), we see that ||M'U/c||g -^ 0- Then, for a subsequence, Mv^ -^ 0 a.e. But by (Ai) the integrand of (6.15) is majorized by \M-'(^f{x,u + TVk)-f{x,u))\'
(6.16)
where the right-hand side converges in L^(Q). Moreover, the integrand converges to 0 a.e. Hence, the left-hand side of (6.15) goes to 0 and (6.14) holds. The continuity of ^\u) is easily observed by the following arguments:
= 2{ui - U2, v)v -2 / v{f{x, ui) - / ( x , U2))dx JQ
<2\\ui-U2\\v\\v\\v^2\\Mv\\,^
\M-\f{x,ui)
-
f{x,U2)W'dxy^\
n Lemma 6.6. Under hypotheses {Ai)-{As), either
the following alternative holds:
(a) there is an infinite number ofy{x) G 1){A) \ {0} such that Ay = f{x,y)
= Xoy
or (b) for each p > 0 sufficiently small, there is an £ > 0 such that ^{u)
>£,
||^||D = P.
Proof. Let Ai > AQ be the next point in the spectrum of .4, and let A^o denote the eigenspace of AQ. Choose M = NQ- HV. By assumption (^2), there exists a p > 0 such that \\y\\v
implies that |7/(x)I < (5/2, y e NQ.
Now suppose u eV satisfying (6.17)
II^IID
< p a n d \u{x)\ > S
6.2. PROOFS
147
for some x G Q. Write (6.18)
u = w^y,
w e M, y e No.
Then for those x e ft satisfying (6.17), 5 < \u{x)\ < \w{x)\ + \y{x)\ < \w{x)\ + 5/2. Hence, \y{x)\ < 5/2 < \w{x)\
(6.19) and
\u{x)\
(6.20)
<2\w{x)\
for ah such x. Furthermore, by assumption (^i),
>\Mv->^o
[ u^dx-cj {\Mu\'^ J\u\<5 J\u\>5
>\\u\\l-\4u\\'-c[
^\Mu\)dx
\Mu\^)dx J\u\>5
(6.21)
> \\w\\% - \o\\w\\^ -c
I \Mw\'^dx J2\w\>5
in view of the fact that \\y\%, = Ao||^|p and (6.20) holds. Note that
In
\Mw\Hx
= o{\\w\\l).
By (6.21), we see that (6.22)
^i^u)>{l-^-o{l))\\w\\l,
\\u\\
If the alternative (b) were not true, then there would be a sequence such that (6.23)
^{uk)^^,\\uk\\v=p.
If p is taken sufficiently small, (6.22) implies that ||I^/C||D -^ ^]\\yk\\v -^ PSince A^o is finite dimensional, there is a renamed subsequence such that Vk -^ yo in A^o. Thus, \\yo\\=P,Hyo)=0,
\yo{x)\<5/2,
xeQ.
CHAPTER 6. SUPERLINEAR
148
PROBLEMS
Then, hypothesis (As) imphes that (6.24)
2F{x,yo{x))
< XoVoixf,
xeQ.
Since J [Xoyoixf
- 2F{x,yo{x))yx
= ^yo)
=0
and the integrand is > 0 a.e., by (6.24), we see that 2F{x,yo{x))
=Xoyo{xf,
xeft.
Let (/)(x) be any function in Co^(O). Then for t > 0 sufficiently small, t-^ (2F(x, yo + t(/)) - Xo{yo + t(l)f - 2F(x, yo) + Ao?/g) < 0. Letting t ^ 0, we see that (/(x, yo) - \oyo)(t>{x) < 0,
x eVt.
Since yo ^ NQ, it follows that the conclusion (a) of the lemma holds.
D
Proof of Theorem 6.1. We define (6.25)
^{u) = \\u\\l -2
f
F{x,u)dx.
Under our hypotheses. Lemmas 6.1-6.5 apply, and (6.26)
{^\u),v)
= 2{u, v)v - 2(/(.,^),.;),
u.veV.
By Lemma 6.6 we see that there are positive constants e, p such that (6.27)
^u)>s,
\\u\\v = P
unless Au = Xou = / ( x , u ) ,
(6.28)
ueV\{0}
has a solution. This would give a nontrivial solution of (6.10). We may therefore assume that (6.27) holds. Next we note that ^±Reo)/R^
[ {F{x, ±Reo)/R^el}eldx -oo Jn as i? ^ oo. By hypothesis (A4), since eo ^ 0 a.e. Since ^(0) = 0 and (6.27) holds, we can now apply the usual mountain pass theorem to conclude that there is a sequence {uk} C V such that (6.29)
= \\eo\\l-2
^Uk) -^c>e,
^'{uk) ^ 0.
6.2.
PROOFS
149
Then (6.30)
^{uk)
= PI-'^
I F{x, Uk)dx -^ c
and (6.31)
{^\uk),Uk)
= 2pl - 2{f{',Uk),Uk)
= o{pk),
where pk = ||t^fe||D. Assume t h a t pk -^ oo, and let Uk = Uk/pk- Since ||i//c||D = 1, there is a renamed subsequence such t h a t Uk ^ u weakly in P , strongly in Lf^^{n) and a.e. in O. By (6.26)-(6.30), ^loci cy
Let Oi = {x G O : u{x) ^ 0 } ,
Liu
UJJL
7
1.
O2 = O \ O i . Then by hypothesis {A4),
2F{x,Uk)^2
2—^k ^k
^00,
^o
X e
ih-
If Oi has a positive measure, then ^
u1 dx >
u1 dx -\-
^
W{x) dx -^ 00.
Thus, the measure of Oi must be 0, i.e., we must have u = Oa.e. Moreover, / i F ( x , Uk) - Ukfjx,
Uk) ^2 ^
/^
-1
But by hypothesis (E),
^k
%
which implies t h a t (/i/2) — 1 < 0, contrary to assumption. Hence, the pk are bounded. Therefore, there is a subsequence which converges weakly in V to a limit i^. For any compact subset OQ C O , t h e imbedding of HQ^{ft) in I/^(Oo) is compact. Thus, we may find a subsequence which converges to u in I/^(Oo). For a subsequence, Uk ^ u Si. e. in OQ. By taking a set of compact subsets of O which exhaust O, we can find a renamed subsequence which not only converges to u weakly in P , but also strongly in I/^(Oo) for each compact subset QQ of Q and also a.e. in Q. We claim t h a t (6.32)
/ F{x,Uk)dx^ JQ
(6.33)
/ f{x,Uk)vdx^ JQ
(6.34)
/ f{x,Uk)ukdx^ JQ
/
F{x,u)dx;
JQ
/ f{x,u)vdx, JQ
/ JQ
f{x,u)udx.
v ^ V;
150
CHAPTER 6. SUPERLINEAR
PROBLEMS
To see this, let ^r{t) be the continuous function defined by
(6.35)
Ut)
r = < [
t, r,
\t\ < r, t>r,
—r,
t < —r.
By (Ai), for a given £ > 0, chose r so large that (6.36)
\f{x,t)\<eM'^W-\
\t\>r
and that (6.37)
\F{x,t)-F{x,£^r{t))\<eM^\t\\
\t\>r.
Then (6.38)
/ (F(x, i^fe) - F{x, u))dx
(6.39)
= /
(F(x, ^fe) - F{x,
U^k))dx
J\uk\>r
(6.40)
+ / {F{x, ^r{uk)) - F{x, Jn
(6.41)
+/
Uu))dx
{F{x,Cr{u))-F{x,u))dx.
J\uk\>r
By (6.37), the integrals of (6.39) and (6.41) are bounded by £ / iMukl'^dx < £c\\uk\\^ < £c. The integrand of (6.40) is majorized by cilM^riukM" + \M^r{uk)\ + \M^r{u)\^ + \M^r{u)\) < c{M^r^ + Mr), which is in L^. Thus, the integral in (6.40) converges to zero. These arguments imply that (6.32) is true. In a similar way, we may prove (6.33) and (6.34). Now, by (6.33), we readily have
Hence, i^ is a critical point of ^ . Noting that
6.2. PROOFS
151
we see that (6.34) implies that \Wk\\v^
/ Jn
f{x,u)udx=\\u\\jy.
Thus, Uk ^ u in V. We now obtain a weak solution of (6.10) satisfying ^{u) = c> £. Since ^(0) = 0, we see that u ^ 0. This completes the proof. D We postpone the proof of Theorem 6.2 until the next section. To prove Theorem 6.3, we shall need the following lemma. L e m m a 6.7. Under the hypothesis (A^), there is aT ^ 0 such that ^(Teo) < 0. Proof. We can assume that ||eo||D = 1. Thus, ^(Teo) = T^ - 2 / F(x, Teo) dx
[ eo{xfdx J\Teoix)\<5 M ^ ( | r e o r + |Teo|)
l\Teo(x)\>5
This can be made negative by taking T sufficiently small. L e m m a 6.8. Under the hypothesis (As), ^{u) ^ oo as
||I^||D
D -^ oo.
Proof. Suppose there is a sequence {u^} C V such that pk = \\uk\\ -^ cxo and ^{uk) < K. We write Uk = Wk ^ T^eo, Uk = Uk/pk, Wk = Wk/pk, Tk = Tk/Pki where Wk^e^. If Ai > AQ is the next point in the spectrum of A^ then \i\\w\\^ < \\w\\l, and w^e^. Thus ^{uk) = \\uk\\l - Xohkf
- 2 / Q{x,Uk) dx Jn
>(l-^)lkfe|||)-2 [
>{l-^)\\wk\\l-2
Q{x,uk)dx
[ W{x)dx.
The only way this would not converge to oo is if ||I(;/C||D is bounded. But then II'^/CIID -^ O5 and \Tk\ -^ 1. Since ||i//c||D = 1, there is a renamed subsequence
CHAPTER 6. SUPERLINEAR
152
PROBLEMS
such that Uk ^ u weakly in P , strongly in I/^^^^(0) and a.e. in O. Since w = 0 and \T\ = 1, we have u{x) = Teo{x) ^ 0 a.e. Hence, \uk{x)\ = pk\uk{x)\ -^ oo a.e. Therefore,
/ Q{x,Uk) dx -^ —oo, showing that ^{uk) -^ oo. This
completes the proof of the lemma.
D
Proof of Theorem 6.3. Let S = infx) ^ . Then we may find a sequence {uk} C V such that ^{uk) -^ S. By Lemma 6.8, we must have ||I^/C||D < C. Hence, there is a renamed subsequence such that Uk ^ u weakly in P , strongly in L'i^^{Q) and a.e. in O. Now,
= \\u\\'^ - 2 /
F{x,u)dx
= hkWv -2((^fe -u),u)v - 2 / F{x,Uk)dx^2 JQ
- \\uk -u\\^ / {F{x,Uk) -
F{x,u))dx
JQ
< ^{uk) - 2{{uk -u),u)v^2
/ (F(x, Uk) - F{x, u))dx. Jn
Similar to (6.32), it is easily seen that / F{x^Uk)dx ^
I
F{x^u)dx.
We therefore have the limit ^(u) < S, by which we conclude that ^\u) = 0 and ^{u) = S. Hence, i^ is a weak solution of (6.4). We see from Lemma 6.7 that 6^ < 0. Since ^(0) = 0, we see that u j^ 0. This completes the proof. D
6.3
The Eigenvalue Problem
Theorem 6.9. Assume that {Ai)-{A4) hold. Then for almost every (3 G (0,1), the equation (6.42) has a nontrivial solution. infinitely many solutions.
Au = (3f{x,u) In particular, the eigenvalue problem (6.42) has
We also make the following conditions. (Ag) There are a (5 > 0 and a A < AQ such that 2F{x,t) < At^,
n.
\t\ < 6, x e
6.3. THE EIGENVALUE
PROBLEM
153
(Aio) Either / F{x, Reo) dx/R^ ^ oo as R ^ oo or /
F{x,-Reo)dx/R'^
oo as R ^ oo.
Theorem 6.10. If we replace hypothesis (As) in Theorem 6.9 with (Ag) and {A^) with (Aio); then (6.42) has a nontrivial solution for almost every
/?e(0,Ao/A). Theorem 6.11. If we replace hypothesis (Ag) in Theorem 6.10 with (All)
F{x,t)/t'^
-^ 0 uniformly as t ^ 0,
then (6.42) has a nontrivial solution for almost every (3 G (0, 00).
We shall also need the following extension of Theorem 6.4. Lemma 6.12. Let 1 < A < B < 00. Under hypotheses {Ai)-{A4), for each tv > 0 sufficiently small (not depending on A), we have (6.43)
^ A ( ^ ) := MMv
- 2 / F{x,u)dx
> (A - 1)/^^
If we replace hypothesis (As) with hypothesis A < B < 00, then we have (6.44)
q>^^u)>lx-^\n\
||^||D = K..
(AQ), assuming 1 < A/AQ <
\\u\\T, = n.
Proof. Let Ai > AQ be the next point in the spectrum of A, and let A^o denote the eigenspace of AQ. We take M = NQ- H V. By hypothesis (^2), there is a ^c > 0 such that ||^||D < n ^ \y{^)\ ^ ^/2, V ^ ^o- Assume that u ^ V satisfies (6.45)
II^IID < /^and \u{x)\ > S
for some x G O. We write u = w -\- y, w ^ M, y ^ NQ. Then for those x G O satisfying (6.45) we have that 5 < \u{x)\ < \w{x)\ + \y{x)\ < \w{x)\ + (5/2).
CHAPTER 6. SUPERLINEAR
154
PROBLEMS
Hence, \y{x)\ < S/2 < \w{x)\. It follows that \u{x)\ < 2\w{x)\
(6.46)
for all such x. By assumption (^i), we have that
u^dx -c [ (|M^|^ + M'^luDdx > X\\u\\l -Xo [ J\u\<5 J\u\>5 > A||^|||,-Ao||^f-c /
{Mul'^dx
f\u\>6
> (A - 1)11^111, + All^lll, - Aoll^f - c /
\Mw\^dx
J2\w\>5
in view of the fact that ||^|||) = Ao||^|p. Thus, by assumption (Ai) again, (6.47)
$^(u)>{X-l)\\y\\l+(^X-^^-c,\\w\\l-'y\w\\l,
We take /^ > 0 to satisfy 1 —
AQ/AI
^x{u) > (A - l)/^2 ^ f x - ^ -
Mr, < n.
> ci/^^~^. This gives cm^-^ -\^l\\\w\\l>{\-
l)n\
where ||I^||D = n. Hence, (6.43) holds. To prove (6.44) under hypothesis (Ag), let T] = A/Ao and A = (?^, 6 ) . Under hypothesis (Ag) we have in place of (6.47) (6.48) for
||I^||D
$,(^.) > (A - ^)\\y\\l + f A - A _ c,\\wr^^\ < t^. We take ^c > 0 to satisfy
T]
\\w\\l
— A/Ai > ci^c^~^. It follows that
^A(^)
> (A - r^)^^ + (^ - ^ - ^ 1 ^ ' " ' - ^ + ^ ) ll^lll)
for
||I^||D
= n. This gives (6.44), and the proof is complete.
D
We now turn to the proofs of Theorems 6.9 and 6.10. We shall prove Theorem 6.10 first by applying Theorem 5.7 and Lemma 6.12.
6.3. THE EIGENVALUE
PROBLEM
155
Proof of Theorem 6.10. We take ^ = P , A = (?^, 6 ) , where ?^ = A/AQ, B > 1 is a finite number, and I{u) = \\u\\l,
J(u) = 2 / F{x, u) dx.
By Proposition 5.6, the sets A± = [0,^6^060], B = {x e V : \\x\\j) = K.} link each other if 6^0 > tv. In our case the condition (Ai) of Theorem 5.7 is satisfied. To verify (As) of Theorems 5.7, we first observe that ^^(i^) = 0 is equivalent to (6.42) with f3 = 1/A. Now by hypothesis (Ag), at least one of the limits holds: ^ 0 0 as 6^0 ^ 00. Hence, for 6^0 sufficiently large, one of the inequalities ^A(±^oeo)/^o' < ©lleolll) - 2 [{F{x,±Soeo)/S^ <0 Jn holds. Thus, ao{X) < 0, A G A. Furthermore, it follows from Lemma 6.12 that (6.44) holds. Hence, 60(A) > (A - ?^)/^^, A G A. This shows that condition (^3) holds. We now apply Theorem 5.7 to conclude that for almost all A G A, there exists an Uk{X) G V such that sup ||i^fc(A)|| < 00, ^^(i^fc(A)) -^ 0 k and that ^x{uk{X)) -^ a{X) > 60(A). Once we know that the sequence {uk} is bounded, we can apply an idea similar to that used in the proof of Theorem 6.1 to conclude that there is a solution of ^^x{u) = 0, ^A('^) = «(A). From the definition, we see that a(A) > {\ — r])H?. Hence, the equation ^^(i^) = 0 has a nontrivial solution for almost every A G A. Since B was arbitrary, the result follows. D Proof of Theorem 6.9. It suffices to choose A = Ao and show that condition (744) implies hypothesis (Aio). To see this, we note by hypothesis (^4) and the fact that eo(x) ^ 0 a.e. that / ^ F ( x , ±6^060) (ix _ as 6^0 ^ 00.
r F(x, ±6^060) . 2 . n
Proof of Theorem 6.11. We let £ > 0 be an arbitrary number. By condition (All), there is a (5 > 0 such that F{x,t)/t^ < e for \t\ < 5 and x G O. By Theorem 6.10, equation (6.42) has a nontrivial solution for a.e. (3 G (0, Ao/e). Since £ was arbitrary, the result follows. D
CHAPTER 6. SUPERLINEAR
156 Proof of Theorem 6.2. By A G (1,B), there exists a ux (A — l)hi^. Choose An ^ 1, that ^x^{un) = 0, ^x^{un) may assume that bo{l) > £ >
PROBLEMS
Theorem 6.9, for each arbitrary B > 1 and a.e. such that ^^(I^A) = 0 and ^A('^A) = «(A) > A^ > 1. Then there exists a sequence Un such = a{Xn) > a(l) > &o(l)- By Lemma 6.4, we 0. Therefore,
We claim that {un} is bounded. Indeed, if ||i^n||D -^ oo, let w^ = u^/Wu^WvThen Wn ^ w weakly in P , strongly in L'i^^{ft) and a.e. in O. If i(; ^ 0 in P , then c
>
2F{x,Un) , I —^ ^—^^
Jn \K
Wn\ dx
>
[
^^^^^^f^\w^\'dx-
Jw^O -^
^n
[
W,{x)dx
Jw=0
OO
and we get a contradiction. However, if i(; = 0 in P , we define a constant tn G [0,1] satisfying ^A^(^n'^n) = max ^A^(^'^n)- For any c > 0 and iD^ = cwn, we have J^ i^(^, w^) dx -^ 0. Thus, ^X^itnUn)
> ^X^{cWn)
= C^K - 2 / F{x,Wn)dx
> C^/2
for n sufficiently large. That is, lim ^Xr^{tnUn) = OO,
(^';^ ( t ^ ^ ^ ) , ^n) = 0.
It follows that, ^Xr^{tnUn) {f{x,tnUn)tnUn ij\X^
tfiUfij
- 2F{x,tnUn))
dx
dx
OO.
By hypothesis (AQ), ^A^('^n) =
/ L{x^Un)dx
> / L{x^tnUn)dx
-^ oo.
However, we have the following estimates which contradict the above conclu-
6.3. THE EIGENVALUE
PROBLEM
157
sion: <
sup
^x^{{l - s)u)
sG[0,l] ,ueA
<
sup
^Q{{1
- s)u)
sG[0,l] ,ueA < c.
Thus, II^^IID < C. Then, ^^u^) -^ 0, ^{u^) -^ a(l) > 6o(l). We can now use the same arguments as those used in the proof of Theorem 6.1 to obtain the desired solution. D Notes and Comments. There were several papers where the authors tried to weaken the A. Ambrosetti-P. Rabinowitz superquadratic condition. We refer the readers to L. Jeanjean [191, 193] and L. Jeanjean-J. F. Toland [196], Z. Liu-Z. Q. Wang [237], M. Willem-W. Zou [378], H. S. Zhou [384]. The theorems of this section were proved in M. Schechter-W. Zou [329]. The readers may go back to the end of Section 5.3 for the notes and comments about the eigenvalue problems.
Chapter 7
Systems with Hamiltonian Potentials Let ^ be a real Hilbert space with an inner product (•, •) and the associated norm || • ||. Let A and B be two bounded subsets of E such that A links B. We describe the situation in which there are two linear, bounded and invertible operators Bi,B2 : E ^ E and a functional H G C^(^,R) whose values are separated by BiB and B2A, i.e., sup H < inf H. Note that BiB and B2A B2A
BiB
become much more uncontrollable. We prove the existence of a critical point oi H without assuming the (PS) type conditions. This theory fits some special elliptic systems.
7.1
A Linking Theorem
Let E have an orthogonal decomposition E = E^ 0 E~ and let /^ > L Consider a family of C^(^,R)-functionals (7.1)
^x{u):=\{Oxz,z)-^{z),
Ae[l,K]
under the following assumptions: (Ai) There exist two bounded linear and selfadjoint operators 0'^^\0'^'^^ : ^ ^ ^ such that OA = A O ^ ^ ^ - O ^ ^ ) , A G [1,/^], where (O^i)^,^) > 0 for dl\ z e E and either {O'^^^z.z) ^ 00 or |(0^^^z,z) + ^ ( ^ ) | ^ 00 as
(A2) ^' is compact.
160
CHAPTER?.
SYSTEMS
WITH HAMILTONIAN
POTENTIALS
(A3) There exist two linear bounded invertible operators Bi,B2 : E ^ E such that the hnear operator B{X,u) := p-B^^e'^^^B2 : E' ^ E' for ah 6J > 0 and A G [1,/^] is invertible, where P~ is the projection over E-. For each p > 0, set (7.2)
S:={Biz:\\z\\=p,zeE+}.
Choose a fixed (7.3)
ZQ G ^ + \ { 0 }
and define
Q := {B2{TZQ + Z) : 0 < r < CT, ||Z|| < M , Z G
^-}
for cr > p and M > p. By dQ we denote the boundary of Q relative to the subspace B2{E~ 0span{zo}). Define Q :={i9 e C([0, l]x E,E) :i9 satisfies (Bi), (62) and (63)}, where (61) i^{t,z) = exp(^u;i{t,z)Os,)z
^W{t,z),
where 5^ G [l,/^],cj^ : [0,1] x
i=l
E -^ [0, +00) is continuous and maps bounded sets to bounded sets; l y : [0,1] X ^ ^ ^ is compact; 1^(0, z) = 0 for any z e E; W{t, z) = 0 for any {t,z) G [0,1] x dQ. (62) i9{t,z) = z,
\/zedQ,\/te
[0,1].
(63) i?(0,z) = z,VzGQ. We note that 'd := id e 6 . Moreover, f3{t,'d{t,u)) G 6 for each 1^,(3 e S. We also recall the following proposition. Proposition 7.1. Le^ E be a Hilbert space and let P : E ^ E be compact. Then for any £ > 0, there exists a PQ : E ^ E such that PQ is compact, locally Lipschitz continuous and | | P ( ^ ) - P o ( ^ ) | | <s,
\/ueE.
Proof. For any i^ G ^ , set Uu:={veE:
\\v - u\\ < 1, \\P{u) - P{v)\\ < s}.
Then {UU}U€E is an open covering of £^ and then has a locally finite refinement {Vi}. Let Pi{u) = dist(u, E\Vi),
7i(^t) = ^ ^ ^ ^ .
7.1. A LINKING THEOREM
161
Then {jiiu)} is a locally Lipschitz continuous partition of unity (cf. Proposition 1.3). By the construction, for each Vi we have di Ui e E such that {P{u) - P{u,),v)
< s\\vl
\/ueV,,veE;
V, C U^^.
Evidently, -f^{u){P{u) - P{ui),v) < e-i,{u)\\vl
\/u,v e E.
It follows that (P(u) -J2^iiu)Piui),v)
< e\\v\\,
Wu,veE.
i
Define i
Then PQ is a convex combination of compact mappings. One readily checks that Po satisfies all the requirements of this proposition. D Theorem 7.2. Assume that {Ai)-{As) S > g > 0 such that ^ A ( ^ ) ^ ^5 ^A(^)
^
Wz e S
Q^
VZ G
dQ
hold and that there exist constants
uniformly for A G [1, /^], uniformly for A G
[1,K];
then for almost all A G [l^f^], there exists a bounded sequence {zn} such that ^ ^ ( z , ) ^ 0;
^A(^n) ^ ^A := inf sup
^ A ( ^ ( 1 , Z))
> S.
Hence, ^\ has a positive critical value > 5 for almost all A G [I, K]. Proof. We first show that (7.4)
i9{l,Q)nS^9,
Wee.
For any 'd{t, z) = exp C^uji{t, z)Os,)z + W{t, z) G 6 , where (jOi{t^ z) > 0, 5^ G [1, ^i^] for z = 1 , . . . , n-,^, define w{t,z) :=
^Ui{t,z). i=l
162
CHAPTER?.
SYSTEMS
WITH HAMILTONIAN
POTENTIALS
For any fixed (t, z), if w{t, z) = 0, then ^{t, z) = z -^ W{t, z); if w{t, z) ^ 0, we let K{t,z) :=
(^uji{t,z)si)/w{t,z).
Then K{t,z) G [l,n] and '^(t, z) = exp(tx7(t, ^)OA(t,^))z + W{t, z). Define 5 := B{t,s,z)
:= P~5]"^exp(^tx7(t, ^2(5^0 + z))OA(t,52(5zo+^)))^2,
for t G [0,1], 5 > 0, z G E~. By assumption (A3), 5 is invertible for any (t, 5, z) e [0,1] X [0, a]xE~. Consider the map H{t, 5, z) : [0,1] x [0, a]xE~ -^ E defined by H{t, 5, z) := stzQ + B-^p-B:[^W{t,
^2(5^0 + z)).
Then H is compact. Let
B^, := {zeE-:
\\z\\ < M}
and define Kt : [0, cr] x B^ ^ R x ^ as follows: i^t(5, z) := ( t | | 5 f i/i(t, ^2(5^0 + z))\\ + 5(1 - t),
z + y(t, 5, z)) .
To prove (7.4), it suffices to show that the equation Ki{s,z) = (p, 0) has a solution in [0, cr] x B^. Obviously, the operator Kt is a compact perturbation of the identity which has the following properties: Ko{s,z) = {s,z), i.e., KQ = z(i. Moreover, for any (5, z) G ^([0,cr] x 5 ^ ) , ^2(5^0 + ^) ^ dQ and hence iy(t, ^2(5^0 + z)) = 0,
i^(t, ^2(5^0 + z)) = B2{szo + z).
If Kt(5, ^) = (p, 0) for some (5, z) G ^([0, cr] x 5 ^ ) , that is, 5tZo + Z = 0,
t||5i-^52(5Zo + ^)|| + 5(1 -t)
= p,
then we get a contradiction since 0 < p < a. By the properties of the LeraySchauder degree: deg(K,,[0,(7] x 5 ^ , ( p , 0 ) ) = deg(Ko,[0,(7] x 5 ^ , ( p , 0 ) ) = deg(z(i, [0, cr] X 5 ^ , ( p , 0 ) ) = 1.
7.1. A LINKING THEOREM
163
It follows that the equation Ki{s,z) = (p, 0) has a solution in [0,cr] x 5 ^ , which implies (7.4). Obviously, by (7.4), we see that dx > S > 0 uniformly for A G [1,K].
_ ddx Since the mapping X ^ dx is non-decreasing, the derivative d^ : ^ ' dX exists for almost every A G [1,/^]. We just consider those A where (i^ exists. For a fixed A G [1,/^), let A^ G [l,/^],An > A and A^ ^ A as n ^ oo, then there exists n(A) such that (7.5)
^1 - 1 < "^^ ~ t^ < d'x + 1. for ^ > ^(^)An — A We show, for almost all A G [1,/^], that there exist 'dn ^ B,/i:o := /i:o(A) > 0 such that (7.6)
||^n(l,^)|| < ko
whenever
^xiM^.u))
> dx - (A^ - A).
For this, by the definition of dx^, there exists i^n ^ ^ such that (7.7)
SUp^A(^n(l,^)) < S U p ^ A j ^ n ( l , ^ ) ) < ^ A . + ( A n - A ) .
ueQ
ueQ
If ^A('^n(l,'^)) ^ dx — (An — A) for some i^ G Q, by assumption (^i), (7.5) and (7.7), we have that (7.8)
(0(l)i^n(l,^),^n(l,^))<^l+3.
Therefore, (7.9)
(0<2^^„(1, w), ^„(1, u)) + $(u) < K(d^ + 3) - dA + 'J
and (0(2)t?„(l, u), t?„(l, u)) + $(u) > -dA - K\d'^ + 2|.
(7.10)
Combining {Ai) and (7.8)-(7.10), we may find a A:o(A) := /CQ, which depends only on A, such that ||'^n(l5'^)|| ^ ^o- By the choice oi'd^ and (7.7), it follows that (7.11) ^ A ( ^ n ( l , ^ ) ) < ^ A j ^ n ( l , ^ ) ) < G^A. + ( A n - A ) < (iA + ( ^ + 2 ) ( A n - A)
for all u ^ Q. For each e > 0, let (7.12)
n,{X) :={ueE:
\\u\\ < A:o + 3, | ^ A ( ^ ) -
^A|
< s}.
164
CHAPTER?.
SYSTEMS
WITH HAMILTONIAN
POTENTIALS
By the definition of dx and (7.6), it is easily seen that ^^(A) ^ 0. Now we claim, for £ > 0 small enough, that (7.13)
•mi{\\$\{u)\\:uen,{X)}
= 0.
If this were not true, then there would exist an SQ > 0 such that (7.14)
Wx{u)\\>eo,
V^*ef2,„(A).
We may assume that SQ < min{2/9, {5 — ^)/4}. Choose n so large that (7.15)
max {((i^;, + 3)(A^ - A), (A^ - A)} < 3sl/4.
Since ^^ is compact, by Proposition 7.1, there exists an operator WQ : E ^ E which is compact and locally Lipschitz continuous such that (7.16)
ll^^(^) - 1^0(^)11 < ^o/4,
yueE.
Xx{u):=Ox{u)^Wo{u),
VueE.
Set (7.17)
Then Xx is locally Lipschitz continuous and, by (7.14)-(7.17), has the following properties:
(7.18)
M ^
> W,iu)\\ + I > ||X.(.)|| > W,iu)\\ - f > 1^
and (7.19)
{^'x{u),X,{u))>l\\'^'x{u)f
for all u G flsoW- Set . . ^ ^ '
(7.21)
J I
ueE:
B:={ueE:
either lli^ll > A:o + 2 or ^ A ( ^ ) < ^A - f or ^x{u) >dx^f
1 / '
\\u\\ < A:o + 1, | ^ A ( ^ ) - ^A| < ^o/4}.
Then B C O^o(^) and A n 5 = 0. Let u;{s) := 1 for 5 G [0,1] and u;{s) := 1/s for 5 > 1. Set dist(i^. A) X{u) := dist{u,B) +dist('U,A)'
7.1. A LINKING THEOREM
165
and consider the vector field (7.22)
Xl{u):=x{uM\\Xx{u)\\)Xl{u).
Note u^ A impfies that u G ^soWthat
By (7.18), (7.19) and (7.22), we conclude
(7.23)
and ||X^(^)|| < 1
{^'^{u),Xl{u))
>0
for ah u e E.
Moreover, for any u e B C ^eoW^ by (7.14), (7.18) and (7.19), we observe that (7.24)
{^\{u),Xl{u))
> min{3£^/4,3£o/5} = Self A.
Let (3x G C([0,1] X E^E) be the unique solution of the initial value problem df3x{t,u) dt
-X^(/3A),
/3A(0,^)=^.
By (7.23), we see that (7.25)
^^A(/3A(t,.)) ^ ot
^^^
^
Moreover, /3A has the following expression: (7.26)
I3x{t,u) = e x p ( ( y " -x{Px{s,u)M\\Xx{l3x{s,u))\\)ds)Ox)u
+ Wx{t,u),
where W\ is a compact map and Wx{t,u) = - j Jo
~e{T)x{lix{T,u))Lo{\\Xx{(3x{T,u))\\)W{f3x{T,u))dT,
e(r) := [exp(^ j ^
-x{Px{s,u)M\\Xx{Px{s,u))\\)ds)Ox\.
Define (7.27)
Plit,u)
:= pxit,Mt,u)),
V i e [0,1],
^ueE.
For any u G dQ, then f3^{t,u) = f3x{t,u) and ^A('^) ^ Q < dx — £o/3. Hence ueA (cf. (7.20)) and X*(^) = 0. Moreover, by (7.25), ^xiMt^u))
< ^ A ( / 3 A ( 0 , ^ ) ) < ^ A ( ^ ) < ^ < ^A - ^o/3.
166
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WITH HAMILTONIAN
POTENTIALS
Consequently, (3x{t,u) G A,X'^{(3{t,u)) = 0. It follows that (3x{t,u) = u, W\{t^u) = 0 and therefore, that /3^ G 6 . For any u e Q, we consider two cases: If ^A('^n(l,'^)) ^ dx — (A^ — A), then (7.28)
^ A ( / 3 J ( 1 , ^ ) ) < ^xiMhu))
If ^A(^n(l,^)) > ^A - (An - A), then \\M^,u)\\
{Xn - A).
< ko by step 2. By (7.15),
sup ^A(^n(l, ^)) < (iA + (2 + d'x){Xn -X)
(7.29)
£o/6,
uEQ
and it follows that '^^(1,'^) ^ ^£o{X). Furthermore, we assume that ^x{l3l{l,u))>dx-{\n-X).
For t e [0,1], by (7.25) and (7.29), (7.30)
dx - (A„ - A)
(7.31)
<^xiM'^,u)) By (7.23), ||/3A(t,^„(l,w))-^„(l,w)||<
f
\\X*x{l3x{t,'dn{hu)))\\dt
Jo
and hence, (7.32)
||/3A(i,^n(l,w))|| < t + | | ^ „ ( l , ^ t ) | | < l + fco f o r t e [0,1].
Combining this with (7.31), we see that /3x{t,'&n{^,u)) G B (see (7.21)) for t e [0,1]. Invoking (7.24), we have that
^xiPxit,M'^,u)))
-
^xiM'^,u))
r' d$x{Ms,M^,u))) ^ ds
I
0
ds '
V;,(/3A(S,^„(1,W))),
Jo ^ 4
•
X*x{px{s,Mhu))))ds
'
7.2. HAMILTONIAN
ELLIPTIC SYSTEMS
167
By (7.15) and (7.32), ^A(/3I(1,^)) 3.^ <^A(^n(l,^))
(7.33)
However, (7.33) and (7.28) contradict the definition ofdx. This completes the proof. D Notes and Comments. The earliest theorem related to Theorem 7.2 was established in P. Felmer [152] under the assumption of the (PS) condition. The generalized result of [152] was applied to study the periodic solutions and heteroclinic orbits for Hamiltonian systems (cf. P. Felmer [152, 153]) and elliptic systems (cf. J. F. Bonder-J. P. Pinasco-J. D. Rossi [60] and D. G. de Figueiredo-P. Felmer [159]). Let H : E ^ H he Si C^-functional having the structure ^{u) = {Ou,u) -\- b{u), where O : ^ ^ ^ is a linear, bounded, selfadjoint operator; b' is compact. If we assume that there exist a subspace ^ C ^ , a set 6^ C ^1 and a bounded set Q C ^ , then a well-known linking theorem due to V. Benci-P. Rabnowitz [55] and Rabinowitz [293] says: • if there exist constants a > co such that inf 5 ^ > a > co > sup^g H and that S and dQ link, then ^ has a critical value c > a whenever ^ satisfies the (PS) condition. This result has widespread applications. See, for example, D. G. Costa-C. A. Magalhaes [109, 110], J.Hulshof-R. van der Vorst [189], E. A. Silva [338] and M. Struwe [352]. Proposition 7.1 is essentially due to V. Benci [51] (see also P. Rabinowitz [293, page 87]). Theorem 7.2 was obtained by W.Zou-S.Li in [395].
7.2
Hamiltonian Elliptic Systems
Consider the following elliptic system /^ 2^x
( Au = u Av = V
in O, in O,
with nonlinear boundary conditions
(7.35)
1^ =Hv{x,u,v)
on aO,
^dv = Hu{x,u,v)
on dVt,
;
168
CHAPTER?.
SYSTEMS
WITH HAMILTONIAN
POTENTIALS
where O is a bounded domain of R ^ with smooth boundary; Hu {^v) denotes the partial derivative with respect to u (resp., v). We introduce the fohowing assumptions. (Bi) There exist p > O^q > 0 such that
\nu{x,U,v)\
< C(l + I^P + |^|P(^+1)/(P+1)),
\ny{x,u,v)\ < c(i + |^|^(^+i)/(^+i) +1^1^) for ah (x, u,v) e dft
xHxH.
(B2) There exist a > 0, f3 > 0 such that ,. . r l-iu{x,u,v)u^l-Lv{x,u,v)v-21-L{x,u,v) lim mt ^ -—^
^ > c> 0
uniformly for x G O. Moreover, \1-L{x^u^v)\ < c{\u\^ + l^;!^) for \{u^v)\ small enough. (B3) p + 1 > Q^ > p,
q^l>(3>q,
.p max {•a
1 > 1/Q^ + 1//3 and
q q{p^l)
v{q^l)
+^ ; ^ 7 +^ l) ^ + I3{p ^ ^+ ^l)' } < i P'a{q
1 N-I'
If A/^ > 4, we assume that p q q{p + 1) p(g_+l) m a x {'a'- ; -p'; ———-; a{q + l)' ———-} p{p + l)' <
A^ + 1 2{N-1)'
Hypothesis (B3) implies that there exist s and t with s,t > 1/4, s + t = 1 such that r7^fi^ ^^••^^>
f7 S7^ ^ • ^'
« - P ^ l ^ r ^ 2
25 - 1 / 2 iV-1 '
/3-« 1 ~ ^ ^ 2
1 - P(g + ^) > i _ 2^ - V^ P{p + 1) 2 N-1 '
2t - 1 / 2 IV^T'
1 _ g(P + l) ^ i _ 2^ - 1/2 a{q + l) 2 N - l '
Theorem 7.3. Assume that Ti : dO, xltxR ^ R satisfies {Bi)-{B3). Then the elliptic system (7.34)-(7.35) has a nontrivial weak solution {u, v) satisfying {u,v) e PF2,(9+i)/9(f2) X PF2,(p+i)/p(f2)^
7.2. HAMILTONIAN
ELLIPTIC SYSTEMS
169
Let A : V{A) C L'^{n) x L'^{dn) -^ L'^{n) x L'^{dn) be the operator defined by dz A{z, z\dQ) := ( - A z + z, — ) , where P(^):={(z,z|ao):zGi/2(0)}. Then V{A) is dense in 1/^(0) x L'^{dQ). The proof is standard. Indeed, let (a, 6) G C(0) X C(^0). For any e > 0, we may find a i^ G (^^(O^) such that ||i^ — a||L2(Q^) is small, where O^ = {x G O : dist(x,^0) > e}. Since ^O is smooth, we can extend i^ to O in such a way that u G C^(0) and that \\^-HL^{dn) is alsosmah. Note that C(0) xC(^O) is dense in 1/^(0) x 1/^(^0), and so is V{A). A is invertible and A~^ is given by A~^{a,b) = {u,u\dQ) with —Au -\- u = a in O,
du —— = 6 on dfl. Of]
By standard regularity theory, A~^ is bounded and compact. By Green's formula, it is easy to check that {Au,v) = {u,Av),
{Au,u) > 0.
That is, ^ is a positive and symmetric selfadjoint operator. Therefore, there exists a sequence of eigenvalues {A^} C R ^ of .4 with eigenfunctions ((/)^, tjjn) G L^{n) X L^{dn) such that 0 < Ai < A2 < • • • < A^ • • • / oo;
(7.38)
cl)neH^{n),
cl)n\dQ=^n;
(/>! > 0 OU O .
From (7.38), we know that {A^} C R ^ and {(l)n,i^n •= ^nldn) ^ 1/^(0) 1/^(^0) are the solutions of the eigenvalue problem -A(j)n
^
+ (/)n = An(/)n
= An(/>n
in O,
on
on.
For 1^ := 2^ ^n{4^ni V^n) ^ 1/^(0) X L'^(OQ) and 5 G (0,1), define the operator n=l
A' :V{A')^L^{n)
xL^{dn)
170
CHAPTER?.
SYSTEMS
WITH HAMILTONIAN
POTENTIALS
with n=l
Let E^ := V{A^)^ which is a Hilbert space with the inner product and norm 1 /2
{Z,W)ES
= {A^z^A^w),
\\Z\\ES = {z,z)^s ,
where (•, •) is the inner product of 1/^(0) x L'^{dft) given by JQ
JdQ
By the results of J. Thayer [369, p. 187] (see also J. F. Bonder-J. P. Pinasco-J. D. Rossi [60, Theorem 2.1], J. L. Lions-E. Magenes [226] and M. E. Taylor [366]):
E^^L^ioni
if.>i, p>i,
l>^^y
Furthermore, the inclusion is compact if the above inequality is strict. Let E := E^ X E^^ where 5 and t come from (7.36)-(7.37). Then ^ is a Hilbert space with norm || • H^; induced by the inner product (7.39)
{{u,v), ((/), V^))^ = {A'u,A'(l)) +
{A'v.A'^).
Moreover, E has a natural orthogonal decomposition E := E^ 0 E~ ^ where ^ + :={{u,A-^A'u) E- := {{u, -A-^A'u)
:ueE'}, : u G E'}.
We introduce the projections P^ : E -^ E^ given by (7.40)
P^{u,v) :=
hu±A-'A^v,v±A-^A'u).
Consider the operator C : E ^ E defined by (7.41)
C{u,v) := {A-'A^v,
A-^A'u).
Write z := (u^v) e E diS z = z~^ -\- z~ with z^ G ^ ^ , then Cz = z~^ — z~. Consider the functional ^ : ^ ^ R defined by (7.42)
^Z):=1{CZ,Z)E-
^
[
Jan
n{x,u,v)
:= \\Z^\\E - \\Z-\\E
- ^ {z).
7.2. HAMILTONIAN
ELLIPTIC SYSTEMS
171
Then, by a standard procedure, ^ is of C^ and ^^ is compact. The derivative of ^ is given by (7.43)
{^\u,v),{cl>,ij))E Jan
Jan
We say that {u, v) e E^ x E^ is an (5, t)-weak solution of (7.34)-(7.35) if {u, v) is a critical point of ^ . Lemma 7.4. If {u,v) is a critical point of^,
then {u,v) G iy^'(^+^)/^(0) x
iy2,(p+i)/p(^)^
Proof. First, j Hu{x,u,v)(t)j Hy{x,u,v)^lj = {) Jan Jan for any ((/>, ip) G E. We choose V^ = 0, (/> G H'^{ft). Then we have (7.44)
{A'u,A^^lj)^{A'(t),A^v)-
(7.45)
(^"(/),^^i;) - / Huix,u, v)(j) = 0, Jan
(7.46)
(^^(/),^M = M^,^) = / (-A(/> + (/>)^ + / ^ ^ . Jn Jn <^V
By (5i),
nu{x,u{x),v{x)) G /:(^+^)/^(6>o). By the basic elliptic theory, we have a i(; G iy2,(p+i)/p(0) such that Aw = w, in O;
^ — = 7Yti(x,i^(x),'u(x)) on ^O.
Thus, (7.47)
0
=
{-Aw^w)(l) Jn
i(;(-A(/) + (/)) + / w/ Hu{x,u,v)(j). n Jan ^V Jan By (7.44)-(7.47), we see that {v - w^Acj)) = 0. Hence v = w. Consider $AW
:=
Ai||^+|||-i||^-|||-^^W(x,u,z;) an 2
D
172
CHAPTER?.
SYSTEMS
WITH HAMILTONIAN
POTENTIALS
where (7.48)
Cxz = Xz^
-Z-.
In view of condition (^3), we may find /i, z/ > 1 such that /i ^ u, fi -\- u < min{/iQ^, Z//3}. Define Bi{u,v) = {p^-^u,p^-\),
(7.49)
B2{u,v) =
{a^-\,a^-\),
where p G (0,1) and a > 1 will be determined as needed according to different situations. Then Bi^B2 : E ^ E are linear, bounded and invertible operators. Lemma 7.5. Given any cj > 0, A > 1, the operator 5(A,C) := P-B^'exp{CCx)B2
: E' ^ E'
is invertible. Proof. For z = {zi,Z2) ^ E, we write z = z~ -\- z~^ with z^ G E^. Then by a simple computation, (7.50)
^,^^z.±Ayz,^z.±Ayz.^^
Let Pi =
(A" + ( - l ) " ) z i + (A" - (-l)")yl-M*Z2 ^ , (A" - {-!)'')A-*A"zi
P2 =
+ (A" + (-1)")Z2 ^
.
Then, by (7.48), £^(z) = {pi,P2). Hence,
exp(CA) = ( . . . . ) ^ + ( . . . . ) ^ +(^-M*.2,^-M^.0(^^ - ^ ^ ) . If z = (i^,-^~M^i^) G ^ ~ , where i^ G ^^, we have by (7.49) that B2Z {a^-^u,-a''-^A-^A'u) and B-^eyip{CCx)B2Z := (qj), where a^-i(exp(AC) + exp(-C)) - a--i(exp(AC) - exp(-C)) Q = f/ =
^
^
'^5
and - a - - i ( e x p ( A C )+exp(-C)) +exp(•. _^ -a-^(exp(AC) + a^-^(exp(AC) -exp(-C))^^_,^^,^^ 2p'
7.2. HAMILTONIAN
ELLIPTIC SYSTEMS
173
By (7.40), it is easy to calculate that p-B^^exp{C/:x)B2Z ~ 2V^^^ >0.
^ y ^ '
2
^^^^
^ ^^^^^^
2
)
Therefore, B{XX) is invertible.
D
Lemma 7.6. T/iere exist p G (0,1) and S > 0 such that ^x{z) > S for all z e S and A G [1,2], where S := {Bi{u,v) : \\{U,V)\\E = p, (t^,'^) G ^ + } . Proof. For z = (i^,^~M^i^) G ^ + , write z := Biz = {p^-^u,p^-^A-'A'u)
:= z+ + z - .
Then by (7.39), ||z||| = 2M^^,^^^) = 2||^|||.. By (7.35), C\z = / (A - l)p^'-^u + (1 + \)p''-^u ((A - l)p'^-^ + (1 + V 2 ' 2
\)p^'-^)A-*A'u ).
and it follows that (7.51)
iCxz,z)E = {A'U, A'U) ((1 + A)p^+'^-2 + i ^ ^ (p2(M-l) + p2(.-l)) j ^
By iBi)-iBs) *(2)
(7.52)
and (7.36)-(7.37), <
c(/9(^-i)"||z||g +
p(''-^)(^+^)||^||?;+Vp(^-i)'^||^|lUp^"'-^^^^+^^INr/0-
By (7.51) and (7.52), for H^H^; = p small enough, there exists a (5 > 0 such that ^ A ( ^ ) > S for ah A G [1, 2]. D Lemma 7.7. There exist cr > 0,M > 0,£o > 0 5^c/i ^/la^ ^ A ( ^ ) < ^/2 uniformly for (A, z) G [1,1 + ^o] x dQ and Q := {52(rzo -\- z) : 0 < r < a, \\Z\\E < M^Z e E~} for some ZQ := (i^o^'^o) ^ E~^ '^'^th \\ZO\\E = l;'^o •= 4^1-
174
CHAPTER?.
Proof. (7.49),
Write z : = {u,-A-^A'u)
B2{rzo^z)
SYSTEMS
WITH
HAMILTONIAN
POTENTIALS
G E', ZQ : = {uo,A-^A'uo)
= (a^~'^{TUo^u),a''~'^A~^A^{TUo
G E^.
By
- u)) := z := z^ ^ z~,
where z^ G E^. By a simple evaluation, we have t h a t z+ : = (7ri,7r2) with ^' =
2 (J^-^A-^A'{TUQ
' + ^) + C r ^ - M - M ^ ( ™ o - U)
^2 =
^
,
and z~ : = (7r3,7r4) with (J^-^{TUQ
-^U)
^3 =
_
7r4t -
- (J''-^{TUQ ^
-G^-^A-^A'{TUO
-
u) ,
+ ^) + C r ^ - M - U " ( ™ o - ^)
^
•
By (7.39),
11^-^111 = 1 -(a^-^A'(TU^ (J^-^A'{TUQ
^u)± ^U)±
( 7 " - U ^ ( ™ o - u), (J^-^A'{TUQ
-
^)).
We m a y assume t h a t A G [1,2]. Note t h a t r < cr, 2(^^1^0,-4^'^o) = lko|||;, | | ^ | | | 2{A'u,A'u) and
We therefore obtain, -(£A^,2)i<; = ir2(A(<7(^-i) + < T ( ' ^ - I ) ) 2 - K " ! - a^-'f)
- K - i +
+\ ( A K - 1 - a^-'f +(A 4
(7.53)
\\z4Ill
1)T((72(''-I)-a2(--i))(^^uo,^'u> 2
+(A-1)((T2''-I+<72'^-1)||Z|U.
7.2. HAMILTONIAN
ELLIPTIC SYSTEMS
175
By condition (^2), (7.54)
/
n{x,z)>c(
[
an
{a^^^-^^\ruo^u\^^a^^''-^^\rvo^vf)-c).
^JdQ
Write 1^ as 1^ := 71^0 + 'u, where u is orthogonal to i^o ^^ L'^ {dft)-sense and 7 G R. By Holder's inequality, we get that (7.55)
(T + 7 ) /
|t^oP = /
(^ - 7) /
I'^oP
{ruo^u)uo
and (7.56)
{rvo + v)uo < c\\rvo + v\\p.
By (7.53)-(7.56),
(7.57)
-cimin{r^(7^(^-i\r^(7^(^-i)} + C2.
Since fi-\- u < minjo^/i, /3z/}, we choose a > 1 so that (7.58)
^(7^+" - ci min{(7^^, a^"} + C2 < 0.
Then we take R > 1 so large that (7.59)
^(7^+^ - (7^+^-2i?2 + C2 < 0.
Ur = a and ||z||£; < i?, then by (7.57)-(7.59), we obtain
(7.60)
+(A - l){a^^-^ +
a^^-^)R.
If IIzII£; = i?, 0 < r < cr, then by (7.57) it is easy to see that ^x{z)
<
^ ^ ( ^ 2 ^ + ^^-) + ^ ^ ^ + ^ A ^ .
2M-2 ^ ^ 2 . - 2 ) ^ 2 _ ^M+--2^2
4 (7.61)
+(A - l){a^^-^
+ (72--i)i? + C2.
176
CHAPTER
7. SYSTEMS
WITH HAMILTONIAN
POTENTIALS
If r = 0, ll^ll^;
^ A ( ^ ) < \ ( A K " ' - a^-^f
- K"' +
a^-^f^R.
Combining this with (7.58)-(7.62), we see that there is an SQ > 0 such that ^ A ( ^ ) < ^/2 uniformly for (A, z) G [1,1 + £o] x dQ. D Proof of Theorem 7.3. By Lemmas 7.5-7.7 and Theorem 7.2, for almost all A G [1,1+ So]i we find a critical point sequence {zx} such that ^A(^A) ^ (5 > 0, ^A(^A) = 0- Since Q is bounded uniformly for A G [1,1+ ^o], we may assume that ^A(^A) < S*, where S* is a constant independent of A. Therefore, there exist sequences A^ G [1,1+ ^o] with A^ ^ 1 and z^ ^ E satisfying ^^ (zn) = 0,^A^(^n) ^ [^5*^*]- It suffices to prove that {z^} is bounded. Write Zn := (un^Vn) := ^^ + z~ with z^ G ^ ^ , then (7.63)
Z^ = -(^Un±A
'A^Vn,Vn±A
M'^n) •
By (^2),
(7.64)
5*
>
HxAZn)--{Hl{Zr,),Zr,)E
> T; I (Kr + K H - c . Since ^^ (z-^) = 0, by (7.63), we have (7.65)
K\\Z:^\\E
=
7,
nu{x,Ur^,Vr^){Ur^^A
'A'Vr^)
^ Jan ^nv{x,Un,Vn){vn^A~^A^Un). By assumption (5i), (7.36)-(7.37), (7.66)
l-Cu{x,Un,Vn){Un^A
^A^Vn)
an
+C b^
MSn|U/(a-p)+c||^n+^
||P(.+l)/(p+l)n
'^X||l
, ^-.^t^^ 'n||/3(p+l)/(/3(p+l)-p(Q+l))
M^i;n|U^
and (7.67)
ny{x,Un,Vn){vn^A an
^A'un)\
^A^U^WE^
7.2. HAMILTONIAN
ELLIPTIC SYSTEMS
177
Combining (7.64)-(7.67), we obtain
\\zt\\l
^
Un\\E^j
^
and it follows that {||z+||£;} is bounded. In a similar way, {||z~ H^;} is bounded. By standard arguments, ^ has a critical value in [(5, 6^*]. D
Notes and Comments. We refer the readers to D. Gilbarg-N.S. Trudinger [174] for the standard regularity theory. The existence of solutions for nonlinear elliptic systems has created a great deal of interest in recent years, in particular when the nonlinearities appear as a source in the systems with Dirichlet boundary conditions (see e.g. S. Angenent-R. van der Vorst [20], T. Bartsch-D. G. De Figueiredo [33], a survey of D. G. de Figueiredo [158], L. Boccardo-D. G. de Figueiredo [62], Ph. Clement-D. G. de Figueiredo-E. Mitidieri [104], D. G. de Figueiredo-C. A. Magalhaes [160], D. G. Costa-C. A. Magalhaes [109], D. G. de Figueiredo [157], D. G. de Figueiredo -P. Felmer [159], P. Felmer-R. F. Manasevich-F. de Thelin [154], D. G. de Figueiredo-E. Mitidieri [162], M. F. Furtado-L. A. Maia-E. A. B. Silva [166], J. Hulshof-R. van der Vorst [189] and W. Zou [387]). We also note, by using fixed-point theory, J. F. Bonder-J. D. Rossi [61] considered (7.34)-(7.35) without variational assumptions. J. F. Bonder-J. P. Pinasco-J. D. Rossi [60] considered (7.34)-(7.35) by using variational methods where the variational structure was established and one nontrivial solution was obtained under much stronger assumptions; their abstract theory was a theorem of P. Felmer [152] with the (PS) condition. The theorem of this section was established in W. Zou-S. Li [395].
Chapter 8
Linking and Elliptic Systems The theory of Chapter 7 was estabhshed to fit the systems with Hamiltonian type potentials. In this chapter, a linking theorem will be given which is designed for systems with gradient type potentials.
8.1
An Infinite-Dimensional Linking Theorem
Let ^ be a real Hilbert space with an inner product (•, •) and associated norm II • II. We assume that E has an orthogonal decomposition E = E~^ ®E~ with both E^ and E~ infinite-dimensional. A and B are two given subsets of E. Let A link B in the sense of Definition 5.1 with respect to T in Section 5.L Let n be the set of those continuous mappings a{t,u) = h{t,u)u + c(t,u),
u e E, t e[0,1)
such that (1) 6(-, •) G C([0,1) X E, B{E)), where B{E) denotes the set of the bounded linear operators from E to E; ' G ^ , b{t, v) is a linear homeomorphism of E onto E (2) for each t G [0,1), U and E~ onto E~ ; (3) for each t G [0,1), b{t, •) G K{E, B{E)); (4)
c{t,u)eC{[0,l)xE,E);
(5) for each t G [0, l),c(t, •) is compact on E.
180
CHAPTERS.
LINKING AND ELLIPTIC
SYSTEMS
Let T* denote the set of those F G T such that F and F~^ are both in H. Then F(t, ^) = (1 - t)u + tuo G T*. Definition 8.1. A set A C E links a set B C E with respect to T* ifAnB and for each F G T*, there is a t e [0,1] such that F(t, A) n 5 ^ 0.
=0
By Propositions 5.4 and 5.5, we have the fohowing two examples. Example 8.2. Let VQ be a vector in ^ + \ { 0 } . Let 0 < p < R, Q := {u = V ^ svo : V e E~, 8 > 0, \\u\\ < R}, A:=dQ,
B:={ueE^
: \\u\\ = p}.
Then A links B with respect to T*m the sense of Definition 8.1. Example 8.3. E~ r\ OBR links E^ with respect to T* for every R > 0 in the sense of Definition 8.1.
Let m > 1 and {^A} be a family of C^{E, R) -functionals having the form ^x{u) :=-{Cxu,u)^^{u),
AG [l,m],
and satisfying the following conditions. (Ai) There are two linear, bounded and selfadjoint operators £^^\ £^^^ : E -^ E such that Cx = A/:^^^ - C^'^^ with A G [l,m], where (C^^K.u) > 0, for all 1^ G ^ and either {C^^^u^u) ^ oo or ^ oo as
ll^ll ^ o o . (A2) ^^ is compact. (A3) Cx is invertible for all A G [1,7TI]. Theorem 8.4. Let E~ be an invariant subspace of Cx^ and assume that A,BcE,A links B with respect to T*, and A is bounded. Suppose that ao(A) := sup^A < ^o(A) := inf ^A, A
VA G [l,m],
^
and ax '•= inf sup ^A(r(5,i^)) < 00. Then for almost all A G [1,7TI], ^^^* ueA,se[o,i] there exists a bounded sequence {zn} such that ^^i^n) -^ 0 and ^xi^n) -^ ^A as n ^ 00. Moreover, if ax = ^o(A), we have that dist{zn,B) ^ 0 as n ^ 00. Finally, ^x has a critical value for almost all A G [1, m] and the critical point lies in B if ax = bo{X).
8.1.
AN INFINITE-DIMENSIONAL
LINKING THEOREM
Proof. Similar to the proof of Theorem 5.7, for almost all A G exist r ^ G T*, ko := ko{X) > 0 such that (8.1)
||r^(5,^)|| < ko
whenever
^x0^n{s,u))
181 [1,7TI],
there
> ax - (A^ - A),
where A^ G (A,m), A^ -^ A. By the definition of ax^, there exists a F^ G T* such that sup
(8.2)
^x0^n{s,u))
ueA,se[o,i]
(i) Assume bo{X) > ao(A). For any £ > 0, define (8.3) n,{X) :={ueE: \\u\\ < A:o + 3, | ^ A ( ^ ) - ^AI < s}. By the definition of aA, (8.1) and (8.2), it is trivial to check that ^^(A) ^ 0. Moreover, (8.4)
mi{\\^'x{u)\\:ueQe{\)}
=0
for £ small enough. To show this, we assume that there is an SQ > 0 such that
(8.5)
\\^'^{u)\\>eo,
yuen^oW.
Decreasing SQ such that SQ < min{2/9, {bo{X) — ao(A))/2}. Choose n so large that (8.6)
(|al|+3)(A„-A)<3£g/4.
Since ^^ is compact, there exists a compact and locally Lipschitz continuous operator WQ : E ^ E such that (8.7)
ll^^(^) - 1^0(^)11 < ^o/4,
V^ G E.
Xx{u) := Cxiu) + Wo{u),
\/u G E.
Set (8.8)
Then Xx is locally Lipschitz continuous and, by (8.5), (8.7) and (8.8), we have the following properties: (8.9) (8.10) (8.11)
\\Xxiu)\\ > W^iu)\\ - So/4 > 3eo/4, for all u e fi,„(A); {<^'^{u),Xx{u))>^^'^{u)f/4, forall^tef2,„(A); \\X,{u)\\<\\<S>',{u)\\+so/4<5\\^',{u)\\/A, V«ef2,„(A).
182
CHAPTERS.
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SYSTEMS
Let
{ (8.13)
either ||i^|| > ko -\-2 or ueE
: ^x{u)
^x{u) > ax ^ f
\\u\\ < A:o + 1, | ^ A H - ^AI < ^o/4}.
Bo:={ueE:
Then Bo C ^^^(A), AQ n 5O = 0. Let tlj{s) := 1 for 5 G [0,1] and tlj{s) := 1/s for 5 > 1. Set (8.14)
i){u):^
(8.15)
Xliu)
^"*("'^^) dist(i^, Bi) -\- dist(i^, Ai)' •.=
i9iuM\\Xxiu)\\)Xxiu).
Since u ^ AQ implies that u G ^^^(A), by (8.10), (8.11) and (8.14), we see that (8.16)
(^';,(^),X^(^)) > 0
and
||X^(^)|| < 1
for ah ^ G ^ .
Furthermore, for any u ^ BQ C ^^^(A), by (8.5), (8.10) and (8.11), we see that (8.17)
{^'^{u),Xl{u))>:isl/A.
Consider the following Cauchy initial value problem: with
/3A(0, U)
(8.18)
= u. The unique solution
/3A G C([0,
—^— = —X^{(3x)
1] x E, E) is given by
(3x(t,u) expl ^
i^(/3A(5,^))V^(||XA(/3A(5,^))||)(i5)/:A)^ + ^ A ( t , ^ ) ,
where Cx is a compact map. By (8.8), (8.14) and (8.16), we see that
(8.19)
^^^(^^f'^»<0, ^ueE,
te[o,i].
Define (8.20)
(3l(t)u:=(3x(t,Vn(t,u))
VtG[0,l],
^ueE.
Combining this with (8.18), we have /3J G T*. For any u e A, if ^A(rn(5, u)) < ax - {\n - A), then by (8.19) and (8.20) (8.21)
^x{Pl{s)u)
< ^x{Vn{s,u))
{K - A).
8.1.
AN INFINITE-DIMENSIONAL
If ^xO^nis.u))
LINKING THEOREM
183
>ax- {Xn - A), then ||r,(5,^)|| < ko. By (8.2) and (8.6),
sup
(8.22)
^A(rn(5,u)) < aA + So/6.
liGA, sG[0,l]
It follows that Tn{s,u) G ^^^(A). Further, we may assume that ^x{l3l{s)u)
>ax-{Xn-X)-
For t e [0,1], by (8.19) and (8.22), aA-(An-A) (8.23)
< < <
^xWl{s)u) ^xO^nis^u)) ax ^ So/6.
By (8.16), it easy to check that \\(3x{t,Tn{s,u)) — r^(5,i^))|| < t. Hence, \\f3x{t,Tr^{s,u))\\
(8.25) By (8.22)-(8.25), (8.26)
^xi(3lis)u)
Thus, (8.21) and (8.26) contradict the definition of ax- This completes the proof of case 1. (ii) bo{X) = ao{X).
Because of the boundedness of A, we have that dA '= max{||i^|| : u e A} < oo.
For any £ > 0, T > 0, we define (8.27)' ^
D{s,T,X):=lueE: V' ' ^ I
t\-^'''^^Wj'/% m ^ A^T ] • |^A('^) - «(A)| < 3£,(i(i^,5) < 4 r J
We now show that D{s^T^X) ^ 0. By (8.2), we choose n so large that (8.28)
sup sG[0,l],iiGA
^ A j r ^ ( 5 , ^ ) )
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CHAPTERS.
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SYSTEMS
Since A links 5 , there exists an (SQ, i^o) ^ [0,1] x A such that r^(5o, i^o) ^ B, that is dist(r^(5o,i^o),^) = 0 and (8.29)
^xiTniso.uo))
> bo{X) = inf^x
= ax > ax - 3s.
B
Thus, ||r^(5o,i^o)|| ^ ^0- Hence, r^(5o,i^o) ^ D{e^T^\). that (8.30)
inf{||^';,(^)|| : u G D{e,T,\)}
=0
It suffices to show
for e,T sufficiently smah.
If this were not true, there would exist ^c > 0, £i > 0, Ti G (0,1) such that (8.31)
11^1(^)11 > 3K.
for u e £)(£!, Ti, A).
Choose K < 1/4, {\a'^\ + 2)(An - X) < Si and (A^ - A) < /^Ti. Let Wo : ^ ^ ^ be a compact and locally Lipschitz continuous operator such that ll^^(^) - 1^0(^)11 < ^ for ah ueE. Let (8.32)
Xx{u) := Cxiu) + Wo{u),
\/u G ^ .
Then Xx is locally Lipschitz continuous. By (8.31)-(8.32), (8.33)
411^1(^)11/3 > 2/.,
for all u G D{£i,Ti, A) and (8.34)
($',(«),X,(^.)>>2||$',(^.)||V3
for all u e D{si,Ti,X).
Let tp{s) := 1 for s G [0,1], tp{s) := 1/s for s > 1 and Ux{u):=i;{\\Xx{u)\\)Xx{u).
Then it is easy to check that (8.35)
(^l(^), Ux{u)) > Qn\
W G 5 ( £ i , Ti, A).
Define
(8.36) ^ ^
D,:=LeE: [
j!^" ^ t5 t) -
Then Di ^ 9 and L)i C 5(£i,Ti,A). Choose a Lipschitz continuous map 7 from ^ into [0,1] which equals 1 on Di and vanishes outside 5(£i,Ti,A). Let Pxi^i"^) be the unique continuous solution of the Cauchy initial boundary value problem ^ ^ ^ ^ ^
=-l{h)UxCPxl
MO,u)
= u.
8.1. AN INFINITE-DIMENSIONAL
LINKING THEOREM
185
Then it is of the form I3\{t^u) = exp
u+
C\{t,u),
where C\ is a compact map. Moreover, (8.37)
j^
< -6/^ -f{f3x{t,u)).
(ii)-l: We claim that ^x{t,u) ^ B for ah t G [0,Ti] and u ^ A. In fact, for 1^ G A, by (8.37), we first observe that (8.38)
^x{Mt.u))
< ^x{u) < ao(A) < b^{\) = ax,
V t G [0,ri]
and (8.39)
^xiMt.u))
< ^x{u) - [ 6K^j{^x{cT,u))da,
Vt G [0,ri].
Jo
If (ii)-l were not true, then there would exist a to G [0, Ti] such that ^x{to,u) G B. Then ^xWx{to,u)) > ax = bo{X) = inf^^A- By (8.37) -(8.39), we must have / 6hi^j{f3x{o-iU))da = 0. Hence, 7(/3A(cr,i^)) = 0 for cr G [0,to], i.e., Jo f3x{cr,u) ^ Di for all a G [0,to]. We must have one of the following three cases: (8.40)
PA(^,^)||
>A:o + 2 + (i^,
(8.41)
|^A(^A(^,^))-a(A)|>2£i,
(8.42)
dist(;5A(cr,^),5) > 3Ti.
Note that \\^x{cr,u) - ^x{cr\u)\\ < \cr - a\. Thus we have PA(^,^)||
< P A ( 0 , ^ ) | | + T i
V(7 G [0,to].
Then (8.40) would not happen. If (8.41) holds, then ^x{Px{cr,u)) < «A - 2 ^ 1 . Hence, Px{o-,u) ^ B. Evidently, (8.42) implies that Px{o-,u) ^ B. Therefore, (ii)-l follows. (ii)-2: We prove that ;5A(TI,r^(25 - 1,i^)) ^ B for oil u e A and 5 G [1/2,1]. To prove this, we consider two cases. Take any fixed u ^ A and 5 G [1/2,1].
186
CHAPTERS.
If px{cr,Tn{2s -l,u)) that
e Difor
LINKING AND ELLIPTIC
SYSTEMS
all a G [0,ri], by (8.35) and (8.37), we have
^A(^A(Ti,r,(25-l,^)))
= ^ . ( r . ( 2 . - i , . ) ) + r-^^A(^.(.,r(2.-i,.)))^^ Jo <$A(r„(2s-l,u))- / Jo
+
which implies that px{Ti,rn{2s
da \K^j0x{'J,rn{'2s-l,u)))da
{a\+2){K-X), — l,u)) ^ B since ax = 60(A).
If there exists a to G [0, Ti] such that Px{to, r„(2s — 1, u)) ^ Di, then one of the following alternatives holds: (8.43)
p A ( t o , r „ ( 2 s - l , u ) ) | | >ko + 2 + dA,
(8.44)
\^xil3xito,Tni2s-l,u)))-ax\>2su
(8.45)
dist(^A(to,r„(2s - l,u)),B)
> 3Ti.
Suppose that (8.43) holds. If/3A(Ti,r„(2s - l,u)) e B, then 60(A) =ax<
$A(^A(Ti,r„(2s - l,u)) < $A(r„(2s -
l,u)).
Then ||r„(2s - l,u)|| < ko- Since \\^x{to,Tn{2s-l,u))-
^x{0,r„(2s
- 1, u))\\ < to,
we see that pA(io,r„(2s - 1, u))\\
+ l,
which contradicts (8.43). Hence, /3x{Ti,Tn{2s — l,u)) ^ B. Assume that (8.44) holds. Note, by (8.2) and the choice of £1, that $A(/9A(io,r„(2s - l),u)))
ei.
Therefore, (8.44) implies that $A(^A(Ti,r„(2s - l,u))) < a(A) - 2£i.
8.2. ELLIPTIC SYSTEMS
187
It follows that ^x{Ti,Tn{2s - l,u)) ^ B since a(A) = 60(A). Assume (8.45) holds. Note that \\Px{t,u) - Px{f,u)\\ < \t - f\. It follows that
>
PA
(to, r^(25 - 1, u)) - w\\ - \ t - tol
for SiW w e B,t e [0,ri]^ Hence, dist{^x{t,Tn{2s - l,u)),B) > 2Ti for ah t G [0,ri]. In particular, Px{Ti,Tn{2s — l,u)) ^ B. This completes the proof of (ii)-2. Define r ^x{2sT,,u),
0<5
[ Px{TuTn{2s-l,u)),
l/2<s
Then r | G T*. But, by (ii)-l and (ii)-2, Tl{s,A) n 5 = 0 for ah 5 G [0,1]. This is a contradiction which finishes the proof of the theorem. D
8.2
Elliptic Systems
Let .4 > /3o > 0, 23 > /io > 0 be selfadjoint operators on 1/^(0) with compact resolvents. Assume that C^(0) C V{A^^'^) C H'^^'^{n) and C^(0) C P(i3^/^) C H^''^{Q) for some m > 0, where O C R ^ is not necessarily bounded, (3o{fio) is the lowest eigenvalue of A{resp. B). Assume that the eigenfunctions of/3o(/io) are not equal to zero a.e. on O. Moreover, we assume that ll^ll, < c\\A^^^u\\2
for ah u G V{A^/^),
\\u\\q < c\\B^^^u\\2
for ah u G V{B^/^),
^.46)
where 2N 2 < ^ < 00,
Let F{x,s,t), fying
f{x,s,t),g{x,s,t)
N
<2m.
be Caratheodory functions on O x R^ satis-
,, , dF f{x,s,t) = — ,
^ ^ OF g{x,s,t) = — .
188
CHAPTERS.
LINKING AND ELLIPTIC
SYSTEMS
Assume (8.48)
F{x,s,t)
> 0, \f{x,s,t)\
+ \g{x,s,t)\ < c{\s\ + \t\ + 1),
for all 5, t G R, X G O. We solve Av = - / ( x , V, w),
(Sp)
Bw = (3g{x, v, w).
Theorem 8.5. Assume that ^—^
> (j)-^{x)v~^{x) — (j)-{x)v~{x),
as t ^ +00, y ^ v,
where a^ = max{±a,0}. Moreover, (/)±(x) > ^ -/3o,
2F(x,0,t)
X G 0 , t G R,
where W{x) G I/-'^(0). T/ien ^/le system (Sp) has infinitely many solutions
{p.v.w).
Theorem 8.6. Assume that f{x,0,tz) t
^-^{x)w~^{x) — ^-{x)w
(x),
as t ^ -\-oo,z ^ w.
Moreover, ^±(x) < ^ / i o ,
2F(x,5,0) > -/3o5^ - i y ( x ) ,
X G 0 , t G R,
where W{x) G 1/^(0). T/ien the system (Sp) has infinitely many solutions {f3,v,w).
Next, we consider the following assumptions: (Bi) F(x,0,t) = o(|tp) as t ^ 0 uniformly for x e n. (B2) F{x, 5, t) = -mo(|5p + |tp) + K{x, 5, t) > 0 for all (x, 5, t), where mo > 2/io and K{x^s^t) = o(|5p + |tp) as |5| + |t| ^ 00 uniformly for x G O.
8.2.
ELLIPTIC
SYSTEMS
189
T h e o r e m 8.7. Assume (Bi) many solutions {(3,v,w).
and {B2).
Then the system
(Sp) has
infinitely
T h e hypotheses of Theorems 8.5-8.7 which guarantee the existence of infinitely many solutions are quite weak. Unfortunately, these theorems do not give any information for any specific f3. We then t u r n our attention to solving (SIS) with (3 = 1. We need some further conditions. For z = (5,t), we write f{x,s,t) = f{x,z),g{x,s,t) = g{x,z),F{x,s,t) = F{x,z),Fz{x,z) = {dF/ds^dF/dt). We assume (B3) liminf I^I^O
Fzix z)z ^ —— > 70 > 2 uniformly for x G O. FyX, Z)
F (x z)z — 2F(x z) — ^— > c > 0 uniformly for x G O, where a G (B4) l i m i n f ^ — ^ \z\^oo
|z|^
(0,2). (B5) Fz{x, z)z — 2 F ( x , z) > 0 uniformly for x G O and \z\ ^ 0. (Be) liminf
Fix z) ; / ^ > c > 0,
T h e o r e m 8.8. Assume (v^w). solution
\o(x z)\ l i m s u p ' ^ \ ' / ' < c, where q satisfies (8.47).
{BI)-{BQ).
Then the system
(Si)
has a
nontrivial
A typical example for operators A and B is A = —A + Vi{x) and B = —A + V2{x) defined on I / ^ ( R ^ ) , where Vi{x) ^ 00 as |x| ^ 00, z = 1, 2 (cf. P. Rabinowitz [295]). Let E = V{^A^I^) X V[B^I^). Then E becomes a Hilbert space with norm given by ||i^p = {^Av.,v) + (Bw^w), u = {v^w) G E. We introduce t h e following functionals ^x{u)
= Xb{w) - a{v) - 2 / F{x,v,w)dx,
u e E, A G [1,2],
JQ
where a{v) = (^'U,'u), 6(i(;) = {Bw,w).
Then ^ A ^ C ^ ( ^ , R ) and
( ^ l ( ^ ) , /i)/2 = X{Bw, /i2) - ( ^ ^ , /ii) - ( / ( ^ ) , /ii) - (^(^), /i2), /i = (/ii, /12) G E. We write f{u),g{u) in place of / ( x , i;, w),g{x, v, w), respectively. It is readily seen t h a t ^xi^) = 0 is equivalent to t h e systems (6^/3) with /3 = 1/A.
190
CHAPTERS.
LINKING AND ELLIPTIC
SYSTEMS
Let E- = {{v,0) : {v,0) e E}, E^ = {{0,w) : {0,w) G E}. Then E~, E^ are orthogonal closed subspaces such that E = E~ ®E^. Define Cxu = 2{-v,Xw), \/u = {v,w) e E,X e [1,2]. Then Cx = A/:^^^ - C^'^^ is an invertible, selfadjoint bounded operator on E for all A G [1,2], where C^^^u = 2(0, w),C^'^^u = 2{v, 0). Also ^'^{u) = Cxu + ^'(i^), where ^\u)
=
-{A-'f{u),8-'g{u))
is compact on E.
Proof of Theorem 8.5. For {0,w) G E~^, we see that ^x{0,w)>Xb{w)-fio\\w\\l-
Therefore, inf ^x>
-
^+
f
W{x)dx.
W{x)dx for ah A G [1, 2]. We claim that JQ
sup
^A -
E-ndBR
—CO as i? ^ oo uniformly for A G [1,2]. Let {v^, 0) be any sequence in E~ such that P1 = a{vk) -^ oo. Then ^x{vk,0)/pi
= -a{vk)-2
/ Jn
F{x,Vk,0)dx/pi,
where v^ = v^/pk- Since a{vk) = 1, there is a renamed subsequence v^ ^ v weakly in E~, strongly in 1/^(0) and a.e. in O such that
^ - 1 = - f m
[ {
<j>.{x){v-{x)f)dx + (/^o +
cl^-{x)){v-{x)f)dx
This is less than zero unless /^oH'^lli = 1- Since a{v) < 1, this would mean that V G E{(3o), the eigenspace of/3o. Thus v ^ 0 a.e. by hypothesis. But then the integral cannot vanish since (/)± > ^ —/3o. Hence, limsup^A(05'^)/tt('^) < 0 a(v)^oo
uniformly for A G [1,2].
8.2. ELLIPTIC SYSTEMS
191
Thus, ^x has the hnking structure described in Example 8.3. By Theorem 8.4, ^x has a critical point for almost all A G [1, 2]. D
Proof of Theorem 8.6. It suffices to interchange the roles of E~^ and E in the proof of Theorem 8.5. D
Proof of Theorem 8.7. By (5i), for any £ > 0, there exists a c > 0 such that F{x,0,w) < £\w\'^ -^c\w\^, where q > 2 satisfies (8.47). Therefore, for any u = (0, w) G ^ + , for ||i^|| sufficiently small and all A G [1, 2], we have that
^AH
=
Xb{w)-2
F{x,0,w)dx
>
6H-2^11^11^-2c||^||^
>
c.
Choose Wo j^ 0 such that Bwo = /io'^o- Define A:=d{u
= u~ -^suo : u~ G ^ " , ||i^|| < i?, i? > 0, 5 > 0},
where i^o = (O^'^o)- We want to show that ^ A U ^ ^ ^^^ some i? > 0 for ah A G [1,2]. Note that ^ A ( ' ^ ~ ) < 0 for all u~ e E~. If this were not true, then there would exist a sequence u^ = S^UQ -\- u~ such that ||i^n|| -^ cxo and ^\{un) > 0. Write u^ = {vn.SnW^). Then
\h{SnWo)
- a{Vn) > 2 / F ( X , Vn, SnWo)dx
Since Ili^nP = b(snWo) -\- a(vn), we may assume that
> 0.
.. ^ ..^
-^ s*b(wo).
Then 5* > 0. Note that
2b{wo) — mo / \wo\ dx = 2fio / \wo\ dx — mo / \wo\ dx < 0. JQ
JQ
JQ
Thus, there is a bounded subset OQ of O such that 2b{wo) — m^o / ^ \wo\'^ < 0.
192
CHAPTERS.
LINKING AND ELLIPTIC
SYSTEMS
Then, Xb{snWo) _ a{vn) _ U
S:
11
..r,
II
119
2 II
26(g^^;o) _ a(i;n) _ —
<
II
119
2b{SnWo)
II
iL. 112
2 II
SfiWQJClX
r 119 /
-t^ [X^Vfi^ SfiWojCtX
a{Vn)
KP^o^2 -
119
r M O / -t^ [XjVfij
mo{\Vnf
II.. — 112 //
ll^nP JQO -^
2s*h{wo) — mos* I
<
0,
+ \SnWof)
^ K{x,
\SnWo\ k n ^ o l d^ x^ -- ^ — ^ '"nil 11^'
V^, SnWo) )dx
/
K(x,i;n,5n^o)^^
JQO
\wo\^dx
which yields a contradiction. We showed that ^x satisfies ah the conditions of Theorem 8.4 with respect to the linking of Example 8.2. Therefore, ^x has the linking structure of Theorem 8.4 and has a critical point for almost all A G [1,2]. This completes the proof of Theorem 8.7. D Proof of Theorem 8.8. By Theorem 8.7, we find two sequences {A^}, {un} such that A^ ^ 1, ^^^(i^^) = 0. Since T{s,u) = 5i^ G T*, we can find a constant c independent of A such that sup ^A^(r(5,^)) < sup ^xAsu) ^ A ^ K ) = inf ^^^* ueA,se[o,i] ueA,se[o,i]
< c.
We show that {un} is bounded. Write u^ = {vn,Wn), then (8.49)
Xnb{wn) - a{vn) - 2 / F(x, Un) < c
and (8.50)
{^'^JUn),Un) = X„b{Wn) - a{v„) -
/ ( / ( x , Un)Vn + g{x, U„)Wn)dx JQ
By
{BS)-{BQ),
there is an i?o > 0 such that
\g{x, s,t)\< c{\s\ + |t|)«-\ F{x, s, t) > c{\s\ + \t\Y,
yxen,\t\
+ \s\ < i?o;
V X e f2, |t| + \s\ < Ro;
= 0.
8.2. ELLIPTIC SYSTEMS
193
f{x,s,t)s^g{x,s,t)t-2F{x,s,t)
> c(|5p + Itp)''/^
f{x, 5, t)s + g{x, 5, t)t > joF{x, 5, t),
Vx G n,\t\^\s\
> Ro;
V X G O, \t\ + |5| < i?o-
It follows that, (8.51) {\Vn\^\Wn\ydx^ \v^Mw^\
< /
/ J\v^\ + \w^\>Ro
{\vX^\Wnndx.
{jo-'^)F{x,Vn,Wn)dx
J\Vr^\ + \Wr^\
l\v^\ + \w^\
J\v^\ + \w^\>Ro
)
/ ( x , ^^)i;^ + g{x, Un)wn - 2F(x, ^^)Jdx < c.
Note that {^xS^^)->^^->^^))
= ^n^('W^n) -
/ g{x,Vn,Wn)wn
= 0, and we
observe that XnKWn) =
/
g{x,Vn,Wn)w,
^52) + /
(\Vn\^\Wn\y~^\Wn\dx.
Choose Q = g(2-a) | ^ | E ^ < 1. By (8.50) and (8.46), \Wn\' \v^Mw^\>Ro 2(i-e)
^•53)
< f /
\wXdx]
I [
Iw^l'^dx]
{\vn\^\wn\y-^\wn\dx (q-l)
< ( /
{\Wn\ + \Vn\ydx]
I f
\Wn\^dx
194
CHAPTERS.
LINKING AND ELLIPTIC
SYSTEMS
and (8.55)
/
\vn\'
J\v^Mw^\>Ro
By (8.49)-(8.55), a{v^) < Xnb{w^) < c 6 ^ K ) + c a ^ K ) + c. Then {||^,||} is bounded. Once this is known, we can use the usual procedures to show that there is a renamed subsequence such that Un ^ u"^ in E and i^* is a nontrivial solution of (^'i). n Notes and Comments. As we have noted in Chapter 7, elliptic systems have been explored by many authors via different theories and methods, see for example T. Bartsch-D. G. de Figueiredo [33], D. G. Costa [106], D. G. de Figueiredo [156], D. G. de Figueiredo-P. L. Felmer [159], D. G. De FigueiredoE. Mitidieri [162], D. G. de Figueiredo-J. Yang [164], J. Hulshof-R. van der Vorst [189], E. A. Silva [338, 339]. We refer the readers to M. Schechter [310, 320] for the systems related to that of this chapter. The results of this section were obtained in M. Schechter-W. Zou [330] in which the methods are different from those of the above papers.
Chapter 9
Sign-Changing Solutions In this chapter, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A linking type theorem is established with the location of the critical point in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities. Under stronger conditions, we show that the existence of signchanging critical points can be independent of the Fucik spectrum.
9.1
Linking and Sign-Changing Solutions
Let ^ be a Hilbert space and let X C ^ be a Banach space densely embedded in E. Assume that E has a closed convex cone PE and that P := PE H X has o
interior points in X, i.e., P =P UdP in X. We use || • || and || • ||x to denote the norms in E and X respectively. We also use dist£;(-, •) and distx(-, •) to denote the distances in E and X respectively. Let ^ G C'^{E, R). For a,b,ce
R, denote
K: = {xeE: ^\x) = 0}; ^^ = {xeE: ^{x) < b}; )Cc = {x e K : ^{x) = c}; X:([a,6]) = {xeK:: ^{x) G [a, 6]}. Since ||
11^1? the following negative gradient flow a for ^ is well
deflned for (t, i^) G R x ^ :
196
CHAPTER 9. SIGN-CHANGING
Suppose (Ai) JCcX
SOLUTIONS
and V^ : X ^ X is in CK
Under this assumption, a{t, x) is continuous in (t, x) G R x X and a{t, x) G X for X G X. With the flow a, we cafl a subset A C E wci invariant set if cr{t,A) C A for t > 0. Definition 9.1. Let V C X be an invariant set under a. V is said to he an admissible invariant set for ^ if o
(a) V is the closure of an open set in X, i.e., V =V UdV; (h) if Un = cr(t^, v) ^ u in E as tn ^ oo for some v ^ V and u e JC, then Un ^ u in X; (c) if Un G /C n P is such that u^ ^ u in E, then u^ ^ u in X; o
(d) for any u G dV \ KL, we have a{t, u) GP for t > 0. Let A C X he di compact set in the X-topology such that A H 5 ^ 0, where S = X\V,
V =
PU{-P).
Let B C E\D be closed and T be deflned as in Chapter 5. Deflne X^ _ jr •~ \
T
r(t, x) : [0,1] X X ^ X is continuous in the 1 X-topology and T{t,V) cV j '
T h e n r ( t , ^ ) = {1 - t)u e T*. Theorem 9.2. Assume (Ai). Let A link B in the sense of Definition 5.1 and ^ satisfy the (PS) condition. Suppose that V is an admissible invariant set of ^ and that (9.2)
ao := s u p ^ < b^ := inf ^ . A
B
Then ^ has a critical value defined by (9.3)
a* := inf sup ^{u). r^T^*r([o,i],A)n<s
Moreover, when 0 ^ KLa*, then JCa* nS^9ifa*>
bo; JCa* nB^9ifa*
= bo.
9.1. LINKING AND SIGN-CHANGING
SOLUTIONS
197
Proof. For any F G T* we have that F([O,l],A)n5^0, smd BnX
F([O,l],A)n5^0,
F([0,1],A)CX
CS. It fohows that F([0,1], A) n 5 n 5 ^ 0. Thus, sup ^ r([o,i],A)n<s > sup ^ r([o,i],A)nSnB > inf ^ r([o,i],A)n<sn5 > inf ^ r([o,i],A)n5 > inf ^
which implies that a* > b^. Case 1: In this case we assume that a* > h^. We suppose that /C^* n 5 = 0 and derive a contradiction. Note that for any u G P \ { 0 } , the vector —^^(i^) o
points toward the interior of V- If ^ has no critical point on the boundary of o
P \ { 0 } , then KLa* CV - By the (PS) condition, there are SQ > 0? ^o > 0 such that (9-4)
' II J " „ >^ l + ||$'(u)|| -<5o
for
ue^-'[a*-eo,a*+eo]\{tCa^)so,
where {ICa*)5o := {u ^ E : dist£;(i^,/Ca*) < So}. By decreasing SQ, if necessary, we may assume that SQ < a^ —bo, and )C[a^ — £o,a^ -\- £o]nS = 9 (otherwise, it is trivial to get a contradiction via the (PS) condition). Let
Then {$'{u),V{u)) (9.5)
> 0 for all u and
($'(u),X^(u)) > ^ for any u G ^-^[a* - £o,a* + £o]\(/Ca05o-
Let e o : = { u e £ ; : | $ ( u ) - a * | < 3£o}, ei:={ueE: | $ ( u ) - a * | < 2£o}, ^, ,^ ._ distg(M, 62) ^^^' '~ disti<;(u, Oi) + distij(u, 62)'
198
CHAPTER 9. SIGN-CHANGING
SOLUTIONS
where B2 = E\So. Then ^{u)V{u) is a locahy Lipschitz vector field on both E and X since X is embedded continuously in E. We consider the following Cauchy initial value problem: ^ ^ ^
= -a
a{0, u) =
ueX,
which has a unique continuous solution cF{t^ u) in both X and E. Obviously,
—^ y
(9.6)
'' < 0.
By the definition of a* in (9.3), there exists a F G ^* such that r([0,1], A) n 5 C ^«*+^o :={ueE:
^{u) < a* + So}.
Therefore, r([0,1], A) is a subset of ^«*+^o U V. Denote A*:=r([0,l],A). We claim that there exists a TQ > 0 such that a{To,A*) C ^«*-^o/4 y p We consider three cases. (i) If 1^ G A* n P , then a{t, u) e V for all t >0. This is due to the assumption that V is an admissible invariant set. (ii) If u e A'^^u ^ (X:a*)35o,^ ^ ^- Since A* C ^«*+^o U P , we see that ^(i^) < a* +£oIf ^(i^) < a* - £0, then ^(cr(t,i^)) < ^{u) < a* - e^ for ah t by (9.6). If ^{u) > a* - £0, then u G ^~^[a* - £o,«* +^o]- Note that \\(j{t,u) -U\\<
/ ||(7(t,l^)||(it < t. Jo or
This implies that (j{t,u)
^ {^a*)35o/2 for ^ ^ [0?
or
]• If there exists a or
ti e [0, —^] such that ^{a{ti,u)) < a* - £0, then ^(cr(—^,i^)) < a* - SQ. Otherwise, a* - £0 < ^{cr{t, u)) < a* + £0 or
for SiWte [0,—^]. Then Cr(t,^) G ^
^[a* - £ o , « * + ^ o ] \ ( ^ a * ) 3 5 o / 2
9.1. LINKING AND SIGN-CHANGING
SOLUTIONS
199
or
for all t G [0,
]. It follows that ^{(j{t,u)) = 1 and {$'{a{t,u)),V{a{t,u))}>eo/5o or
for a l H G [ 0 , ^ ] . Therefore,
<
/•^
a*+60-
{^'ia{s,u),Via{s,u)))ds
Jo
<
a
- - .
Hence, (9.7)
^(^(To,^)) < a* - y
for
any To >
^ .
(hi) If 1^ G A*,!^ G {JCa*)35o,u ^ V, then ^{u) < a* + SQ. If moreover, ^(i^) < a* - So, then by (9.6), ^(cr(t, u)) < a* - So for ah t > 0. Assume that ^{u) > a* — £o- Then u G ^~^[a* — So^a* -\- SQ]. If there exists a sequence {t^} and ^0 ^ ^ such that a{tn,u) -^ ZQ in E, then cr has to travel at least (^o-units of time, and an argument similar to that of (ii) provides the proof. o
If there exist a sequence {tn} and ZQ G P such that a{tn,u) ^ ZQ in E and o
therefore in X, then there exists a IN such that a{tN,u) G P . The remaining situation is when (9.8)
dist^; f cr([0, oo), ^), X:[a* - SQ, «* + ^o]) := ^i > 0.
By the (PS) condition, there exists an e* > 0 such that
for u G ^~^[a* - £ o , « * +£o]\(^[«* - ^ o , « * +^o])5i> a* — SQ for all t. Then by (9.^
Similarly, we suppose that ^{a{t,u)) (9.10)
(7(t,^)G^-^[a*-£o,a*+^o]\(X:[a*-£o,a*+^o])5i.
Therefore, or
(9.11)
^{a{^,u))
/^f^
= ^{u)^
I
(i^(cr(5,^)) < a * - 2 £ o .
200
CHAPTER 9. SIGN-CHANGING
SOLUTIONS
By combining (9.7) and (9.11) for cases (ii)-(iii), we see that for any u G A*,i^ ^ P , there exists a T^ > 0 such that (J(TU,U) G ^«*-^o/2y p g y continuity, there exists an X-neighborhood Uu such that ^(T,,^n)c^"*-^°/^UP. Since A* is compact in X, we get a TQ > 0 such that cr(ro, A*) C ^«*-^o/4y p We define r
a(2ro5,^),
5G[0,^],
r*(5,^) = <^ [ (7(ro,r(25-i,^)), 5G[^,i]. Then, T* G T*. If 5 G [0, ^], we have that
r*(5, A) n 5 c cr(2ro5, A) n 5 c ^"° n 5 c ^«*-^o/4. If 5 G [^,1], then r*(5,A)n5 c c c
cr(ro,r(25-i,A))n5 (7(ro,A*)n5 (^"*-^°/^uP)n5
It follows that G(r*([0,1], A) n 5) < a* - £o/4, a contradiction. Case 2: Assume a* = 6o- If ^a* H 5 = 0, then there are positive numbers ^15^2,^3 such that (9.12)
||^'(^)||>£i
for | ^ ( ^ ) - a * | <£2 and dist£;(^, 5 ) < £3.
It holds true if we decrease £2 and £3. Thus, we may assume that £2 < £?£3/2(l+£i). Let (9.13)
Mi:={ueE:
dist^(^, B) < ^ , |^(^) - a*| < ^ }
and (9.14)
M2 := {^ G ^ : dist^(^, B) < ^ , |^(^) - a*| < y } .
Then M2 ^ 0. In fact, if we choose F G ^* such that sup ^(u) < a* + £ 2 / 3 , r([o,i],A)n<s
9.1.
LINKING
AND SIGN-CHANGING
SOLUTIONS
201
then we can find a i^o G r ( [ 0 , 1 ] , A) n 5 n 5 ^ 0 since A finks B. Note tfiat since a* = 6o, we fiave bo < ^{uo) <
sup ^{u) < a* + £2/3, r([o,i],A)n<s
tfiat is, uo e M2 C Ml. Let dist£;(i^,^\Mi)
eiM
dist£;(i^, E\Mi)
+ dist£;(i^, M 2 ) '
and consider tfie Caucfiy initial value problem
[ ai{0,u)
=
u e E,
wfiicfi fias a unique continuous solution (Ji{t,u) easily seen tfiat
in E. By (9.12)-(9.13), it is
Let u e ^«*+^2/3^ If tfiere is a t i < — sucfi tfiat cri{ti,u) a*
^ M2, tfien eitfier ^{cri{ti,u))
<
or dist£;(cri(ti,i^), 5 ) > —. Note tfiat ||
tfius we fiave tfiat dist£;(cri(t,i^),5) > — , and fience, (Ti{t^u) ^ B for all
te[0,|]. If a i ( i , u) e M2 for all t G [0, £3/4], then £3
$((7i(^,u))
=
^{u) + j
<
a* ''^ 3
<
a
'
d^{a,{t,u)) 4 ( ^^^' 1+ei)
- - .
Tfiat is, eitfier ^ ( c r i ( ^ , i ^ ) ) < a* - — or (Ji{t,u) eacfi u e A^
E""*^^.
^ B for afi t G [0, y ] and
202
CHAPTER
9. SIGN-CHANGING
SOLUTIONS
Next we show t h a t ai{t,u) ^ B for dl\ u ^ A and t G [ 0 , ^ ] . cTi(£3/4,1^) ^ B. Further, by (9.15),
First,
nt
^2
-L + ^ 1 Jo rt
^2
<
a* -
/
/
^i(cri(5,^))(i5.
-L + ^ 1 Jo If cri(t,i^) G 5 , then ^{ai{t,u))
>ho=a*^
and we must have £^i{(Ji{s^u)) = 0
for 5 G [0,t]. This implies t h a t (Ti{t^u) ^ M2 and either ^{(Ti{t^u)) or dist£;(cri(t,i^),5) > —. Both cases imply cri{t,u)
< a*
^ B. By t h e definition
of a*, we may choose a F i G ^ * such t h a t sup ^
—^iV.
'^3
Therefore, cri( — , F i ( t , i ^ ) ) G V and hence, cri( — , F i ( t , i ^ ) ) ^ B, since
5 n P = 0. Let
F * ( t , ^ ) = <^ (7i(^,Fi(2t-l,^)),
^
Then it is easy to check t h a t F* G T*. But t h e above arguments also imply t h a t F2([0,1], A) n 5 = 0, which contradicts t h e fact t h a t A links B. This completes the proof of t h e theorem. D
9.2
Free Jumping Nonlinearities
We consider the semilinear elliptic boundary value problem -Ai^ = / ( x , i ^ ) , (9.16)
in O,
; 1^ = 0,
on dfl,
9.2. FREE JUMPING NONLINEARITIES
203
where O C R ^ is a bounded domain with smooth boundary dft; f{x,t) Caratheodory function on O x R such that
is a
'
> a a.e. x ^ U as t
(9.17) > b a.e. X ^ U as t -
t Let E be the Fucik spectrum (see Section 5.4 of Chapter 5 for details). Note that we only know the existence of the curves Cn and C12 in the square (A/_i, A/+i)^. It is quite a challenging problem to determine the Fucik spectrum and the position of (a, b) with respect to this spectrum. On the other hand, since the eigenfunctions associated with A/ (/ > 2) of —A are sign-changing, a natural problem is whether the solutions of (9.16) are signchanging if resonance occurs around Xi {I > 2). In this section, we shall show • how to deal with the case when (a, b) is not restricted to the inside of the square (A/_i, A/+i)^ described above, no matter whether or not resonance occurs; • how to get more information concerning the solutions. These results allow (a, b) to be independent of the Fucik spectrum. With some additional assumptions there is no need to involve the Fucik spectrum in dealing with problems with jumping nonlinear it ies. (Bi) / G C\n X R), |/^(x,^)| < c(l + 1^1^-2) for a.e. x G O and ah u e K, where 2 < p < 2* and there exists a constant Co > 0 such that f(x,t) — f(x, s) t —s
^
w
^
(B2) a,b > Xk for some integer k > 2. (B3) 2F{x,t) > Xk-it^ for ah (x,t) G O x R, where F{x,t) = J^
f{x,s)ds.
(B4) /(x,0) = 0 and 2F{x,t) < Kot'^ for all x e Q and \t\ < 5o, where 5o > 0, hio ^ (A/c-i,A/c) are constants. f(x t)t — 2F(x t) (B5) liminf'^^ ' \ ^—^^ > c> 0 uniformly for x e Q; here a G (1,2) is a constant.
By assumption (^2), the point (a, 6) may or may not lie on any curves Cn or C12 and may even lie outside the square (A/_i,A/+i)^ for all / > k. The
204
CHAPTER 9. SIGN-CHANGING
SOLUTIONS
points a or 6 may be situated across multiple eigenvalues Xi {I > k). In particular, we permit a = b = Xi {Wl > k -\- 1). This means that resonance at infinity can occur at any Xi {I > k -\-l). Assumptions (^3) and (^4) contain the case when lim
^— = Xh-i, a resonant case at the origin. Let Ei denote
the eigenspace corresponding to A/(/ > 1) and N^ = EiU - - - U E^. Define (9.18)
^{u) := - / \Vu\'^dx - / F{x,u)dx,
u G
HI{Q).
We have Theorem 9.3. Assume that f{x,t) satisfies (9.17) and that {Bi)-{B^) hold. Then equation (9.16) has a sign-changing solution u^ with ^{u^) > 0.
The next case includes double resonant, oscillating and jumping nonlinearities. (9.19)
n ^
_^ ^^^^^) ^^
^ ^Q
as t ^ ±00,
where A^ < b±{x) < A^+i (A: > 2). Theorem 9.4. Suppose that (5i), (^3), (^4) and (9.19) hold. Assume that (Be) min{6+(x),6_(x)} ^ Afc; (B7) no eigenfunction of —A corresponding to X^ or A^+i is a solution of —Au{x) = b-^{x)u'^{x) — b-{x)u~{x). Then equation (9.16) has a sign-changing solution u* with ^{u*) > 0.
Let E := HQ{Q) be the usual Sobolev space endowed with the inner product and norm {u,v)=
{\/U'\/v)dx,
\\u\\ = (
\\/u\'^dx]
,
u^veE.
Let X := CQ{^) be the usual Banach space which is densely embedded in E. The solutions of (9.16) are associated with the critical points of the C^functional ^{u) = -\\uf 2
-
f F{x,u)dx, Jn
u G Hl{Q).
9.2. FREE JUMPING NONLINEARITIES
205
By the theory of ehiptic equations, )C = {u e E : ^'{u) = 0} C X. positive cones in E and X are given respectively by
The
PE
:= {u e E : u{x) > 0 for a.e. x G 0 }
and P := {u e X : u{x) > 0 for every x G O}. It is well known that PE has an empty interior in E and P has a nonempty o
interior P= {u e X : u{x) > 0 for all x G ^^dyu{x)
< 0 for all x G ^O}, o
where z/ denotes the outer normal. Therefore, P =P UdP. We rewrite the functional ^ as Hu) = ^WuWl - ^ ( i ( C o + l y + F(x,^))(ix, /
\ 1/2
where ||i^||£; := ( /^(|Vi^p + (Co + l)\u\'^)dx] Then the gradient of ^ at i^ is given by ^\u)
, which is equivalent to ||i^||.
= ^ - ( - A + (Co + l ) ) " ' ( / ( x , u ) + (Co + 1)^) :=u-
Ju,
where the operator J : E ^ E is compact and J{X) C X. In particular, by the strong maximum principle, J\x, the restriction of J to X, is strongly o
order preserving; that is, for any u — v G P\{0}, we have Ju — Jv GP . Since / ( x , 0) = 0, the ±P are invariant sets of the negative flow of the vector — ^ ^ It is easy to check that V is an admissible invariant set. Lemma 9.5. Assume that {Bi)-{B^) hold. Then ^ satisfies the (PS) condition. Proof. Let {un} be a (PS) sequence, that is, ^\un) -^ 0,^(i^^) -^ c. By Theorem L41, it suflices to prove that {un} is bounded in E. In fact, by (^5), there exists an i?o > 0 such that -f{x,t)t
— F{x,t) > c\t\^ for all x G O and
\t\ > RQ. Because of (9.17), we may assume that \f{x,t)t\ < d? for x G O and \t\ > RQ. Then, for n sufliciently large, we have the following estimates:
c+ ll^nll
= ( /
> —C-\- C /
+ /
)(-f{x,Un)Un-
\Un\^dx.
F{x,Un))dx
206
CHAPTER 9. SIGN-CHANGING
SOLUTIONS
Choose Co = (2 - a){N + 2)/(2AT + 4 - Na). Then CQ G (0,1) and
/
\Un\'^dx ,\>Ro „
2CQN
(2Ar + 4)
• /
<-{ J\u^\>Ro
^
\ 2JV + 4
^
< ( c + c||w„||)^^^^||w„f^». Consequently,
\\Un\\
=
{^'{Un),Un)
<
\\Un\\ -^C^C
^{j
+ /
)f{x,Un)Undx
\Un\ dx '\u^\>Ro
<
\\UJ^C^{C^C\\UJ)'^'
'°^/"||^nf^°.
Since a G (1, 2) and CQ G (0,1), we have that 2(1 — co)/cr + 2co < 2. Thus, we see that {||i^n||} is bounded. D Rewrite ^ as
Hu) = huf-U\u-\\l-h\\u+g-fp{x,u)dx,
ueH'om, J S2
where H{x,u) := / h{x, t)dt; h{x, t) = / ( x , t) - {bt^ Jo
-at~).
Let El denote the eigenspace of A/(/ > 1) and Nk = EiU - - - U Ek- Then L e m m a 9.6. Assume {B2). Then ^{u) -^ —00 as \\u\\ -^ oo,u ^ N^. Proof. Without loss of generality, we assume that a > b. For u = U- -\-uo G Nk with U- G Nk-i,uo G Ek, then * N = hwf
- U\u-\\l
- h\\u+\\l - f J S2
H{x,u)dx.
9.2. FREE JUMPING NONLINEARITIES
207
We have that *W
2'
'"
"^
[ H{x,u)da Jn
2
< i ( l - - ^ ) | | ^ _ f + i(l-A)||^,||2 1 {a — b)\\u II2 — / ~2 Q
H{x,u)dx
^
^k-1
^k-l
^
-ha-b)\\u-g-
f
^k
Hix,u)dx
Therefore, there exists an e > 0 such that ^{u) < -s\\u\\^ - I
for all u G Nk' Recall that lim
H{x,u)di
h(x,t) , , , ^(u) ^— = 0, thus we have limsup -—— < —e, hm sup -—J-
1.1
\t\^oo
M ^. M ^ IklHoo ll'^ll
t
which implies the conclusion of the lemma.
D
Lemma 9.7. Suppose that (9.17) (or (9.19) ) and {B4) hold. Then there exist po > O5 Co > 0 such that ^{u) > CQ for u G Nj^_-^ with \\u\\ = po-
Proof. By (9.17) (or (9.19)) and (^4), we may choose a 5i > A^ so large that 2F(x,t) < sit'^ for \t\ > 5o, x G O, where 5o comes from (^4). Choose 52 : = 2 ^ 1 . T h e n ,
(9.20)
2F{x,t) < S2t^ - sisl
for \t\ >so,xe
n.
For any u G Nj^_-^, we write u = v -\-w with v e E^ ® Ek+i © • • • © Ei-i and w G iV/"ii, where / is large enough so that Xi > /ooiA
(9.21)
(52 + A / )
/ii :=
;j
2 , (Afe + ^ o )
w -\
IO51 +H ^— +h 1051 Xk — 1^0
2
V —
77^
,
N
F{x,v-\-w).
^ ^0
. Let
208
CHAPTER 9. SIGN-CHANGING
SOLUTIONS
If IU ' +1(;| < 5o, then by condition (^4) and the choice of A/, we have that /il
>
(9.22)
W +
V
--K.o{v^w)
.
(52 + A/) - 2/^0
^ ^
i ^ + 4 ^((g2 + Az-2/^o)(Afe-/^o))V^
2 , (Afe + ^ 0 ) - 2/^0 2
I
I
^ -^ol^^l V
> 0. If IU ' +1(;| > 5o, then by (9.20), we get that (9.23)
/il > /i2 + /i3,
where /n o/i\ (9.24)
A/ - 52 2 , (Afe - ^0) 2 /i2 := — z — ^ + ^ ~A ^ ~ n^vw, o 4
(9.25)
/is := — ^ — ^
^
V - (52 - t^o)vw + ^ - ^ .
We claim that /i2 is always greater than or equal to zero. In fact, if K.o\w\ > 0, then (9.26)
M2 > ^^^w^
+ ( ^ ^ | « | - ^oH)\v\ > 0.
Otherwise, by the choice of A/, we have that /A/ —52
(9.27)
4:Hin , 9
Xk —1^0 9
^2 > ( ^ ^ - l - ^ ) « ^ ' + ^ ^ « ' > 0.
Also, we have that /is
. >
(Az + 5 2 ) - 2 5 2 2 I 1/ ^ ^ - \S2 - f^0\{\v\ (Az+52)-252
2
/
-(^2-^0)H^ + ^ ^
(A/ - 552 + 4/^o) 3(52 - no) 2 I
M I , '^I'^O ^ \w\)\v\ ^ ^—
M l
^ 2
^("^2 - ^0)
2 I '^I'^O
^2
Set (9.28)
Qi := {x en:\v^w\<
5o},
O2 := {x G O : 1^; +1(;| > 5o}.
1^;!
9.2. FREE JUMPING NONLINEARITIES
209
Since dim.Ni_i < oo, we may find a constant C/_i such that (9.29)
max|i;| < Ci_i\\v\\
foidllv
e Ni_i.
Let (9-30)
So := —^ r T 7 ^ ( l - T^)8(52 - /^0)Cf_/ Xk^
Then (^o > 0. By (9.22)-(9.28), we have / /j^idx
=
/
>
/
/j^idx -\- /
/j^idx
fiidx
JQ2
2
2
JQ2
If meas02 > SQ, then ^ M x > _ -3(52 v - ^ -^ ' ^o)|| - ^ | | ^ ||2 f +, rSl4 ^^,.
(9.31)
If meas02 < (^o, then by (9.29)-(9.30), (9.32)
f fiidx
>
-^^^^^^Cf_i||i;fmeas02
Combining (B4) and (9.20)-(9.27), we have ^(u)
=
l{\\vf+\\wf)-1
>
\\\vf
+ \\\wf
> \{l-^J\\vf
F{x,v + w)dx + \Xk\\v\\l + \xi\\w\\l -
+ \{l-'p\wf
+ J^f^,dx
>
l n i i n { ( l - g ) , ( l - g ) } | H | 2 + ^^;.idx
>
\{l-j^)\\uf+
f ^^ldx.
jj{x,u)da
210
CHAPTER 9. SIGN-CHANGING
SOLUTIONS
By (9.31), if meas O2 > (^o, then (9.34)
^u)
(9.35)
>
i ( l - g ) | H | 2 - f c - ^ | H | 2 + f^5o
>
i ( l - g ) | H | 2 - f c - ^ | H | 2 + f^<5o.
By (9.32), if meas O2 < (^o? then (9.36)
^u)
> \{1 - g ) | | ^ . f - A ( i _ g ) | | , | | 2 > _L(i _ g ) | | „ | | 2 .
By (9.34)-(9.36), we may find po > 0 and CQ > 0 such that ^{u) > CQ for u G Ni^_^ with ll^ll = po. • Proof of T h e o r e m 9.3. Invoking condition (^3), we readily have ^(u) < 0 for all u G Nk-i. By Lemmas 9.5-9.7, there exist RQ > po > ^ such that ao := sup^(i^) < 0 < Co < 60 •= inf ^(i^), A
B
where A:={u = v^syQ:ve Nk-i.s B:={ue N^_, : ||^|| = po}
> 0, ||^|| = Ro} U [A^^^-i n BR,],
and yo G Ek satisfying \\yo\\ = 1- Theorem 9.2 implies that there is a critical point i^* satisfying ^^(i^*) = 0, ^(i^*) = a* > 60 > 0- Obviously, 1^* ^ 0 and either i^* G 5 or i^* G 5 . The second alternative occurs when ^{u*) = bo := inf^ ^{u). Both cases imply that u* is sign-changing. D L e m m a 9.8. Under the hypotheses of Theorem 9.4, ^{u) -^ —00 for u e N^ as \\u\\ -^ 00. Proof. Note that Hu) = ^\\uf-
f {^h^{x){u^f
^h_{x){u-f
^ H{x,u))dx,
ueE,
where H (x^u) := /
h{x,t)dt;
h{x,t) = f{x,t)
— (b-^{x)t~^ — b-{x)t
j.
Note that min{6+(x), 6_(x)} > and ^ A^, and recall the variational characterization of eigenvalues {A^}. We then have the following estimates for any
9.2.
FREE
JUMPING
NONLINEARITIES
211
ueNk.
= lhf-
[ H{x,u)dx
-\{\
+[
'^^Jb-{x)>b+(x)
-\W--^\ (b-{x)
lb-^{x) — b-{x)]{u'^)'^dx ^
— / Jn
Jb_(x)
-I
^ I
2
H{x,u)dx
b^{x)u^dx
^ 2
2
)'^dx
^
h-{x)v?dx
2 Jb_ix)
--
'
— b-^{x)]{u
Jb_(x)>b+ (x) ^
- /
2
^
b+ix)u'dx
1 2
)(b+{x){u+f + b.{x){u-f)dx
Jb-(x)
Jb-{x)>b+{x) Jb-{x)>b+{x)
/
b-{x)v?dx
— I
H{x,u)dx
Jb-{x)
-, / inm{b-^{x),b-{x)}u^dx 2 JQ
<-(5||^f- /
— /
H{x^u)di
JQ
H{x,u)dx;
t h e lemma follows immediately.
P r o o f of T h e o r e m 9.4. By Lemma 9.8 and ( ^ s ) , there exist RQ > po > 0 such t h a t ao := sup^(i^) < 0 < Co < 6o •= inf ^(i^), A
B
where A and B are defined as in t h e proof of Theorem 9.3. To apply Theorem 9.2, we just have to check the (PS) condition. Assume t h a t {un] is a (PS) sequence. Otherwise, we may assume t h a t ||i^^|| -^ oo as n -^ oo. Let w^ = '^n/ll'^nll- Then ||t^n|| = 1 and, for a renamed subsequence, Wn ^ w weakly in E, strongly in 1/^(0) and a.e. in O. Moreover, {^\Un),v)
= {Un,v) -
/ f{x,Un)vdx
and f [Wn^v) — /
f{x,Un)v ax ^
0.
^ 0
212
CHAPTER
9. SIGN-CHANGING
SOLUTIONS
By (9.19), we see t h a t —Aw = b-^w~^ — b-W~. Since
we see t h a t
/ (b^{w^f
^b_{w-fyx=1.
This implies t h a t w ^ 0. Let w = W- -\- W-^ with W- G Nk^W-^ G A^^ and w = !(;+ — W-. Take g'(x) = &+(x) when w{x) > 0 and q{x) = b-{x) when i(;(x) < 0. Then we have t h a t —Aw = q{x)w. Hence
Ih+f- - I k - f
= / q{x){w^f Jn
- / q{x){w_) Jn
It follows that 0
<
\\w+f - Xk+^\\w+\\l
< Jn =
\\w.f- 1 q{x){w.f
<
Ww.f-Xk f{w-f
<
0.
Jn
T h a t is, | | i ^ ± P = J^q{x){w±)'^. T h e only way this can happen is q{x) = A^ when W-{x) ^ 0 and q{x) = A^+i when W-^{x) ^ 0, and therefore, either Wis an eigenfunction of A^ or W-^ is an eigenfunction of Afc+i. This implies t h a t W-{x)w-^{x) = 0. Thus q{x) = 6+ = A^ when W-{x) > 0; q{x) = b- = Xk when W-{x) < 0; q{x) = 6+ = A^+i when W-^{x) > 0; q{x) = 6_ = A^+i when w^{x) < 0. Then —Aw- = XkW- = 6+i(;l — b-wZ, —Aw-^ = A/c+ii(;+ = b-^w^ — b-W^. Hence, w± = 0. This is a contradiction and t h e (PS) condition follows. We obtain t h e conclusions of Theorem 9.4 as in t h e proof of Theorem 9.3. D
N o t e s a n d C o m m e n t s . T h e assumptions in (9.19) mean t h a t t h e problem is double resonant. An earlier paper on this line is H. Berestycki-D. G. deFigueiredo [57] (see also M. F . Furtado-L. A. Maia-E. A. B. Silva [166]). At t h e end of t h e Chapter 5, we gave historical notes and our comments.
9.2. FREE JUMPING NONLINEARITIES
213
Their results are closely related to the Fucik spectrum. They did not make the stronger assumptions on ^{u) that are made here. Sign-changing solutions have attracted much attention in recent years (see T. Bartsch [29], T. Bartsch-K. C. Chang-Z. Q. Wang [31], T. Bartsch-Z. Q. Wang [42], A. Castro-J. Cossio-J. M. Neuberger [83, 84], A. Castro-M. Finan [85], E. N. Dancer-Y. Du [126], S. Li-Z. Q. Wang [218, 217], Z. Q. Wang [374]). Also there are some interesting results in G. Chen-W. Ni-J. Zhou [100], Z. Ding-D. G. Costa-G. Chen [137], D. G. Costa-Z. Ding-J. Neuberger [108], J. Neuberger [266] and J. Neuberger-J. W. Swift [267] for numerical methods for sign-changing solutions. These have suggested various types of sign-changing solutions. In T. Bartsch [29] (see also T. Bartsch-Z. Q. Wang [42]), the author established an abstract critical theory in partially ordered Hilbert spaces by virtue of critical groups and studied superlinear problems. In S. Li- Z. Q. Wang [217], a Ljusternik-Schnirelman theory was established for studying the sign-changing solutions of an even functional. Some linking type theorems were also obtained in partially ordered Hilbert spaces. We also refer readers to T. Bartsch-Z. Liu-T.Weth [38] for the existence of sign-changing solutions. Concerning the theory of ordered Banach spaces, we refer to the paper of H. Amann [8]. As for the properties of flow invariant sets, see H. Brezis [65] and K. Deimling [130]. Theorems of this section were given in M. Schechter-Z. Q. Wang-W. Zou [331]. The readers may also consult T. Bartsch-Z. Q. Wang [44] (on superlinear Schrodinger equations), T. BartschT. Weth [46, 47] (on superlinear elliptic equations and singularly perturbed elliptic equations), E. N. Dancer-S. Yan [127] (on sign-changing mountain pass solutions), M. Schechter-W. Zou [333] (on asymptotically linear Schrodinger equations), Z. Zhang-S. Li [382], W. Zou [390](on sign-changing saddle point) and W. Zou [388] for more results.
Chapter 10
Cohomology Groups Let ^ be a real Hilbert space with an inner product (•, •) and associated norm || • ||, and let / G C^(^,R) be a strongly indefinite functional, i. e., / is unbounded from below and from above on any subspace of finite codimension. It is well known that the Morse index of any critical point of / must necessarily be infinite. In this case usually one can not expect to obtain any useful information from the usual Morse theory. In order to overcome this difficulty,a more advanced theory is needed. In sections 10.1-10.4 of this chapter, we first introduce the W. Kryszewski-A. Szulkin infinite-dimensional cohomology theory and a Morse theory associated with it (see [200]) and then we develop some methods to compute the groups precisely. Applications to Hamiltonian systems and beam equations will be given.
10.1
The Kryszewski-Szulkin Theory
Let X be a metric space, and let A be a closed subset of X. We denote the Cech cohomology of the pair (X, A) with coefficients in a fixed field J^ by H'^{X,A). Let {Gn)'^=i be a sequence of Abelian groups. Define oo
n=l
and introduce the equivalence relation ~: (<^n)^i ^ (/5n)^i
if and only if an = (3n
for almost all n > 1.
Define the asymptotic group [(v4n)^i] by the formula oo
n=l
216
CHAPTER 10. COHOMOLOGY
GROUPS
Definition 10.1. The sequence ( ^ n ) ^ i ^s called a filtration of E if it is an increasing sequence of closed subspaces of E such that E = ( U ^ ^ ^ ^ ) Let {dn)'^=i be a sequence of nonnegative integers and 8 = { ^ n , ^ n } ^ i Definition 10.2. For a pair {X,A) of closed subsets in E,A C X, the qth S-cohomology group of (X, A) with coefficients in a field T is defined by Hl{X,A)
:=
Um^^-{XnEn,AnEn)r n=l
qeZ.
Evidently, each group Hg (X, A) is in fact a sequence of cohomology groups of the spaces {X H E^^A H E^) approximating {X,A). Since JF is a field, Hg (X, A) is in fact a vector space over J^. ^Y) Definition 10.3. A continuous function f : (X, A) -^ (F, B) (or f : X is filtration-preserving if f{X H En) C En for almost all n. In this case, f is called admissible. A filtration-preserving continuous function induces a homomorphism
r:H*s{Y,B)^H*s{X,A), where
r[K)~=i] := [(/*K))~=i] with fn ••= f\ixnE„„AnE„,),
a „ G H*+''"iY nE„,Bn
E„).
P r o p o s i t i o n 10.4. We have the following properties. (1) (Contravariance^ / / i d is the identity on {X^A), then id* is the identity on H*{X,A) and if f : (X, A) ^ (F, 5 ) and g : {Y,B) -^ {C,D) are admissible, then {g ^ fY = f* ^ g* (2) (Naturality) / / / : (X, A) -^ {Y,B) is admissible, then (^*(/U)* = /*^*, where (5* := [S^] with
51 : i/^+^- {A n En) -^ i/^+^-+i(X ^En^A^
En)
is the the usual coboundary homomorphism in the Cech cohomology theory. (3) (Exactness) For each pair {X^A)^A C X, of closed subsets of E, let i : A C X and j : X C (X, A) be the inclusions. Then ••• ^ Hl{X,A) is exact.
^ HliX)
^ HliA) ^ Hf\X,A)
- •••
10.1. THE KRYSZEWSKI-SZULKIN
THEORY
217
(4) (Strong excision) For any two closed subsets A, B of E, the inclusion (A, Ar\B) C {AVJ B^B) induces the excision isomorphism Hl{A, AnB)
^^ Hl{A U B, B).
(5) (Homotopy invariance) If f,g : (X, A) -^ {¥, B) are admissible and homotopic by an admissible homotopy, that is, if there is a homotopy G between f^g such that G([0,1] x (X H En))) C E^ for almost all n, then
r = 9*(6) (Exact sequence of a triple) Let B C A C X be closed subsets of E and i : (A, B) C (X, B) and j : (X, B) C (X, A) be the inclusions. Then there is a homomorphism (5* : Hg{A,B) -^ Hg~^^{X,A) such that the cohomology sequence > Hl{X,A)
C Hl{X,B)
^ Hl{A,B)
^ i:f|+^(X, A) ^ • • •
is exact. These can be easily proved by the ordinary cohomology; we refer the readers to A. Dold [138], E. H. Spanier [346]. In fact, Proposition 10.4 was given in W. Kryszewski-A. Szulkin [200]. Let Pn be the orthogonal projection of E onto E^. Definition 10.5. Let I G C^(^,R) be a strongly indefinite functional. We say that I satisfies the (PS)^ condition with respect to S = {E^^dn} if whenever a sequence (uj)^-^ is such that Uj G E^^, for some nj^Uj -^ oo,sup^- \I{uj)\ < oo and PnT\uj) ^ 0 as j ^ oo, then {uj)^i has a convergent subsequence.
Let /C = K:{I) :={ueE:
I'{u) = 0}.
Basic Assumption. In this section, we always assume that (1) / satisfies the (P6')* condition with respect to
218
CHAPTER 10. COHOMOLOGY
GROUPS
Definition 10.6. Let Q C E\JC. A mapping V : Q ^ E is called a gradientlike vector field of I on Q if (1) V is locally Lipschitz continuous; (2) \\V{u)\\ <
IforallueQ;
(3) there is a function r] : Q ^ R+ such that {r{u),V{u)) u e Q;
> r]{u) for all
(4) inf r]{z) > 0 for any set M C Q which is bounded away from KL and such that sup \I{z)\ < oo. zeM Since a gradient-like vector field V of / on Q is always associated with a function r], we sometimes use (V, r^) to denote it. Definition 10.7. A gradient-like vector field V for I on Q is called 8-related if for any M e Q which is bounded away from JC and satisfies sup |/| < oo, M
we have V{MnE^)cEr,. The following lemma gives the existence of the gradient-like vector field. Lemma 10.8. Let Q be an open subset of E. If I ^ C^{E,IV) satisfies the (PS)^ condition, then there exists an S-related gradient-like vector field V of I on Q\K. Proof. Let Qk'={ueQ:
\I{u)\ < k, dist(^,X:) > 1/k},
keN.
Then Qk C Qfe+i,
U^iQfc = Q\JC.
Note that Qk j^ 9 for some k = ko and we may define an{k) := mf{\\PnI\u)\\
:ueQk^
K},
n > 1, A: > A:o.
Let a{k) := - liminf Q^^(A:). Then a{k + 1) < a{k)^ and moreover, by the (PS)* condition, a{k) > 0 for k > ko. For any u G Q\/C, define (10.1)
r{u) := minjA: > ko : u e Qk},
i^{u) := -a{r{u)
+ 1);
10.1. THE KRYSZEWSKI-SZULKIN (10.2)
Du:={weS:
THEORY
219
{I'{u),w) > ^ c ^ ( r H ) } ,
where 6' = { i ^ G ^ : | | i ^ | | = l } . Then Du is open in S^ and by the definition of an and o^, we see that Du is nonempty. Let Sn := Sr\En. Since cl{yj^^iSn) = S^ we may define K.{U) := min{n > 1 : D^ f^Sn ^ 0}. Let w{u) be an arbitrary point of Du n S^(^u)' Since V is continuous and (3r(ii) is open, there exists an open neighborhood U{u) of i^ such that (10.3)
(10.4)
(10.5)
U{u)^Q^^u)-Qr{u)-2]
{I\y)Mn))>\c^{r{u)),
||r(^)-r(^)||
yeU{u);
^Gt/(^).
Hence, by (10.1)-(10.3), we have (10.6)
either r{y) = r{u)
or r{y) = r(i^) — 1
whenever y G t/(i^).
Consider the open covering {U{u)}ueQ\K:'', by Proposition 1.3, it has a locally finite Lipschitz continuous partition of unity {Xj}j^j. For each j ^ J there exists an Uj G Q\IC such that suppAj C U{uj). Define V : Q\IC -^ E by the formula (10.7)
V(^) := ^ A , ( ^ ) ^ ( ^ , ) ,
^ G Q\)C.
We show that V is what we want. Firstly, we note that V is locally Lipschitz continuous and ||V(^)|| < 1 for y G Q\1C. Moreover, for any y G Q\/C, if Xj{y) ^ 0, then y G U{uj) and by (10.6), r(^^) < T{y) + 1. Hence, by (10.4),
(10.8)
(r(^),V(^)) = ^ A , ( ^ ) ( r ( ^ ) , ^ ( ^ , ) ) >
la{r{y)^l)=^{y).
Secondly, if M C Q\IC and M is bounded away from /C and s u p ^ |/| < oo, then there exists Si k > ko such that M C Qk- Hence, (10.9)
inf i9(z) > -a(k + 1) > 0.
220
CHAPTER 10. COHOMOLOGY
GROUPS
Finally, choose no so large that an{t) > a{t) for all n > UQ and ko < t < k. Therefore, (10.10)
||P„J'(y)||>a(t)
whenever n > no, /CQ < t < k^y ^ Qk C\ E^. Let n > no and y G M n EnIf Xj{y) ^ 0, then y G U{uj) and r(7/) < T{UJ). Moreover, r{y) < k since P
T(y)
M C Qk. Let e := n " . r , , , G 5 , . By (10.10) and (10.5),
= llp.r(^)ll-(r(^)-r(^,),0
>
-Q^(r(^^-))-
It follows that (f G Duj riSn- By the definition of hi{u), ^{^j) ^ ^- Therefore, w{uj) e E^i^uj) ^ En and V(^) G ^n- This implies that V(M n E^) C ^nTherefore, V satisfies all conditions in Definitions 10.6 and 10.7. D Definition 10.9. Let M be an isolated compact subset of KL. A pair (B, B~) of closed subsets of E is said to be an admissible pair for I and M with respect
to 8 if (1) 6 is bounded away from JC\M, 6 ~ C dQ and M C int{Q); (2) I\Q is bounded; (3) there is a neighborhood N of S such that there is an 8-related gradientlike vector field V (called admissible field) for I on N\M; (4) S~ is the union of finitely many closed sets, each of which lies on a C^-manifolds of codimension 1; (5) V is transversal to each of these manifolds at points of Q~ ; (6) the flow T] of —V can leave 6 only via 6 ~ and if u e 6 ~ , then r]{t,u) will leave Q, i.e., r]{t,u) ^ 6 for any t > 0.
Lemma 10.10. Assume I G C^(^,R). Let a < 6, B := /~^([a,6]) and 6 ~ := I~^{a). If 6 is bounded away from JC\{JC H int(6)), then ( 6 , 6 ~ ) is an admissible pair for I and /C H int(B).
10.1. THE KRYSZEWSKI-SZULKIN
THEORY
221
Proof. We first note that there is an open neighborhood UofQ such that U is bounded away from JC\JC H int(6). By Lemma 10.8, there exists an 0 whenever u G B~, then (B, B~) is an admissible pair. D Given u ^ E,£ > 0, set B{u, s) := {x e E : \\x - u\\ < e},
B{u, s) := {x e E : \\x - u\\ < e},
S(u, s) := {x e E : \\x - u\\ = s}.
We have Lemma 10.11. Let U be an open neighborhood of the isolated critical point p of I. Then there exists an admissible pair (B, B~) for I and p satisfying ecu,
I\e-
< I{p) := c.
Further, there is anso > 0 such that B{p, SQ) C int{Q). For any u G 6'(p, £o)n I^, there is a t > 0 such that r]{t, u) G B~, where r] is the flow of —V. Proof. Choose 5 > 0 small enough such that B{p, 5) C U. Let V : B{p, S)\{p} E be an S- related gradient-like vector field with function r]. Then p := M{(3{u) : S/2 < \\u - p\\ < 5} > 0. Choose £ e (0, p6/4). Let 61,62 > 0 he such that 62 < 6i/2 < 6/4 and B{p, 61) c{ueE
: \I{u) -c\<
s}.
Set M := B{p, 6). Let ^ : M ^ [0,1] be a locally Lipschitz continuous function such that ^{u) = 0 in a neighborhood of p and ^{u) = 1 for 62 < \\u — p\\ < 6. Consider the Cauchy initial value problem: ^ ^ ^
= -aa{t,
u))V{a{t, u)),
<7(0,
u)=ueM,
which has a unique solution a. Define B := {cr(t, u) :t>0,ue
B{p, 61), I{a{t, u)) > c - s}
and B~ := B n /~^(c - s). Then (B, B~) is what we want. Since mf{(3{u) : ^61 < \\u—p\\ < 6} > 0, there exists an £0 > 0 such that if u G S{p,6i) and I{u) < c, then I{a{t,u))) < c — SQ whenever ||cr(t,i^)|| = ^.
222
CHAPTER 10. COHOMOLOGY
GROUPS
Choose 62 sufficiently small so that I{u) > c — SQ for each u G B{p,S2). Therefore, a{t,u) cannot enter B{p,S2). Since ^{u) = 1 and r]{u) is bounded away from 0 when 62 < \\u — p\\ < S, I{a{to,u)) = c — £ for some to and r]{to,u) = cr{to,u) G 6 ~ . D Lemma 10.12. Assume that I G C^{E,Il) and that p is an isolated critical point of I. Suppose that ( 6 1 , 6 ^ ) and ( 6 2 , 6 ^ ) are two admissible pairs for I and p. Then Proof. Since ( 6 1 , 6 ^ ) and ( 6 2 , 6 ^ ) are two admissible pairs, we have a neighborhood Ui of B^ and a vector field Vi on Ui\{p}, z = 1,2. By Lemma 10.11, there is an admissible pair (B, B~) for / and p such that B C int(Bi) H int(B2). Thus, we just have to show that H^iSi.S^) ^ H^{e,e-). By Lemma 10.11 and its proof, we get a gradient-like vector field {F^jSp) which is admissible for both (B, B~) and (Bi, B^). Since B~ := Bn/~^(c—e) for some small £ > 0, where c = /(p), the flow rj of —F cannot re-enter B after leaving it. Choose (5 > 0 so small that B{p^ 6) C int(B). Let dist(^,5(p,(5/2)) dist(^, B{p, 5/2)) + dist(^, Ui\B{p, 5)) Then ^ : Ui ^ [0,1] is a locally Lipschitz continuous function. Consider the Cauchy problem: ^ ^ ^
= -a
<7(0,
u)=uee,.
Note that since ^F preserves the filtration on Bi, we see that (10.11)
cr(t, 1^) G B i n ^ ^ if 1^ G B i n En, (J(t, 1^) G B i .
Since ^F is locally Lipschitz continuous and bounded, for any fixed i^ G Bi, either there is a unique t = t{u) G [0,+00) such that a{t{u),u) G B ^ or cr([0, +00), 1^) G Bi. In the latter case, we set t{u) = +00. Then the function u -^ t{u) is continuous on the set {i^ G Bi : t{u) < 00}. Let (10.12) L:={cr(t,^) : t > 0 , ^ G B - } n B i ,
Q := B U L, Q' := Q n 9]^.
It is easy to check that (Q, Q~) is an admissible pair for / and p. Note that (10.13)
I{a{t,u))-I{u) < f C{a{s,u)){I{a{s,u)), Jo
-F{a{s,u)))ds
< - / C{cr{s,u))(3F{cr{s,u))ds. Jo
10.1. THE KRYSZEWSKI-SZULKIN
THEORY
223
For any u e L of (10.12), since
H*£{L,Q-) = 0.
By (10.14), the exactness of the cohomology sequence and the excision property, we observe that (10.15)
HI{Q,Q-)^
H*s{Q,L)%"
Hl{e,e-).
Let Qo = Q U Bj". Then (Qo, G)j") is an admissible pair for / and p. Hence, (10.16)
F|(Q,Q-)l'F|(Qo,er).
We will show that
(10.17)
i/|(Qo,er)'^'ff|(ei,er).
For any i^ G Bi, if (j{t,u) ^ Q, then by (10.13), I{(j{t,u)) < I{u) —t^F, where Then either ^F •= infei\Q (3F{U) > 0. Choose T = (sup^^ / — infei I)/PF' G{t,u) eQfoiT t ( i ^ ) . Define a mapping C : [ 0 , r ] x B i -^ Bi by
[
(T{t{u),u),
te[t{u),T].
Since the function t{u) is continuous, we get that ^ is a deformation of the pairs (Bi,Bj") into {Qo,S]^) and C([0,r] X Qo) C Qo,
filtration-preserving
C([0,T] X B r ) C B ^ .
Further, if i : (Qo, B^) -^ (Bi, B^) is the inclusion and (T •= C{T, •), then (T^i
— id
on
(Qo,B^),
i o C,T — id
on(Bi,B^).
It follows that (Qo, B^) and (Bi, B^) are homotopy equivalent by filtrationpreserving homotopies. Hence, we get (10.17). Combining (10.15) and (10.16), we get the conclusion. D We now can introduce the definition of an
224
CHAPTER 10. COHOMOLOGY
GROUPS
Definition 10.13. Let I G C^(^,R) and p be an isolated critical point of I. Let (6, 6~) be an admissible pair for I and p. The q-th 8-critical group of I at p is defined by
C|(/,p):=if|(e,e-),
geZ.
By Lemma 10.12, Cg{I,p) is well defined. Moreover, the following continuity property will be a key role in further applications. Lemma 10.14. Let I G C^(^,R) and p be an isolated critical point of T Let ( 6 , 6 ~ ) be an admissible pair for I and p. Let I e C^{E,Il) with sup \I\ < oo. e Assume that there exists a c> 0 such that sup\\I\u)-r{u)\\ e
Further, assume that I has only one critical point p inQ and that 6 is bounded away from JC{I)\{p}. Then (6, 6~) is an admissible pair for I and p. Proof. Take a neighborhood U of S such that U is bounded away from )C{I)\{p} and sup^ |/| < oo. Let V : U\{p} ^ ^ be an admissible vector field for / with function r]{u) (cf. Definition 10.6). Let B{p, S) C int(6). Note that U\B{p,S) is bounded away from /C(/), fj = inf{?^(i^) : u G U\B{p,S)} > 0. Let £ G (0,?^). By shrinking [/, we may assume that s u p | / | < oo,
u
s u p ||/^(l^) - I\u)\\
< £.
u
It follows that
= {I'{u),V{u)) +
(I'{u)-I{u),V{u))
> r]{u) — £
>0. Therefore, p G B{p, 5) and U\B{p, 5) is bounded away from /C(J). Similar to Lemma 10.8, we may construct an
Theorem 10.15. (Continuity Theorem) Le^ {/A}AG[O,I] ^ C^{E,IV) be a family of functional that satisfy the (PS)* condition. Assume that there exists an open set U such that px e U is the unique critical point of Ix in U for each A and sup
ueu,xe[o,i]
|^A('^)| < oo.
10.1. THE KRYSZEWSKI-SZULKIN
THEORY
225
/ / the mapping A ^ /^ is uniformly continuous in u e U, then C|(/A,PA) = Q ( / I , P I ) ,
VAG[0,1].
Proof. For any A G [0,1], choose a ball Bx such that px e Bx C U. By Lemma 10.11, there exists an admissible pair ( B A , B ^ ) , B A ^ Bx for Ix and Px. By Lemma 10.14, there exists an e > 0 such that for each t G [0,1], \t—X\ < £, (BA, B ^ ) is an admissible pair for It and pt. Hence, Cg{Ix,Px) = Cg{It,pt)The conclusion follows. D A pair (B, B ) of closed subsets of E is called a globally admissible pair for / and JC with respect to 8 if (B, B~) satisfies the conditions of Definition 10.9 with A = /C and M = ^ . Lemma 10.16. There exists a bounded globally admissible pair (B,B ) for I and KL. Proof. Choose a, b such that a < I{u) < b for all u ^ KL. Invoking Lemma 10.8, let V : E\]C -^ E he d. gradient-like vector field related to 8. Let [/i, U2 be bounded neighborhoods of /C such that Ui is closed and U2 open, Ui C / 7 2 C / - i ( [ a , 6 ] ) . Let d\st{u,Ui) dist(i^, Ui) + dist(i^, E\U2) and consider the Cauchy problem do-{t,u) dt
-C{a{t,u))V{a{t,u)),
a{0,u)=u.
Define B := {cr{t,u) :t>0,ue
U2,1{cr{t,u)) > a},
B~ : = B n / ~ ^ ( a ) .
It is easy to see that (B,B~) is a globally admissible pair and B,B~^ are bounded. D Definition 10.17. Let (B, B~) be a globally admissible pair for I and JC with respect to 8. Define the 8-cohomology critical groups by
clii,Jc):=Hiie,e-). Lemma 10.18. The 8-cohomology critical groups C|(/,/C) are well defined.
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CHAPTER 10. COHOMOLOGY
GROUPS
Proof. Choose a, 6 such that JC C int/~^([a, 6]). Let 6o = /~^([a, 6]), 6 ^ = I~^{a). Then by Lemma 10.10, (6o,B^) is a globahy admissible pair. If ( 6 , 6 ~ ) is another globally admissible pair with a globally admissible field V, we may assume that B C BQ. Hence, V is also a globally admissible pair (Bo,B^). Similar to Lemma 10.12, we have
i7|(B,B-)-i7|(Bo,Bo-). D Theorem 10.19. (Continuity Theorem) Let {/A}AG[O,I] ^^ a family of C^functionals satisfying the {PSy condition. Suppose that the mapping A ^ /^ is uniformly continuous on bounded subsets of E and there exist a bounded set N and a constant C > 0 such that JC{Ix) C N and sup^ |/A| ^ C for all A G [0,1]. Then C^{IX,K:{IX)) is independent of X e [0,1]. Proof.
Let ( B A , B ^ ^ ) be a bounded globally admissible pair for Ix and Similar to Lemma 10.14, ( B A , B ^ ^ ) is a globally admissible pair for Is and 1C{Is) as long as |5 — A| is sufficiently small. Since BA is bounded and sup |/A| < oo, we see that sup I/5I < 00. Note that lC{Is) C N and
X:(/A).
BA
BA
II^^HII>II^;HII-II^^M-^;HII>O for 1^ in a neighborhood of A^\int(BA). Consequently, lC{Is) C BA- By using a partition of unity, it is easy to construct a gradient-like vector field Vs • E\JC{Is) -^ E such that Vs = VA in a neighborhood of ^BA- The conclusion follows from the connectedness of [0, 1]. D Notes and Comments. The theory of this section was due to W. KryszewskiA. Szulkin [200]. We refer the readers to K. C. Chang [96] and J. Mawhin-M. Willem [252] for a Morse theory of the singular relative homology groups. The readers can also consult the surveys of Morse theory given by R. Bott [63] and E. H. Rothe [302]. The earlier ideas can be found in M. Morse [259], J. Milnor [258], R. Palais [269], R. Palais-S. Smale[272] and S. Smale [344]. In A. Szulkin [357], a kind of cohomology group was obtained using the K.Geba-A. Granas theory, where more strict assumptions were needed (e.g., it was assumed dim(^^+i 0 En) = 1). A generalized cohomology groups was defined in A. Abbondandolo [1, 2] with more general functional properties.
10.2. MORSE INEQUALITIES
10.2
227
Morse Inequalities
Set oo
oo
n=l
n=l
[z] =nz/0z and [Z+] := |[(e^)^^i] e[Z]:en>0
for almost all n\.
Let (X, A) be a pair of closed subsets of E such that for each q e Z there exists an Uq such that
dimi:f^+^-(xnK,AnK) < oo whenever n> Uq. Then we define
(10.18)
dimsHl{X,A)
:= [(dimi7^+^-(X n K , A n K ) ) ^ = i ] G [Z+].
If dimi:f^+^-(X n K , A n E^) = d for ah n sufficiently large, we write diiRg Hg{X,A) := [d]. Moreover, if d = 0 for almost all g' G Z, we say that (X, A) is
(10.19)
M|(e, e-) := J2 dim^ Cl{I,p,),
q G Z,
i=l
and (10.20)
/3|(e, e - ) := dim£: H^e,
6"),
^ G Z.
Setting
q= — oo
and
oo
P£{t,e,e-):= J2 /3|(e,e-)t«, we have
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CHAPTER 10. COHOMOLOGY
GROUPS
Lemma 10.20. Suppose that I G C^{E,'R) satisfies the (PS)* and that (6, 6~) is an admissible pair for I and A := {pi,P2, • • • ,Pm}- ^/ ^^^ Pi ^^e 8-finite and I{pi) = I{p2) = • • • = I{pm), then m
^dim^C|(/,p,) = dim^i7|(e,e-),
V^ G Z.
i=l
Proof. Let ( 6 j , 6 j ) be an admissible pair for / and p j , l < j < m. We may assume that the 6^ are pairwise disjoint. Then ( U ^ ^ B j , U ^ ^ B j ) is an admissible pair for / and A. By arguments similar to those used in the proofs of Lemmas 10.11 and 10.12, we have
Hiie,e-) m
-0if|(e„e-) m
J= l
The conclusion follows from the definitions in (10.18).
D
Theorem 10.21. (Morse Inequalities) Assume that I G C^(^,R) satisfies the {PSy and that (6, 6~) is an admissible pair for I and AQ := {pi,P2, • • • ^Pk}If all Pi are S-finite, then the pair (B, B~) is S-finite and there is a polynomial oo
Q{t) = y^ttgt^ such that aq G [Z+] for all q and — oo
Miit, e, e-) = p|(t, e, e-) + (i + t)Q(t), or equivalently,
^(-ir^M|(e, e-) > ^(-i)«-i/3^(e, e-), qe z. — oo
—oo
Proof. Let D ^ B ^ Ahe closed subsets of E. Let i:{B,D)(l{A,D);
j : {A, D) C {A, B)
be the inclusions. Then there exists a homomorphism (5* :Hl{B,D)
^H^^\A,B)
10.2. MORSE INEQUALITIES
229
such that the cohomology sequence (10.21)
>Hl{A,B)^
Hl{A, D) ^ Hl{B, D) ^ HI-^\A,
B) ^ • • •
is exact. Denote the range of the map by IZ and dim£: H^{-) by /3|(-). Suppose that the pairs {A,B), {A,D) and {B,D) are if-finite. Then by (10.21),
/3|(A, D) = dims n{f) + dim^ 7^(^^), /3|(5, D) = dims nii"^) + dim£: 7^((5^), /3|(A, B) = dims n{S^-^) + dim^ n{f). It follows that (10.22)
/3|(A, B) + /3|(5, D) = (31{A, D) + dim^: n{5^) + dim^:
n{5^-^).
Let
— oo
and oo
(10.23)
Q(t, A, B, D) := ^ dim£: n{S'^)t'^. — oo
Since dim£: 7^((5^) = [0] for almost all q, by (10.22) we have (10.24)
Ps{t,A,B)^Ps{t,B,D) = Ps{t,A,D)^{l^t)Q{t,A,B,D).
Let ci < C2 < ''' < Cm he the critical values of / | e . Choose m-\- 1 numbers Gi such that (10.25)
ao := inf / < ci < ai < C2 < • • • < ftm-i < c^ < «m •= s u p / , e e
Define e^ := (6 n r O U e ~ , e^ := {ueSi'.
I{u) > a^_i},
z = O, l, 2, . . . , m;
e," := {u G e^_i : I{u) > a^_i}
for i = 1,..., 771. Note that B ^ = B, BQ = B~ and exc
Hl(Qi,Qi_^)
^ Hl(Qi,Q-),
i = l,2,...,m.
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CHAPTER 10. COHOMOLOGY
GROUPS
If u e S n I~^{ai), we have that {r{u),V{u)) > 0. It is easy to see that (6^, 6~) is an admissible pair for / and the critical points pj satisfying I{pj) = Cj. By Lemma 10.20, m
(10.26)
m
Mi{e,e-) = ^M|(e,,e-) = ^/?|(e,,e-). i=l
i=l
It follows that each pair (B^, 0~_^) and (B^, B~) are S- finite. The exactness of the cohomology sequence of the triple (B^, B^_i, B^_2) implies that (B^, B^_2) is also S- finite. By induction, (B,B~) = (B^,Bo) is also S- finite. If in (10.24), we replace A by B ^ = B, 5 by B^, L) by B^_i, we obtain Ps{t, B, B,) + Ps{t, B„ B,_i) = Ps{t, B, B,_i) + (1 + t)Q{t, B, B„ B,_i). Summing up, we find m
J2 Ps{t, Oi, Oi-i) = Ps{t, e, e-) + (1 + t)Q{t), where Q{t) has coefficients a^ G [Z+] and a^ = [0] for almost all q. Finally, multiplying (10.26) by t^ and summing over q, we have
M£{t,e,e-) m
= ^P^(t,B„B-) m
= ^P^(t,B„B,_i) i=l
= P^(t,B,B-) + (l+t)Q(t). D Theorem 10.22. Assume that I G C^(^,R) satisfies (PS)*. Suppose that a
= Ps{t, I\ n
+ (1 + t)Q(t),
where Q{t) is as in Theorem 10.21. Proof. Choose (B,B~) := (/~^([a, 6]),/~^(a)). It is an admissible pair by Lemma 10.10. Invoking exc
,
F^(e,e-) - H*s{i\n,
10.3. THE SHIFTING THEOREM
231
we see that the conclusion of the theorem fohows immediately.
D
Notes and Comments. The results of this section are due to W. KryszewskiA. Szulkin [200]. We refer the readers to K. C. Chang [96] and J. MawhinM. Willem [252] for the Morse Inequalities of the singular relative homology groups. An earlier version was due to J. Milnor [258], and the earliest results were due to M. Morse [259].
10.3
The Shifting Theorem
Let L be a bounded linear selfadjoint operator on a Hilbert space E with range 1Z{L) and null space J\f{L). We denote by M~{L) the Morse index of L, by M^{L) the nullity of L (i.e., M^{L) := dimJ\f{L)). Suppose that L is a Fredholm operator. Let Q^ : 7^(L) ^ 7^(L) n E^ be the orthogonal projection of 1Z{L) onto 1Z{L) H E^, and Pfi ' E ^
En
be the orthogonal projection of E onto E^. Proposition 10.23. Pn — Qn '^ciy be regarded as an operator Moreover, P^ — Qn -^ 0 in C{1Z{L), E) as n ^ oo.
ofC{lZ{L)^E).
Proof. Obviously, {PnU — QnU^v) = 0 for all v G 7^(1/) ^ E^. It is easy to verify that En = 7^(L) n ^ ^ © PnJ^{L) and that 7^(L) n En and PnAf{L) are orthogonal. Hence, PnU — QnU G PnJ\f{L). So, we may find Si Wn ^ -^i^) such that PnU — QnU = Pfi^Jfi- Since dinx/V(I/) < oo and Pn -^ id uniformly on compact sets, we have that ll^^ll < 2||P^^^|| = 2\\PnU - Qnu\\ < 4\\u\\. It follows that
\\PnU-Qnuf = {PnWn.PfiU=
QnU)
(PnWn.u)
= {PfiWn
-Wn,u)
<\\{Pn-id)Wn\\\\u\\.
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CHAPTER 10. COHOMOLOGY
GROUPS
Therefore, P^u — Q^u -^ 0 uniformly in i^ G 7^(1/) with ||i^|| < 1, whence the proposition. D We denote by C{E^ E) the space of bounded linear selfadjoint operators from E to E and by ^0(^5 E) the space of Fredholm operators of index 0. Definition 10.24. A map L : E ^ E is called A-proper ifL = A-\-B with A a bounded linear Fredholm operator of index 0, A{En) C E^ for all n and B compact. Lemma 10.25. Let L G C{E,E) be a self-adjoint A-proper operator. Then there exists a c> 0 and UQ > 1 such that if n > UQ, \\PnLu\\ > c\\u\\ holds for allu G 7^(L) nEn. Proof. Assume that for any A: > 1 there are an n/c > A: and Un^ G 'Tl{L) H En^ such that \\Pn^LunA\ < \\unA/^ properness of L implies that which is a contradiction.
implies PnM^^^
-^ 0. The A-
^^ -^ y ^ ^ ( ^ ) ^^^ \\y\\ = 1. But Ly = 0, ll'^nfc II D
Define the
M-{L):=
lim (M'{Q^L\niL)nEj
- d.
Lemma 10.26. Assume A G C{E,E) n ^^{E.E), A{Er^) C E^^ for almost = dn-\- k for almost all n all n, and B G C{E^E) is compact. If M~{A\E^) and some k e Z, then Mg {A -\- B) is well defined and finite. Proof. Let L := A-\- B, and let Q : ^ ^ ^ ( ^ ) be the orthogonal projection,
Fr, := (7^(L) n ^^+1) n (7^(L) n ^ ^ ) ^ ,
w^ := ^n+i n E^,
H^:=Pn+i-Pn'^E^W^. For any u G E^^ if u is orthogonal to PnN'{L), then 0 = {u, PnW)
= {u, w)
for ah w G N{L). It follows that, for u G 1Z{L), (10.27)
En = {n{L) n En) © PnAf{L),
10.3. THE SHIFTING THEOREM
233
and the sum is orthogonal. Hence (10.28)
K+i
(10.29)
= =
(7^(L)nK)©F^©P^+lA^(L) (7^(L) n Er^) © PnJ\f{L) © Wr^
and the sums are orthogonal. Note that L is A-proper. Thus by Proposition 10.23 and Lemma 10.25, it is easy to show that there exist CQ > 0 and no such that no. \\QnLu\\ > co||^||,V^ G 7^(L) nEn,n> For n > no and u G ^{L) H ^n+i, write (10.30)
U = V^W
= V^ PnX + Z,
where v G (7^(1/) H En),w G F^, x G M{L),z DnU = Qn+lLu
G VF^, and define
- {QnLQnU +
HnAHnU).
Combining this with (10.30) and noting that A{En) C E^, we have
= {L{v -\- w),v -\- w) — {Lv, v) — {Az, z) = 2{L{v + z), Pnx) + {LPnX, Pnx) + 2{Bv, z) + {Bz, z). Since B is compact, z G W^ C F ^ , and LP^x -^ 0 uniformly in x G M{L) H 5(0,1), and it follows that ||-Dn|| ^ co/2 for large no- Hence, for all A G [0,1] and u G 'Tl{L) H ^n+i?
||Q,+iL^ - AT^^II > IIQn+ii^^ll - A||T,^|| > co||^||/2. Further, recall that 7^(1/) H F^ is orthogonal to W^. Hence, we have (10.31)
M-(Q,+lL|7^(L)^^.+J = M-{{QnLQn^HnAHn)\n{L)nE^^,) = M-(Q,L|7^(L)^^J+M-(A|^J.
On the other hand, A{Ej^) C Ej^ implies that (10.32)
dn+i -dn =
M-{A\E^^^)
-
M-{A\EJ
=
M-{A\wJ-
By (10.31) and (10.32), we see that M~((5^I/|7^(2.)n£;^) — d^ is Si constant for almost all n. It is finite since M~{QnQA\^(^L^f^En) ^ d^ -\- k, dim.Af{A) < oo and B is compact. D We have
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CHAPTER 10. COHOMOLOGY
GROUPS
Theorem 10.27. (Nondegenerate Case) Assume that I G C^(^,R) and that p is an isolated critical point of I satisfying I{u) = I{p) ^ -{L{u - p),u - p) +(/)(^), where L is an invertible A-proper operator and (j)\u) = o(||i^ — j9||) as u ^ p. If Mg (L) is well defined and finite, then [^],
cui,p)
forq =
[0], If\M^{L)\
M^{L),
•
= +00, then Cl{I,p)
otherwise.
= [0], V^.
Proof. Consider the family of functionals Ix{u) := I{p) + ^{L{x-p),x-p)
+ (1 - A)(/)(^),
A G [0, l],ue
E.
Since L is invertible and A-proper, there exist CQ > 0 and no > 1 such that llP^Li^ll > co||i^||,
u G En,n > no.
Choose £ > 0 such that \\
yueB{p,e).
Then it is easy to check that Ix satisfies {PSy Theorem 10.15,
on B{p^e). Therefore, by
Cl{h,p)=Cl{I,p). Next, we compute the left-hand side of the above identity. Without loss of generality, we may assume that p = 0. Hence, Ii{u) = ^{Lu,u). Let
By Lemma 10.10, (6, 6~) is an admissible pair for /i and 0. If n > no, IIIE^ is a nondegenerate quadratic form (since L is invertible) and {Qr\En,S~r\En) is an admissible pair for IIIE^ and 0. We compute i!f*(B HE^, S~ HE^) first. By using an equivalent inner product, we may assume that
h{u) =
l\\u^'-l\\u-\\^
where u"^ G E^, En = E^ ® E~. For i^ G / I n En, define K.{u) = m i n | / ^ > 0 : /(i^++ (1 - K)U-) G [-1,1]}.
10.3. THE SHIFTING THEOREM
235
Then the mapping (A,i^) ^ i ^ + + (l -Xn{u))u-,
A G [0,1]
is a strong deformation retraction of
(ilnEn.i^^nEn)
onto{enEn,e-nEn).
Similarly, the mapping (A,^)
-^ ( 1 - A ) ^ + + ^ " , A G [0,1]
is a strong deformation retraction of
{ll n E^,l{^ n E^) onto {ll n ^ - , / r ' n E-) = {E-,E-\B{o, 1)). Hence, (6 H ^ ^ , 6 ~ r\ E^) is a homotopy equivalent of ( 5 , ^ 5 ) , where 5 is the closed unit ball in E~. Thus,
H%enEn,e- nEn) ^H%B,dB). Since d i m ^ ~ = M~{PnL\E^),
we see
m^^-{e n K , e - n K) ^ ^ if ^ = M - ( P , L | ^ J - (i, and 0 otherwise. By the definitions of Hg and Mg (L), we achieve the proof of the theorem. D
Assume that there is a neighborhood U of the critical point p such that / G C^(t/, R). Then we may express / as (10.33)
I{u) = I{p) + ^{L{u-p),u-p)
+ (/)H.
Assume that the operator L is Fredholm. Write u = p^x^y,
xeAf{L),
yeU{L).
Let Q : E ^ ^ ( ^ ) denote the orthogonal projection (onto 1Z{L)). Then I\p ^x^y)
= Ly^ d^'{p + X + 7/),
^(p) = 0,
I"{p) = L.
Since I/|7^(L) is invertible, it follows from the implicit function theorem that there exist a (5* > 0 and a C^-function y = fi{x) : 5(0, (5*) n J\r{L) -^ n{L)
236
CHAPTER 10. COHOMOLOGY
GROUPS
such that /i(0) = 0,/i'(0) = 0 and that (10.34)
Q r ( p + x + /i(x)) = 0.
Define (10.35)
M^)
•=
/(p + x + / i ( x ) ) - / ( p )
=
-(L/i(x), /i(x)) + (/)(p + X + /i(x)).
Suppose 0 is an isolated critical point of (/)o, and let C^((/)o, 0) := H^{S, B~), where (B,B~) is an admissible pair for (po and 0 in M{L) with a trivial filtration of it.
Let X G M{L),y G 1Z{L),\ G [0,1], and assume that x ^y e 5(0,(5*). We construct a family of functionals: /(A,p + x + 7/) (10.36)
= /(p) + l{Ly,y)
+ iA(2 - A)(L/i(x),/i(x))
+A(/)(p + X + /i(x)) + (1 - A)(/)(p + X + 7/ + A/i(x)). Then /(O, -) = I and (10.37)
/ ( l , p + x + 7/) = /(p) + i(L7/,7/)+(/)o(x).
After several computations, we observe that (10.38)
/^(A,p + X + 7/) = LT/ + (1 - X)Qcl)\p + X + 7/ + A/i(x))
and that
(10.39)
lUX,P + x + y) = A(2 - X){Lii{x), n'{x)-) + \{(t)'{p + X + ii{x)), • + ix'{x) •) +{l-\){(t)'{p + x + y + XiJL{x)), • + Xii'{x) •)
Denote /(A) = J(A,-),
Ae[0,l].
Lemma 10.28. There is ar > 0 such that /(A) satisfies the {PSy for each A.
on B{p, r)
10.3.
THE SHIFTING
THEOREM
237
P r o o f . Let (uk) be a (PS)* sequence: Uk = p -\- Xk -\- Vk ^ S{p,r) En,,PnJ'{\uk) -^ 0,nfe ^ oo. By (10.38),
H
PnJy{\p^Xk^yk) = PfikLyk + (1 - \)PnuQ4>'{P ^Xk^Vk^
A/i(xfc))
= • fk
Let Ck := ?/fe - (1 - X)fi{xk). /fc = (1 - \)PnkLll{Xk)
We get
+ Pnfc^efc + (1 - \)PnkQ4>'{P + ^fe + /^(^fe) + Sfc).
By (10.34), Lii{xk)
+ Q(/>^(p + ^fe + /^(^fe)) = 0.
Hence, (10.40) fk = PrikLCk^
(1 -X)PnkQ[
^ fi{Xk) +efe) -(/)'(p + Xfe+/i(Xfe))j.
Note t h a t Pfik^k -ek
= {id-
Pnk){p^Xk^{l
-X)fi{xk)).
Therefore, by Proposition 10.23, Puk^k — e/c ^ 0 and Quk^k — Ck^O. by Lemma 10.25, we have t h a t WPn^LekW (10-41)
>
Hence,
WPn^LQu^ekW - WPnuL^Qnu^k - ek)\\
>
c\\Qnf,ek\\ -
>
c\\ek\\ - {\\L\\ ^ c)\\{Q^,ek
||L||||(Q^,efe-efe)|| - ek)\\
for almost all k. Furthermore, since (j) G C^(t/, R ) and (/)^^(p) = 0, we may choose r sufficiently small so t h a t (10.42)
^llefell > ||(/)^(p + Xfe+/i(xfe) + efe)-(/)^(p + Xfe+/i(xfe))||.
Combining (10.40), (10.41) and (10.42), we have ^l|efe|| + | | M I > c | | e , | | - ( | | L | | + c ) | | Q ^ , e , - e , | | . This implies t h a t Ck -^ 0. convergent subsequence.
Since dimN{L)
L e m m a 10.29. C | ( / , p ) ^ C | ( / ( l ) , p ) .
< oo, we see t h a t (uk) has a D
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CHAPTER 10. COHOMOLOGY
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Proof. Let r be as in the proof of Lemma 10.28. If necessary, we may choose r so smah that JC{I) H B{p,r) = {p}. We claim that p is also an isolated critical point of the family {/(A)}. Suppose that p -\- x -\- y G B{p,r) and r ( A , p + x + 7/) = 0. By (10.38), Qr^(A,p)|^(^) = L|^(^). Hence, by the implicit function theorem, (10.38) has a unique solution y = y{X,x) if r is sufficiently small. Moreover, by (10.34), y = (I — A)/i(x). Therefore, (10.34) and (10.39) imply /^(A,p + x + ( l - A ) / i ( x ) ) = A(2 - X){I\p + X + /i(x)), /i'(x)-) + (id - Q)(l)\p + x + /i(x)) = (id-Q)(/)'(p + x + /i(x)). Note that in B{p,r), I'{p ^x^y)
= 0^x^y
= 0,y = /i(x), {id - Q)(i)'{p + x + /i(x)) = 0.
Hence, we must have x = 0,7/ = 0. This means that /C(/(A)) H 5(j9, r) = {p}, and the claim is proved. By applying Theorem 10.15, we get the conclusion of the lemma. D Assume that p = 0,I{p) = 0. Since {71{L) H ^ n ) ^ i is a ffitration of 1Z{L), there is an admissible pair (Bi,Bj") for {Ly,y) in 7^(1/) such that Bi is bounded. Let Vi(^) be the corresponding admissible field. Let (B2, B^) be an admissible pair for (j)o and 0, B2 C 5(0, S) r\Af{L) and V2{x) be the corresponding admissible field. Choose mo so that Pmo \j\f{L) • •^(^) ^ Pmo-^iL) is a linear isomorphism and define 6)3 • = 5 m o ® 2 ,
©3 • = 5 m o ® 2
and (10.43)
(6, e - ) := (Oi + 03, (Or + 63) U (61 + 63 ) ) .
We have
Lemma 10.30. (B, B ) of (10.43) is an admissible pair for / ( I ) and 0. Proof. For each u e E, write u = x -\-y = Pmo^ + / , where x, e G M{L) and y,f e n{L). Choose an £ > 0 so smah that 5(0, £)n7^(L) C int(Bi), 5 ( 0 , £ ) n M{L) C int(Bo). Let TT : R ^ [0,1] be a Lipschitz continuous function such that 7r(t) = 0 for t < e/2 and 7r(t) = 1 for t > e. Define V3(^):=^(||/||)Vi(/)+^(||e||)P^,V2(e),
10.3. THE SHIFTING THEOREM
239
It suffices to sfiow tfiat V4 is an admissible field in a neighborhood of ^ B . Let ^ 1 , 6 , ^4 be the flows of - V i , -V2, -V4. Then
satisfies the items 4-6 of Definition 10.9. Since / G 7^(i^) H ^ ^ whenever u e En and n > mo, V4 is
= 7^i\\f\\){V^if),Ly) + ni\\e\\){V2ie),<|>oi^)) +7r(||e||)((P™„-id)V2(e),iy + 0^(x)> >7r(||e||)ryi(/) + 7r(||e||)r?2(e)-£™„, where 6^^ ^ 0 as mo -^ 00 and r]i,r]2 are as in Definition 10.6. We may assume the neighborhood U has been chosen in such a way that u ^ U if \\y\\ < £ and ||e|| < £. Taking TTIQ large enough, we see that (V4(i^), Ly^(j)o{x)) is positive and bounded away from 0 on [/. Hence ( 6 , 6 ~ ) is an admissible pair. D Based on Lemmas 10.28, 10.29 and 10.30, we can finish the proof of the following theorem. We will use the notation as usual: (^1,^2) X {Bi,B2) = ( A I X 5 i , (Ai X B2) U {A2 X 5 i ) ) .
Theorem 10.31. (Shifting Theorem) Assume that U is a neighborhood of an isolated critical point p of I ^ C^(t/, R) and that the operator L in (10.33) is A-proper. If Mg (L) is well defined and finite, then
/ / \M^{L)\ = +00, then Cj{I,p)
= [0],
Vg.
Proof. Let mo be as in Lemma 10.30. Then (61 + Os) nE„ = (Oi n E„) + 0 3 .
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CHAPTER 10. COHOMOLOGY
GROUPS
We compute the cohomology of
((Bi n E^) + 63, ((er n E^) + 63) u ((Bi n E^) + e^)). Topologically this is equivalent to
(ein^^,ern^^)x(e3,e^). Let 5 be a closed ball of dimension dim^ := M~{QnL\^(^L^f^E^)- Then (61 H En, Bj" n En) is homotopically equivalent to {B, dB) for almost all n. By the Klinneth formula, we have
i7^+^-((einK,ernK) x (63,63")) ^i7^+^-((5,a5)x (63,63-)) ^ [i:f*(5,a5)0i:f*(63,6^)]^+^-
If Mg{L) is finite, then q-\-dn — dim^ = q — Mg (L) for almost all n. Hence, we get the first conclusion . If \Mg {L)\ = oo , then q -\- dn — dim^ < 0 or q^dndim^ > dimA/'(L) for almost ah n. Then m+d^-^'^^^^Q^^ Q-) = 0. This completes the proof. D We have the following critical groups for local maximum and minimum. Theorem 10.32. Assume that I satisfies the hypotheses of Theorem 10.31 and Mg (L) is finite. (1) If(j){u) > (j){p) = 0 for all ueU,
then
(
[^],
[
[0],
forq =
M^{L),
Clil.p) = (2) If 4>{u) < 4>{p) = 0 for all ueU, ( CI{I,P) = \ [
[^], [0],
otherwise. then
forq =
M^{L)+M°{L),
otherwise.
Proof. (1) According to the positive and negative spectrum, we split 1Z{L) as following: Tl{L) = E~^ 0 E~. Then we may find a constant CQ such that
10.3. THE SHIFTING THEOREM
241
Write fi{x) = fi^{x) + / i - ( x ) e E^ ® E'. Af{L), we get (10.44)
(/)o
=
By (10.35), for all x G 5(0, (^o) H
i(L/i+(x),/i+(x)) + (/)(p + x + /i+(x)) + 2 ^^'^"(^)' '^"(^^^ + (/)(p + X + /i(x)) -(/)(p + x + /i+(x)).
Let (9(t) : = p + x + / i + ( x ) + t / i - ( x ) , t G [0,1]. Then (10.45)
(/)(p + X + fi{x)) - (/)(p + X + /i+(x)) 0
dt
1
((/)'((9(t)),/i-(x))(it 0 1
{cl)\0{t)) - (j)'{p + X + /i(x)), /i-(x))(it 0
+ ((/)^(p + x + /i(x)),/i-(x)) 1
{^\e{t))-^'{p^x^^{x)),^-{x))dt 0
-(L/i-(x)),/i-(x))
(by (10.34)).
Note that (j)"{p) = 0. We choose 6 sufficiently small so that (10.46)
||(/)^(^(t))-(/)^(p + x + /i(x))||
r
[^],
iovq = M^{L),
[
[0],
otherwise.
(2) If (l)(u) < 0 in [/, then 0 is a local maximum of (^Q. The conclusion follows similarly. D
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CHAPTER 10. COHOMOLOGY
GROUPS
Notes and Comments. The results of this section are due to W. KryszewskiA. Szulkin [200]. We refer the readers to K. C. Chang [96] and J. Mawhin-M. Wihem [252] for the Shifting Theorem and critical groups of local maxima and local minima of the singular relative homology groups. The earlier version of the Shifting Theorem was due to D. Gromoll-W. Meyer [175]. The family of (10.36) was introduced by E. N. Dancer [121], where the characteristics of critical groups of a non-local-max (min) was obtained in the case of a finitedimensional space. Readers may consult A. Dold's book [138] on algebraic topology.
10.4
Critical Groups of Local Linking
Assume that E has a decomposition E = E~^ 0 E~. The functional / is said to satisfy the local linking condition at 0 if there exists a constant p > 0 such that I{u) < /(O) for ue E- with ||i^|| < p, I{u) > /(O)
for ueE^
with ||^|| < p.
Theorem 10.33. Let I G C^(^,R) have a local linking at 0 and satisfy the (PS)* condition. Assume that I maps hounded sets into hounded sets. If 0 is an isolated critical point of I, /(O) = 0, E^ = {E~ H E^) 0 {E~^ H E^) and dim(^~ n En) = qo -\- dn for almost all n, then C|«(J,0)^[0]. Proof. Suppose that 0 is the only critical point of / in a ball 5(0, p). Let 0 < ri < r < p and let ( 6 , 6 ~ ) , 6 C 5 ( 0 , r ) , be an admissible pair. Particularly, we may assume that (B, B~) has those properties in Lemma 10.11. Define Bi := {r]{t, u) eS
:t>0,ue
S{0, n ) n
E-}.
Since / < 0 for i^ G ^~with ||i^|| < r i , for u G Bi, there exists a unique t{u) such that r]{t{u),u) G B~. According to the Definition 10.9, t{u) depends continuously on u. Define r]{st{u),u),
if u e Bi, s G [0,1],
u,
if 1^ G B~, 5 G [0,1].
r]i{s,u):--
10.4. CRITICAL
GROUPS OF LOCAL
LINKING
243
It is a filtration-preserving strong deformation retraction of Bi U B Therefore, (10.47)
onto B
i7|(B,B-)-i7|(B,BiUB-).
For 1^ G ^ + , let r{u) := minjri, dist(i^, Bi U B )}, B2 := {u- ^u-^ eE-
®E^ : \\u- \\ < r(^+)}.
Hence B2 is open, E^ C B2 and (Bi U B") n B2 = 0. Set Fr := (5(0, r) n E-) © (5(0, r) n E^). Define inclusion mappings i and j as follows: i :
{B{0,ri) n E-, S{0,ri) n E-) -^ ( B , B i U B - ) , j:
(B,BiUB-)^(F„FAB2).
Then (10.48)
i7|(F„FAB2) ^i7|(B,BiUB-) ^i7|(5(0,ri)n^-,5(0,ri)n^-),
where i*,^* are the induced homomorphisms. Then the mapping
2sriu + (1-25)^-+^+, max{||i^~||,r(i^+)}
se [0,1/2],
riu~ + (2-25)^+, max{||i^~||,r(i^+)}
5 G [1/2,1],
is a deformation of (F^, F^\B2) onto (5(0, r i ) n F , 6^(0, ri)nE ). It preserves the filtration since u^ G E^ whenever u'^ -\- u~ G ^n- Moreover, V2\[0,l]x(B(0,ri)nE-
,S(0,ri)nE-)
is a homotopy between 7^2(1, O^OO and the identity on (5(0, r i ) n F ~ , 6^(0, r i ) n F ~ ) . 7^2 is also a homotopy between (j o i) o 7^2(1,-) and the identity on {Fr, Fr\S2)- Hence, the inclusion mapping j o z is a homotopy equivalence by
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CHAPTER 10. COHOMOLOGY
GROUPS
filtration-preserving homotopies. Thus, z*oji'* is an isomorphism (see (10.48)). Note i7f ( 5 ( 0 , r i ) n ^ - , 5 ( 0 , r i ) n ^ - ) = [ ^ ] , thus we have
n Notes and Comments. The idea of local linking and related results can be found in H. Brezis-L. Nirenberg [70], K. C. Chang [94], J. Liu-S. Li [233], S. Li-M. Willem [219], J. Q. Liu [230], M. Ramos-S. Terracini-C. Troestler [300], and E. A. B. Silva [336, 337]. If dim ^ ~ < oo, the characteristics of the critical groups with applications were given in S. Li-A. Szulkin [215], K. Perera [278] (Homological local linking), [279] (for asymptotically linear elliptic problems at resonance) and also K. Perera [277]. Theorem 10.33 of this section is due to W. Kryszewski-A. Szulkin [200].
10.5
Computations of Cohomology Groups
Let / G C^{E, R) be a strongly indefinite functional which satisfies the {PSy condition with respect to 0 such that aoo(5 -\- t) < c(aoo(5) + «oo(0) ^^^ ^^^ s,t G R+, • there are two constants a > 0, f3 > 0 such that
10.5. COMPUTATIONS (10.49)
a <
OF COHOMOLOGY *-^^
GROUPS
245
for all s, t e R+.
A special case of ttoo is aoo(0 = ^^ ^^^^ ^ ^ (^^ !)• Assume (Ai)
(AJ)
II J^HII < c ( l + aoo(ll^ll)) for any u e E',
liminf - —
> 0,
where A^dt) = / aoc{s)ds;
||-0||-oo A o o ( l l ^ ^ l l ) liOGker(L)
Jo
(A3)
/ satisfies the {PSy
condition;
(A4)
I\E^ satisfies the (PS) condition for each large n.
We obtain the following theorem about the precise computation of the
(As) and {A4). Then
(1) {A2) implies that [^],
forq = M^{L) + M^{L),
CI{I,)C)-[0],
otherwise.
(2) {A2) implies that [J^], forq =
M^{L),
ciiix)[0],
otherwise.
Proof. (1) Note that ker(L) C E^, then E^ = E^ Q E' Q ker(L) and there exists a constant n > 0, which is independent of n, such that
{Lu^,u^)>K.\\u^f
for
u^eE^.
We write u = u'^ -\-u~ -\-u^ with u"^ G E^^vP G ker(I/) and set:
(10.50) M:={u:ue E^, ||.+ f " ^ l l - - f " ^ ^ ^ S " "^1'
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CHAPTER 10. COHOMOLOGY
GROUPS
where the parameters r, T are to be determined later. The normal vector on the boundary dM of V is given by no :=no{n,u)
r u^ = u~^ - du~ - 2'^'(ll^^ll)^;^'
where (10,51)
. ^ ^ ,
m - f ^ .
Next, we show that I\En has no critical point outside V for appropriate r and T. Indeed, by {Ai) and for e > 0 small enough, we first observe that (/'(u),no> = {Lu^, u^) — d(Lu~ ,u~) + (J'(u), no) >K\\u+f + dK\\u-f -c{l+a^{\\u\\Mu+\\+d\\u-\\+Ti}'{\\u^)) > K\\u+f + dK\\u-f - c{l + a^iWu^'W) + \\u+\f-'
+
\\u-f-'){\\u+\\+d\\u-\\+Td'{\\u^))
-ceT'\^'(\\u%f-cs-'al{\\u%-c. By (10.51) and the definition of ttoo and a simple calculation, it is easily seen that
for t > 0. Choose r > (10.52)
, then
cer'\^'{\\u^)\^+cs-'al{\\u^) < cer^ (l + |k0||2)4l|k0||^^ + ll^ II ^ + 11" II (1 + ||U0P)4 +ce' - o 2 i ll + II„,0II2 | | w 0 | | 2 IV
-
„ K
2 l + |k0||2 ^ ^ -
n( l +1 II„,0II2\3'^ ||uO||2)3V
IL.OII ||yO||
" " " l l " II''
10.5. COMPUTATIONS
OF COHOMOLOGY
GROUPS
247
Consequently, for sufficiently large T, we have that
(/>),no)>f(||^+f-^|Kf-r^|M)_, (10.53)
>^T-r - 2 >0.
Let —V denote the negative
< Imww+f - \^\\u-f+j{u')+j{u)
- j{u')
+ J{u') +
+c£||u+ + u - f + c £ - i ( l +aoo(||w°||) + \\u+\f-' + \\u-\f-'^^ Hence (10.54)
I{u) < ( i | | L | | d - i m + ce + c e d ) | | u - f + c e - i ( l + d ) | | u - f ( / 5 - i ) + ( ^ | | L | | r + c£T)^(||uO||) + ce-V^('^-i)( +ce-'al{\\u^)
+
k°ll
J{u')+c
<-.^n\\u-f+cs-'{l+d)\\u-f(^-'^
In view of {A2 ) and the definition of ttoo, we observe that lim
„ ,|| „||, = - 0 0 .
By (10.54), I{u) -^ —00 as |u*^ + w~|| -^ oo uniformly for n. However, by (10.50), \\u+f
for
ueM.
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CHAPTER 10. COHOMOLOGY
GROUPS
Consequently, I{u) = -{Lu+,u+)
+ -{Lu-,u-)
+ J{u)
>-\\L\\\\u-f-c{l+a^{\\u^\\) + \\u+f-' + \\u-f-^)\\u++u-\\+J{u^) >-\\L\\\\u-f-cs\\u+ + u-f -cs-\l
+ a^(||««||) + ||^.+ f C^-i) + | | ^ . - f (/'-D) + J(^,0)
> - | | u - f (||i|| + ce + ced) - (ce-^d + c £ - i ) | | u - f (/^-i) -ceT^{\\u^\\) - ce-'r^(^-'\\\u^\\) - ce-'aU\\u^\\) + J{u') - c, which imphes that ||i^~ + i^^|| ^ oo as I{u) -^ —oo. Now we choose a > 0 such that /C = /C(/) C {z G ^ : \I{u)\ < a}. Then the above arguments imply that there exist b > a and Ri > R2 > 0 such that Mi:={ueM:
\\u^^u-\\
> Ri}
cr^nM CM2:={ueM:
| | ^ V ^ " | | > R2}
Obviously, there exists a geometry deformation retraction "^ of M2 onto Mi. By the (PS)* condition, we may assume that JC{I\E^) C M\I~'^[—b,—a]. Thus the flow of —V provides a strong deformation retraction r] of / ~ " H Ai onto I~^ nAi. Then 1!^ * ?^ is a strong deformation retraction of I~^ H Ai onto Ml. Also by the flow of —V, we obtain a strong deformation retraction of / " n Er^ onto ( / - " n ^ ^ ) U Ai . Therefore,
^ m{{r'' n K) u Ai, /-" n K)
{
T^
\i q = dim E~ + dim ker(I/),
0,
otherwise.
Since (/~^([—a, a]),/~^(—a)) is an admissible pair for / and /C(/) and d i m ^ ~ = Mg (L) -\- dn, we see that (10.55) implies Hl{I,JC{I)) - ^,,M-(L) + MO(L)[-^].
Vg e Z.
(2) Assume that (^2^) holds. We consider T:={ueE:
\\u-f
- d\\u+f - T^\\U^)
< T}.
10.5. COMPUTATIONS
OF COHOMOLOGY
GROUPS
249
Then the normal vector on dT is no := no{n,u) = u
— du'^ —-i}W\u^\
u^
2 ^" "^K||-
By a similar argument, there exist r and T such that (rH,no)<-^T + c<0 and that I{u) ^ oo
if and only if
||i^+ + '^^11 ^ oo
uniformly for n. Therefore, we may find a > 0 such that 1C = 1C{I) ^{ueE:
\I{u)\
and
/-" n ^, c Er,\r. Moreover, there exist Ro> {)^h > a such that I^nr
^D:={uer
: ||^++^^|| < i ? o } c / ^ n f .
It is easily seen that there exists a strong deformation retraction 7 of E^ onto D U dT. By the {PS)* condition, I\En has no critical value in [a, 6] for a, 6 and n large enough. Hence,
x:(/|^jcr\/-i[a,6]. Since /l^;^ satisfies (P^') (see (^4)), we see that the fiow of —V induces a strong deformation retraction r] of {En\T) U L) onto I^ H ^n. Combine ?^ and 7, giving
[ 7(2r,^),
0
(?^*7)(r,^) := <
[ 7?(2r-1,7(1,^)),
\
Thus we get a strong deformation retraction of E^ onto I^ r\ E^. Since / - " n ^^ C ^n\T,
and (r(^),no) < 0,
we can use the fiow of —V again to construct a strong deformation retraction of En\T onto / - " n En. It follows that
^m{En^En\r) T,
q=
M^{L)^dn,
0,
otherwise.
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CHAPTER 10. COHOMOLOGY
That is, Hl{I,]C) theorem.
^ ^q^M-{L)[^]^
GROUPS
^^ ^ ^ This finishes the proof of the D
From the next two theorems, we obtain precise computations of the 8cohomology critical groups at zero. We write u = u'^ ^u~
^u^
with u^ G E^
and u^ G E^.
Theorem 10.35. Assume that I G C^(^,R) satisfies the {PS)* condition. Suppose that J\0) = 0 and that J maps bounded sets into bounded sets. Then we have the following alternatives. (1) If there exists a p > 0 such that {I\u)^u^ with \\u\\ < p, then (
[^],
— u~ — u^) > 0 for all u ^ E
forq =
M^{L)+M°{L),
cui,o) = l [
[0],
otherwise.
(2) If there exists a p > 0 such that {I\u)^u^ with \\u\\ < p, then (
[T],
[
[0],
— u~ -\- u^) > 0 for all u ^ E
forq =
M^{L),
otherwise.
Proof. (1) Without loss of generality, we may assume that 0 is an isolated critical point of I in Up := {u ^ E : \\u\\ < p}. For any A G [0,1], we consider the perturbation h{u) = hLu,u)
+ (1 - X)J{u) + lx{\\u+f
- \\u-f
-
Note that (/;(«), ^ * + - ^ * - - « o > = {Lu, u^ — u~ — u^) +(1 - \){J\u),u+ -u- - u^) + X\\u^f > {Lu,u^ — u^ — u~) +A||^^f + min {0, {J\u),u-^ - u' - u^)} >0
\\uY)-
10.5. COMPUTATIONS
OF COHOMOLOGY
GROUPS
251
for u G Up\{0}. We conclude that Ix has a unique critical point 0 e Up. Furthermore, SUP{|/A('?^)| :U eUp.Xe [0,1]} < oo and A ^ /^ is continuous uniformly for u e Up. By Theorem 10.15, we have that C|(/,0)-C|(/o,0)-C|(/i,0). Let Liu = Lu-\-u'^ — u~ — u^. Then Li is a bounded linear Fredholm operator of index 0. Hence Li is an invertible A-proper operator. Note that L{En) C E^. Thus it is easy to verify that By Theorem 10.27,
This completes the proof of case (1). (2) The proof is analogous to case (1) by setting 1_ _ . 1 h{u) = -{Lu^u) + (1 - X)J{u) + - A ( | | ^ + f + ll^^f - l l ^ - f ) . D Theorem 10.36. Suppose that I G C^(^,R) satisfies the {PSy condition and that J maps bounded sets into bounded sets, J'{u) = o(||i^||) as \\u\\ ^ 0 . For p > 0, /^ > 0, let u = u^ ^u~ Mp^^:=
{ueE
^u^ e E,
||^++^-|| <^ul
0< ll^ll
Then one of the following alternatives holds true. (1) If there exists p > 0, ^c G (0,1) such that {I'{u)^vP) > 0 for u G Mp^^^ then [:F], forq = M^iL), C|(J,0) [0], othe
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CHAPTER 10. COHOMOLOGY
GROUPS
(2) If there exist p > 0, /^ G (0,1) such that {r{u),u^) < 0 for u e Mp^^, then M^{L)+M\L), [^], forq = [0],
otherwise.
Proof. (1) Define the perturbation h{u) = ^{Lu,u)
+ (1 - X)J{u) + ^XWu'f,
ueE,
Xe% 1].
If 1^ G Mp^ ^, then i^^ ^ 0 and
= (1 - A)((L^,^0) + ( J ^ H , ^ 0 ) ) + All^of >0 for ah Xe[0,l].lfue{ueE: smah enough, we have that
\\u\\ < p}\Mp^^ and i^ ^ 0, then for p > 0
{I'^iu),u+-u-) = (Lu, u+ - U-) + (1 - X){J'{u), u+ - U-) \\J'iu)\\ \\u+ +u-
>ii...+..-inc-(i^Ml« hi
\\u
>0. It fohows that 0 is the unique critical point of /A in {u e E : \\u\\ < p}. By a similar argument, C|(J,0) ^ C|(/o,0) ^ C|(Ji,0) ^ <5,,M-(L)[-^].
Vg e Z.
(2) The proof is analogous to case (1) by introducing h{u) = l{Lu,u)
^ {1 - X)J{u)
-hWu^f. D
Notes and Comments. The computations of the singular relative homology groups have been studied by many authors (see e.g. K. C. Chang [96] and the references cited therein). We also refer the readers to recent papers
10.6. HAMILTONIAN
SYSTEMS
253
by K. C. Chang-M. Y. Jiang [98], T. Bartsch-S. Li [37], V. Benci [52], K. C. Chang-N. Ghoussoub [97], N. Hirano-S. Li-Z. Q. Wang [185], S. Li-J. Liu [212], S. Li-W. Zou [220], J. Liu-S. Liu [234], K. Perera [275, 276, 277], K. Perera-M. Schechter [284, 285, 287], W. Zou [392], W. Zou-S. Li [394] and W. Zou-J. Liu [396]. The results of [37, 212, 220, 396] can not be used directly to deal with strongly indefinite functionals. In A. Szulkin [357], the cohomology group was defined as the limit of a cohomology sequence. In A. Abbondandolo [1, 2], a generalized cohomology, similar to W. KryszewskiA. Szulkin [200] but with more general functional properties, is constructed. The
10.6
Hamiltonian Systems
Consider the existence of nontrivial 27r-periodic solutions to the asymptotically linear Hamiltonian system (HS)
z = Jn'{z,t),
ZGR^^,
where / 0 '^ ''~\id
-id 0
is the standard symplectic matrix; Ti G C'^{R?^ x R, R) is 27r-periodic in t] Ti' denotes the gradient of Ti with respect to the first 2A^ variables. Assume that there exist 5 > 0, c > 0 such that (Bo) \Hzz{z,t)\ < c(l + \z\') for ah {z,t) G R^^ x R. Suppose that there exist two symmetric 2A^ x 2A^ matrices A(t) and Ao(t) with continuous 27r-periodic entries such that
(10.56)
n{z, t) = ]-A{t)z ' z + G{z, t),
where G'{z^t) = o{\z\) uniformly in t as \z\ -^ oo and
(10.57)
1-L{z, t) = ^Ao{t)z . z + Go{z, t),
where GQ{z,t) = o{\z\) uniformly in t as \z\ -^ 0. We denote by • and I * I the usual inner product and norm in R^^.
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The Hamiltonian system (HS) satisfying (10.56) and (10.57) is called asymptotically linear both at infinity and at zero. Moreover, it is called nonresonant at infinity if 1 is not a Floquet multiplier of the linear system z = JA(t)z; nonresonance at 0 is defined in a similar way by replacing A{t) with Ao(t). We assume (Bi) \G'{z,t)\ < c ( l + aoo(k|))
for a h z G R ^ ^ and t G R;
(B^) liminf ^.^^^' f; := a^{t) h 0 uniformly for t G R. \z\^oo
Aoo{\z\)
Here and in the sequel, we write a{t) >z 0 if a{t) > 0 and strict inequality holds on a set with positive measure; ttoo, ^oo are the control functions given in the previous section. Let ho : R ^ -^ R ^ be a control function of Go such that
(10.58)
2 < /3o < ^ J 4 T ^ 70 for t smah, ^o(^)
where Ho{t) = J^ ho{s)ds, and /3o, 7 are constants. Obviously, ho{t) = t^ with (5 > 1 is a simple example. Moreover, although ho is defined only for small t > 0, we may assume without loss of generality that it has been extended so that (10.58) holds for ah t G R+. Suppose (B3)
\G'o{^,t)\ < cho{\z\)
for \z\ small;
-\-G^ (z f) z liminf ^^-f-^— := b^{t) h 0 uniformly for t G R. \z\^o Ho[\z\) In order to state our results, we shall need the notation of the
/ Jo
{-Jz - Az) ' zdt,
where A = A(t) is a symmetric 2N x 2A^-matrix. Denote this index by i~{A), the nullity of the quadratic form by i^ (A) and let
i^{A) =
-i-{A)-i\A).
Note the Maslov-type index or the Conley-Zehnder index (cf. H. AmannE.Zehnder [10], C.C. Conley-E.Zehnder [105], Y. Long [241] and C. Liu [229])
10.6. HAMILTONIAN
SYSTEMS
255
of the fundamental solution 7 : [0, 27r] -^ Sp{2N) of the equation i(t) = JA{t); here Sp{2N) denotes the group of symplectic 2N x 2A^-matrices and a matrix T is called symplectic if T^JT = J. If we denote the Maslov-type index of A by (i,n), then i = i~{A) and n = ^^(A). Theorem 10.37. Assume thatH G ^^(R^^ x R , R ) satisfies (BQ), (Bi) and one of the conditions (Bf). Then (HS) has a nontrivial 2ii-periodic solution in each of the following two cases: (1)
(B^) andi-{A)
(2)
{Bt)
^ [r(Ao), r (AQ) + zO(Ao)];
and i+{A) i [z+(Ao), z+(Ao) +
i\A^)].
Theorem 10.38. Assume that H G ^^(R^^ x R, R) satisfies (Bi) and (Bs). Then (HS) has a nontrivial 27T-periodic solution in each of the following four cases: (1) (2) (3) (4)
{B+), {B+), (B^), (B^),
{B+) (B4-) (Bt) (S4-)
and and and and
i-{A) i-{A) riA) i-{A)
+ i^{A) ^ r (AQ) + i'{Ao); + i^{A) + r ( A o ) ; ^ z-(Ao) + iO(Ao); ^ i-{Ao).
If the difference between the
(^2+), {Bt), {B^), {B-),
{Bt) {BD {Bt) {BD
and and and and
\i+{A) \i+{A) | r (A) | r (A)
^ + -
i+{A^)\ i'{A^)\ z+(Ao)| i'{A^)\
> > > >
2N; 2N; 2N2N.
Theorem 10.40. Assume that H G ^^(R^^ x R, R) satisfies {BQ), {BI), one of the conditions {B^) and A{t) = Ao{t) = 0 (hence 1-C{z,t) = G{z,t) = Go{z,t)). Furthermore, let l-[\z,t) = o{\z\) uniformly in t for \z\ -^ 0. Then {HS) has at least two nontrivial 27r-periodic solutions in each of the following two cases: (1) condition {Bt) holds and either there exists a (5 > 0 such that 1-L{z^t) < 0 whenever \z\ < 6 or (^3), {B^) are satisfied;
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(2) condition {B2) holds and either there exists a S > 0 such that 1-L{z^t) > 0 whenever \z\ < S or (^3), (B^) are satisfied.
Let E := ilf 2 (^'^R^^) be the Sobolev space of 27r-periodic R^^- valued functions of the form 00
(10.59)
z(t) = ao +/_^(a/c cos/ct + 6/c sin/ct),
ao, a/c, 6/c G R^^,
k=l 00
such that Yl ^(l^feP + l^feP) < ^^- Then ^ is a Hilbert space with a norm k=l
II • II induced by the inner product (•, •) defined by (10.60)
(z, z') := 27rao • ag + TT ^
k{ak • a'j^ + 6^ • ^fe)-
Let Fk := {a/c cos /ct + bk sin kt : ak^bk G R ^ ^ } ,
A: > 0,
and n
n
^n •= ^ ^ f e = {^ G ^ • ^(0 = «o + y^(afc COS H + 6fc sin kt)}. k=0
k=l
Then ( ^ n ) ^ i is a filtration of E. Denote (10.61)
S' = {En,dn}
with
dn := N{1 ^2n)
=-dimE^.
If 5(t) is a symmetric 2N x 2A^-matrix with continuous 27r-periodic entries, then the operator B given by the formula {Bz,w) := / Jo
B{t)z'wdt
is compact. By Lemma 10.26, the operator LB given by />27r
(10.62)
(L^z, w) := /
is A-proper and M^{LB) (10.63)
{
( - J i - B{t)z) • i(;(it
is well defined and finite. Denote i-{B):=M^{LB), i+{B):=M+{LB) f{B) •=M^{LB)
= =
M^{-LB), dimker(Lij)-
10.6. HAMILTONIAN
SYSTEMS
257
Then we have i-{B)^i^{B)^i^{B)
=0.
Since M^{LB) is in fact the number of linearly independent 27r-periodic solutions of the linear system z = JB(t)z, therefore, 0 < M^(LB) < 2A^. It is well known that under condition (5i), z{t) is a 27r-periodic solution of (HS) if and only if it is a critical point of the C^-functional ^z)
= \ J^"{-Jz
- A{t)z) . zdt - /Q'" G{z,t)dt
(10.64) =
^{Lz,z)-^{z),
which can be rewritten as ^z)
= \ J^"{-Jz
- Ao{t)z) . zdt - /o'" Go{z, t)dt
(10.65) =
^{Loz,z)-^o{z),
where {Lz, z) =
{-Jz - A{t)z) ' zdt, Jo
{Loz,z)=
/
{—Jz —
Ao{t)z)'zdt,
/o PZTT
PZTT
r2-
^f{z)=
i Jo
G{z,t)dt,
'^o{z)=
Go{z,t)dt,
z e E.
Jo
Moreover, ^ G C'^{E,Il) if (BQ) is satisfied. By (10.56)-(10.56), it is easy to check that (10.66)
vl>'(^) = 0(11^11) as | | z | | ^ ^ ,
vl>^(z)=o(||z||)as||2||^0.
In particular, 0 is a trivial solution of (HS). Let L :=
LB,
LQ
:=
LBQ
and introduce a new filtration S := {E'^, (i^}^]^, where E'^ := {R{L) H E^) 0 ker(I/) and d^ = A^(l + 2n) as before. Then L, LQ are A-proper with respect to 8 because they are such with respect to 8^ defined in (10.62). Then (10.67)
M^,iL) =
M^{L)^i-{A)
and (10.68)
M^,{Lo) = M^{Lo) = i-(Ao).
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CHAPTER 10. COHOMOLOGY
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We will compute the critical groups C | ( ^ , 0 ) and C|(^,/C(^)). Therefore, we first show how conditions (Bi) and (^2 ) ™ply the {PSy condition with respect to 8. Lemma 10.41. Assume that (^2 ) holds. Then correspondingly, ±
liminf
f^''G(z,t)dt ^Q „ / >Q.
2;Gker(L)
Proof. Note that dimker(I/) < oo, we see that the norm || • || and the L ^ norm are equivalent on ker{L). Moreover, recall that z G ker{L) has the unique continuation property. Therefore, S\\z\\ < \z{t)\ < c||z||, for some S,c> 0 and all t. Recalling the definition of ttoo in (10.49), we have that ^'" ^oo(l^l) ,, 0 ^oo(lkll)
f'-
-
\z\a^i\z\)
Jo INII«oo(||^||)
(Krdt+cf
J\\z\\<\z\
ll^ll
p^dt A\z\\>\z\
Fll
Hence, by the definition of ttoo (see (10.49)) and condition ( 5 ^ ) , for any s > 0 small enough and \\z\\ large enough, we obtain the following estimates: f'^ Jo
G{z,t)
Aoo(|z|)
^ o o ( N ) • ^oo(lkll) r27r
> I a^{t)f'^^j'^ldt-cs Jo
^oo(lkll)
-yi,i>ii,ii > /
'Moo(ikii) a^{t)P.dt^
J\z\>\\z\\
f
P\\
> f a^{t)dt^ J\z\>M
4|<||,|| J\z\<\\z\\
f J\z\<M
''^oodkiD a^{t){P.ydt-cs P\\
a^{t){Sydt-cs
r27T pZTT
> c /
h
a^{t)dt — ce
>0. Since a^(t) ^ 0 and £ > 0 is arbitrary, the conclusion follows immediately. D
10.6. HAMILTONIAN
SYSTEMS
259
Lemma 10.42. Suppose that (Bi) and (Bf) hold. Then ^ satisfies the (PS)* condition with respect to 8. Moreover, under these hypotheses ^{E^ satisfies the {PS) condition for each n. Proof. Assume that (^2^) holds. Let {zj) be a (P6')*-sequence, i.e., Zj G E'^.^^{zj) is bounded, P'^.^'{zj) -^ 0 and Uj ^ oo as j ^ oo (P^ is the orthogonal projector onto E'^). By Lemma 10.25, we may find a c > 0 and no > 0 such that \\P'^Lz\\ > c\\z\\ for ah z G R{L) n E^ and n > UQ. For z G ^ 4 , write z = w ^ z^ e R{L) n ^n © ker(L). Then (10.69)
p;,^^\zj)
= p;,^Lwj -
p;,^^\zj)
0.
By the definition of ttoo in (10.49), we have that ^ ^ 4 < c,{-r~' for s > aoo{t) H^ t > 0. Therefore, similar to the proof of Lemma 10.41, we get that r27r
aooilz^'lMdt 0 27r
a^{\\z^)\y\ /o
< car.
z
Oil
\y\\
On the other hand, it is easy to see that cit < Aoo{t) < C2t^ for large t. Moreover, if z = zi + ^2 G L^([0, 27r]) and w G L^([0, 27r]), then 27r
G'{z,t) 'Wdt 27r
< c
[I +
ttood^ll)
+ «oo(k2|)j k l ^ ^
0 ^27r ot-l
(10.70)
-^
aoo{\z2\)]\w\dt 27r
-c / /o
aoo(|^2|)k|^^,
where || • ||Q, denotes the usual norm in I/^([0, 27r]). Hence, by combining with the Sobolev embedding theorem we get that
< WPL.Lw,
\\Wn
I C K - l
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CHAPTER 10. COHOMOLOGY
GROUPS
Therefore, \\wj\\ < c(l + aoo(||^j ||))- By a similar argument to (10.70) and by the mean value theorem,
>-c\\w,f-^{z,) = -c|Kf - / Jo
{G{zj,t)-G{zlt))dt-^{z<^)
> -c\\w,f - c(l + Wwjir' + a„o(||z«||))||^,|| - *(z«) >-c(l+a^(||^0||))-vl>(4). Note that ^~^*^
^
>ct^-"
^oo
as t ^ o o .
Therefore, if \\z^\\ -^ oo, by Lemma 10.41, we have
«§o(lk^l > —c
-^
aU\\z^\ -*(^?) -^
^oo(||4|
OO,
as j -^ oo. This contradicts the boundedness of ^{zj). This means that ||^^||, and hence ||^j||, is bounded. Recalling the compactness of ^ ^ we see that (zj) has a convergent subsequence. D L e m m a 10.43. Assume that (Bs) and (Bf) hold. For any sequence (zn) G E,Zn = z^ -^Wn, write z^ G ker{Lo),Wn G (ker(I/o))^. / / \\zn\\ -^ 0 and ||^0||
,, ^,, ^ 1 as n ^ oo, then we have that
10.71
hm mf
—
n^oo
> 0.
Hc)i\\Zr,\\)
Proof. By the definition of ho in (10.58), it is easily seen that (10.72)
(-)^° < ^ ^ 5
Ho[S)
< (-)^° S
for t > 5 > 0
and 5, t small.
10.6. HAMILTONIAN
SYSTEMS
261
Since ho may be extended in such a way that (10.58) holds for all t > 0, we may assume that the above inequality holds for all 5, t > 0. Let z = w -\- z^ G (kerl/o)^ 0 kerl/Q. Since w G I/^([0, 27r]), for each si > 0 there exists an R{si) > 0 such that measjt G [0, 27r] : \w{t)\ > R{si)\\w\\} < e\ for any w G E. Set r ^ = {t G [0,27r] : |i(;(t)| < R{ex)\wW. Then meas([0, 27r]\r^) < £i. Noting that /^ ^ h'^(t)dt > 0, we may choose £i > 0 so small that
(10.73)
/
b^{t)dt >l
[
h^{t)dt > 0.
Since kerLo is finite-dimensional, we may assume |^n(OI — c(i?(£i) + c)||z-^|| as long as t G r^. For any £2 > 0, by {H^ ), we have that (10.74)
±^M£!il!l_i!i>5±(i)-e2 J^O[\Zn\)
for all t G r ^ and n large enough. Since Ho{t) is increasing, Ho{\zn\) > Ho{\\zn\\) for |zn| > ll^nll- Since ^ ^
(10.75)
^ 1, we see that
'^"^^^' >
k^WI-K(t)|
,0 1 ^6\\zn-R{s,)\\wr,\
as t G r ^ and n ^ 00, where 5 is as in the proof of Lemma 10.41. Combining (10.72)-(10.75), we have that
for t G r^, \zn{t) < \\zn\\ and n large enough. On the other hand, by (10.72) we get that (10.77)
Ho{\Zn\) ^Odl^nll) r^
< C.
CHAPTER 10. COHOMOLOGY
262
GROUPS
By (10.73), (10.75), (10.76) and (10.77),
r^
(10.78)
> c
b^{t)dt-C£2
^^. > c / =
dt
Ho{\\zn\\j
b^{t)dt-C£2
C-Jo C£2-
Further, we may assume that (^3) holds for ah z. Indeed, suppose that (^3) is satisfied whenever \z\ < SQ. Since ho may be extended such that (10.58) holds for all t, we have by (10.58) that ^(i)/3o-i < M l < ^ ( i ) 7 o - i 7o s ho{s) (3o s
forant>.>0.
It follows that ho{t) > ct^^~^ for t > SQ. Combining this with the asymptotic linearity of the Hamiltonian, we see that (10.79)
\G',{z,t)\
for some c > 0 and \z\ > SQ. Keep in mind that (^3) holds for all z. Thus we see that (10.80)
±G'Q{Zn,t)
' Zn\ ^ chQ(\Zn\)\Zn\
Ho{\Zn\)
-
^
Ho{\Zn\
Noting that meas([0, 27r]\r^) < £1, we have that (10.80) implies
±Go(^n,0 Zfi
(10.81)
-dt
[0,27r]\r^
^o(ll^nll) Ho{\Zn\)
< c [0,27r]\r^
dt
Ho{\\Zn\
If U.,1 < WZr. then 5°j!^"[^. < 1. Otherwise, by (10.72),
i^odl^nl
(10.82)
Ho(\Zn\)
, \Zn\ ,^„
10.6. HAMILTONIAN
SYSTEMS
263
Using (10.81)-(10.82) and the Sobolev embedding of E into ^^^^([O, 27r]), we obtain ^Go[Zn,t)
(10.83) /[0,27r]\r^
• Zji^.,
^odl^n
for n large enough. Combine (10.78) and (10.83). Then we have /
~irT\\—\[\^^ >c-ce2-
Jo
eel > 0-
^o(lknll)
Since Si and £2 are arbitrary, we obtain the conclusion of (10.71). Lemma 10.44. Set
{
D
z = z^ ^w e ker(Lo) © (ker(Lo))^, zeE: 0 < \\z\\
and \\w\\ < K.\\Z\\
Suppose that (Bs), {B^ ) hold. Then correspondingly there exist p > 0 and tv ^ (0,1) such that ±{^\z),z^)
for all z G Q(p, K.).
<0
Proof. If the conclusion were not true, then for any n, there would exist Zn = z^ -\- Wn G ker(I/o) 0 (ker(I/o))^ such that
0 < \\Zn\\<-, U„ <
n
|k„||<-||^„||,
±($'(Z„),4>>0.
This implies that bO
0,
^
^ 1
as n ^
00
and />27r
/
±G'oiZn,t) • zyt
Jo
= T($'(^„),4) <0. Consequently,
(10.84)
limsnp^"\^';»^""f • ; " ^ ^ < 0 . n^oo
/lo(||^n||)Fn||
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CHAPTER 10. COHOMOLOGY
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By (10.72) and the definition of /IQ, (10.85)
Mi!iQ^ <emax{(M)/3o-i^(M)7o-i|.
By (10.85) and (10.79) we obtain that
hoi\\Zn\\)\\Zn\\
0
(10.86)
^''OVII^nllJ
^
^Jo
ll^nll^
IknII
^ 0
as n ^ oo. By (10.86) and Lemma 10.43, we obtain the fohowing result: y
. r Jo ^^oi^n.t)
n^oo
• z^dt
/lodl^nlDll^nll
_ ^.^ .^^ JQ"" ±G'o{Zn,t) n^oo
' Zndt
/lodl^nlDlknll
>0. This contradicts (10.84) concerning the upper limit.
D
Based on the above lemmas, we can compute the
(1) {Bt) ^ cu^,o) (2) (B^) ^
[T]
iovq =
i-{Ao)+i''{Ao),
[0]
otherwise;
[J^]
iovq =
[0]
otherwise.
; i-iAo),
Cl{^,0)
P r o o f . (1) Consider t h e following perturbation of ^ :
$,(z) := $(z) - iA||z«|| = l(Loz - Xz\z)
- *o(^),
10.6.
HAMILTONIAN
SYSTEMS
265
where A G [0,1] and z = z^ ^ w e ker(Lo) © (ker(Lo))^ = E. Let Q(p, K.) be as in Lemma 10.44. If z G Q(p, /^), then z^ ^ 0 and (10.87)
( ^ l ( z ) , z^) = (^^(z), z^) - A(z^ z^) < 0.
U z e {z e E : 0 < \\z\\ < p}\Q{p,f^), then \\w\\ > K\\Z\\. We write w w~^ -\- w . Then there exists a constant c such t h a t ±(I/ot(;^,t(;^) > c||i(;^|p. Therefore,
{^'^{z),w^ -w
)
= {LQW, W~^ — w~) — ( ^ o ( z ) , i(;^ — w~)
> \\w^ + w
r •i' •f i'
\\%{m \\w^ -\- w
> \\w^ -\- w~ (10.^
K.\\Z\\
> 0
for sufficiently small p and \\z\\ < p. Therefore, (10.87)-(10.88) imply t h a t there exists a neighborhood UQ of 0 such t h a t 0 is t h e unique critical point of ^ A ^^ UQ := {z : \\z\\ < p} for ah A G [0,1]. Since \\P;^Low\\ > c\\w\\ whenever w G R{Lo) H ^ ^ (cf. Lemma 10.25) and n is large enough, it is easily seen t h a t ^x satisfies t h e {PSy condition on UQ. Moreover, sup | ^ A | < oo, and the mapping A ^ ^ ^ is Uo
uniformly continuous in z G UQ. By Theorem 10.15, Cg{^x,0) of AG [0,1]. Therefore, (10.89)
C|(^,0)
is independent
=C|(^i,0).
Note t h a t kerLo is finite-dimensional and LQ is A-proper. Thus it is easy to check t h a t the operator LQ defined by LQZ = LQZ — z^ is invertible and A-proper. Since M~{P^LOZ\E^) is the Morse index of t h e quadratic form {Loz.z)
= (Low.w)
-
{z^,z^),
where z G E'^^ it is easy to see t h a t this form is nondegenerate for almost all n, and E^ = R{LQ) n En Q P^ ker(I/o). Therefore, z = w^z^=w^z^e R{Lo) n En® Pn ker(I/o) and w — w = z^ — z^. Since P^y -^ y uniformly for y on bounded subsets of ker(I/o) and w — w e R{Lo), it follows t h a t sup{||i(; -w\\
: z = w ^ z^ = w ^ z^ e En, \\z\\ = 1} ^ 0
as n ^ oo. Therefore, for n large enough, M~{P^Loz\En) is t h e sum of G R{LQ) H En and —{z^,z^),z^ G t h e Morse indices of the form {LQW,W),W
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CHAPTER 10. COHOMOLOGY
GROUPS
ker(I/o). Hence, according to the definition of
[J^]
iovq =
[
[0]
otherwise;
i-{Ao)+i''{Ao),
By (10.89), we accomplish the proof. (2) The proof is analogous by introducing ^ A ( ^ ) •= ^{Loz-\-Xz^, z) — '^o{z). We omit the details. D Next we turn to the computation of the critical groups C|(^,/C(^)). It is a direct consequence of Lemmas 10.41-10.42, Theorem 10.34 and condition (Bi). Lemma 10.46. Suppose that (Si) and one of the conditions of {B2 ) hold and that K, = /C($) is finite. Then [JP] ioiq = i-{A)+i^{A),
(1) (i?+)^c|($,/c) (2)
; [0] [J']
otherwise; for<7 = i-(A),
[0]
otherwise.
(B^)^Cl(^X)
Based on t h e computations of the critical groups C | ( ^ , 0) and C | ( ^ , /C), we can prove t h e main results. Proof of Theorem 10.37. (1) By Lemma 10.46, (^2^) implies that r C|($,/C) = <^ [
[T] [0]
ioxq =
i-{A),
otherwise.
If 0 is the only critical point of ^, then C|(^,/C) = C | ( ^ , 0 ) . It follows from the Shifting Theorem 10.31 that
10.6. HAMILTONIAN
SYSTEMS
267
where ^o is defined on a subset of ker(I/o). Since dimker(I/o) = z^(Ao), C | ( ^ , 0 ) = [0] whenever ^ ^ [ r (AQ), r (AQ) + z^(Ao)]. So by our assumption, Cg ^ (^,0) = [0] ^ Cg ^ ^(^,/C), providing a contradiction. (2) Since i~ (A)-\-i^(A)-\-i~^(A) = 0, by a similar argument, the conclusion follows from Lemma 10.46-(1). D Proof of Theorem 10.38. It follows from Lemma 10.45 and Lemma 10.46 that C | ( ^ , 0 ) ^ C|(^,X:) for some q, hence /C ^ {0}. Proof of Theorem 10.39. We only prove the case (1) as an example. The other cases are similar. Since
r [^], iovq = i-iA)+i'>iA), (B+)^C|($,/C)=<^ [ [0],
otherwise,
and f [^],
forg = r ( ^ o ) + i ° ( ^ o ) ,
[ [0],
otherwise,
we see that there exists a nonzero critical point ZQ. Suppose there are no others. Then by Theorem 10.31, C|($,zo) = [C«-'^°(*o,0)] for some ro G Z and some functional ^o defined on a space Z with dim Z < 2N. In this case, the Morse inequalities read 2N-2
where 6^ G [Z],Q^ G Z. That the sum on the left-hand side above contains at most 2N — 1 nonzero terms follows from the fact that if C^(^,0) = 0, then § has a local minimum at 0 and Cf ( § , 0) = 0 for p ^ 0, and if C | ^ ( § , 0) = 0, then ^ has a local minimum there and C | ( ^ , 0 ) = 0, for p ^ 2N. By comparing the exponents, we can find mi and m2 such that Q^ + mi = i~{A)
-^i^{A)
and a^m2
= r ( A o ) + z^(Ao) ± 1,
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CHAPTER 10. COHOMOLOGY
GROUPS
where mi, m2 G { 0 , 1 , . . . , 2N — 2}. Therefore, |z+(^)-z+(^o)| = \i-{A) + i'>{A)-i-{Ao)-i\Ao)\ = \mi -m2±l\<2N -1, which is a contradiction.
D
Proof of Theorem 10.40. We only prove case (1). Since A = AQ = 0, we have i~{0) = —N and i*^(0) = 2A''. Consequently, by Lemma 10.46, r [T],
foi q = N,
[ [0],
otherwise.
On the other hand, by Theorem 10.32 and Lemma 10.45, [:F],
foi q =
[0],
otherwise.
-N,
C|($,/C) If ^ has only one nontrivial critical point, then by the Morse inequalities, 2N-2
t-^ + Yl ^^^"^' = t^ + (1 + t)Q(t), where 6^ G [Z],Q^ G Z. Similarly as in the proof of Theorem 10.39, we get a contradiction. D
Notes and Comments. Morse type indices for Hamiltonian systems had been introduced in H. Amann-E. Zehnder [9, 10], V. Benci [51] and S. Li-J. Liu [213] where computational formulas for these indices are also discussed. Readers may see books (or papers) of A. Abbondandolo [2], T. Bartsch-Z. Q. Wang [43], K. C. Chang [96, Section IV. 1] and Y. Long [241, 242] for more details of Maslov type (or Conley-Zehnder) indices and their applications. The first result on the existence of a nontrivial periodic solution of (HS) with superlinear potentials was due to P. Rabinowitz [290]. Other results on this case can be found in P. Felmer [152], S. Li-A. Szulkin [216], Y. Long-X. Xu [243], P. Rabinowitz [292] and the references cited therein. For the asymptotically linear case of (HS) (nonresonant at infinity), the first result was due to H. Amann-E. Zehnder [9, 10]. It was assumed that
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
269
Hzz is bounded and A^AQ are time-independent. In C. C. Conley-E. Zehnder [105], Hzz is bounded and (HS) is also nonresonant at zero. In S.Li-J. Liu [213] A^AQ are time-independent and in Y. Long [241], Hzz is bounded. For (HS) resonant at infinity, it was assumed in A. Szulkin [357] that A{t) is a constant matrix. K. C. Chang [95], K. C. Chang-J.Liu-M. Liu [99] and L. Pisani [289] considered the strongly resonant case. In W. Kryszewski-A. Szulkin [200], the case of (HS) with resonance at infinity was studied under the hypotheses that G\z, t) is bounded and G(z, t) ^ oo (or —oo) uniformly in t as \z\ -^ oo. G. Fei [148] studied {HS) under the assumption that A(t)^ Ao(t) are so-called finitely degenerate, which is a strong condition (see also J. Su [355]). We refer the readers to A. Abbondandolo [2], T. Bartsch-A. Szulkin [40](surveys), D. Dong-Y. Long [140], G. Fei [149], G. Fei-Q. Qiu [150], Y. Guo [178], M. Jiang [197], Y. Long [240], Y. Long-E. Zehnder [244], J. MawhinM. Willem [252] and W. Zou [393] for other results on Hamiltonian systems. In particular, if H is positive, more geometrically distinct periodic solutions can be obtained, see the earlier papers of H. Amann-E. Zehnder [10] and V. Benci [51], and generalizations in M. Degiovanni-L. Olian Fannio [129] (by cohomological index theory and a variant of pseudoindex of V. Benci [51]), A. Szulkin [358] (by relative limit index), and M. Izydorek [190] (by equivariant Conley index). If H is even, see V. Benci [51]. The theorems of this section were obtained in A. Szulkin-W. Zou [360].
10.7
Asymptotically Linear Beam Equations
Consider the existence of nontrivial solutions of the nonlinear beam equation:
{
BU : = Utt + U:cxxx = f{x,
t,u),
for 0 < X < TT, t G R ,
1^(0, t) = u{iT, t) = Uxx{0, t) = Uxx{7T, t), for t G R, u{x^ t + 27r) = u{x^ t), for 0 < X < TT, t G R, where / satisfies the following hypotheses. (Ci) / G C([0, TT] X R 2 , R), / ( x , t + 27r, uj) = / ( x , t, u) and / ( x , t, 0) = 0 for all x^t^uo. (C2) There exists an £0 > 0 such that ( / ( x , t, oj) — / ( x , t, u)\(iji} — u) > So{uj — z/)^
for all (x, t, uo).
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CHAPTER 10. COHOMOLOGY
GROUPS
Set g{x,t,u)
= f{x,t,u)
— bu,
G{x,t,u)
= /
g{x,t,s)ds;
Jo
go{x,t,u;) = f{x,t,u;)-boio,
Go{x,t,u;) = /
go{x,t,s)ds,
Jo
where b^bo ^H are two constants. Let O := (0, TT) X (0, 27r). From now on, for two functions a, b defined on O, we write a{x, t) :< b{x, t) (or a(x, t) >z b{x, t)) to indicate that a{x,t) < b{x,t) (resp. a{x,t) > b{x,t)) with strict inequality holding on a set with positive measure. (C3) \g{x,t,uj)\ < c(l +aoo(|^|)) for all x,t,6J, where a^o is the control function defined in Section 10.5; ( C j ) liminf ^ ^ % ^ ^ (C^)
lim
^—^
:= a^{x,t)
h 0 uniformly for {x,t) G Q;
= 0 uniformly for x^t^uj. Moreover, there exists a po >
0 such that ±Go(x,t,C(;) > 0 for \(JO\ < po and all {x,t) G O. Let (10.90)
a{B) = {f -k^
:j
eN.keZ}
denote the spectrum of the beam operator B. Assumptions (C2) and (C3) imply that b > 0. (C2) and (C5 ) imply that bo > 0. Evidently, by (C3) and the definition of ttoo, we have that lim |CL;|^OO
^(^'^'^)=0 UJ
uniformly for (x, t) G O. Therefore, in this case, equation (B) is asymptotically linear both at infinity and at zero. If 6, bo G cr{B), then equation (B) is resonant both at infinity and at zero. Theorem 10.47. Assume {Ci)-{Cs) holds. Then (B) has a nontrivial (weak) solution in each of the following cases: (1)
b i a{B), bo i (J{B) and (0, b] n a{B) ^ (0, bo] H a{B);
(2)
b i a{B), bo G a{B), (C^) and (0, b] n a{B) ^ (0, bo) n a{B);
(3)
b i a{B), bo G a{B), (C+) and (0, b] n a{B) ^ (0, bo] n a{B);
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
271
(4)
b e a{B), (C4-), 5o i a{B), and (0, b) n a{B) ^ (0,60] n a{B)-
(5)
b e a{B), (C4-), 5o e a{B), (C^),
(6)
b e (7(B), (C4-), 5o ^ (7(B), (C5+), and (0, b) n a(B) ^ (0,60] n a(B);
^7;
bea{B),{C^),bo<^a{B),
(8)
bea{B),{C+),boea{B),{C^),
and {0,b]na{B)
(9)
b e a{B),{C+),bo
and (0,5] n a ( B ) ^ (0,5o] n a ( B ) .
and (0,5) n (7(B) y^ (0,5o) n a(B);
and {0,b]na{B)
e aiB),iC+),
^
{0,bo]na{B); ^ {0,bo) na{B);
Define (10-91)
minj/c^ - j4 . ^2 -f^bo > OJ G N, A: G Z}; ^ - _ - m^a_xr j/ /. c2 ^ - j 4.4.Z.2 Z}. . J.2 _ .4-A j ^ ^ ^ ^ Qj eN.ke
{
We assume go{x,t,uj)
(Ce) —/^^ < a < ^°^^' '^^ < b < tv , for cj ^ 0 and (x, t) G O, where a and b are constants; (C^) there exists CTQ > 0 such that ±^0(^5^5^)^ ^ 0 for any (x,t) G O and ICJI < CTQ.
Theorem 10.48. Assume (Ci)-(C3), ( C j ) , (Ce) and (C^). Then (B) has a nontrivial (weak) solution ui in each case of Theorem 10.47 with (C^) holding instead of (C^). Moreover, if ui is nondegenerate (i.e. C|(/,i^i) = ^q,qi[^],'^Q G Z, where I is the corresponding functional of (B) given in (10.93)), then (B) has at least two nontrivial solutions.
Theorem 10.49. Assume a^oit) = t^ with a G (0,1). Then Theorems 10.47 and 10.48 remain true if (Cf) are replaced by / ^ + \ 1. (jog(x^t^(jo) — 2G(x^t^(jo) ^ , . ^ .„ 1 r ^ \ (Cg) hmsup •—^— := a [x^t) ^ 0 uniformly for [x^t) G
n. Let E be the Hilbert space of functions 00
u{x,t) = ^
00
^
j = l k= — oo
Ujksmjxe'^\
Uj^^k = Ujk
272
CHAPTER 10. COHOMOLOGY
GROUPS
satisfying p^\k\
p=\k\
equipped with the inner product p^\k\
p=\k\
Let (10.92)
J\f() = {ueE
: u{x, 0 = E ^^^ sin j W ^ ^ } , p=\k\
then J\f is the generalized null space of the operator B subject to the boundary and periodicity conditions. A/" is a closed subspace of 1/^(0), A/'Q^ ^^^^ 1/^(0) is compact. A function u is said to be a (weak) solution of {B) if u e E and u{^u + (Pxxxx)dxdt = / / ( x , t, u)(pdxdt,
\/(p G ^ .
Consider the following functional /: (10.93)
I{u) := - / {ut -ulJdxdt^
where F(x,t,^) = /
/
F{x,t,u)dxdt,
/ ( x , t, 5)(i5. Then / G C^(^, R). Let 0 < Ai < A2 < • • •
be the positive eigenvalues of {B) and let e i , e 2 , . . . be the corresponding eigenfunctions chosen in such a way that Cn = sinjxcosH
or
e^ = sinjxsinH,
j ^ — k^ = \n-
Then (e^, e^) = 0 for m ^ n. Define ^n = {i^ G ^ : i^(x, t) =
\ .
'^jfe sin jxe^ ^} 0 s p a n j e i , . . . , e^}.
Then ( ^ n ) ^ i is a filtration of E. Let «f := { ^ n , ^ } ^ i - Rewrite / as I{u) —~
{u^ — u^x-\-bu^)dxdt-\-
= -{Lu,u)
^J{u),
/
G{x,t,u)dxdt
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
273
then 1/ is a bounded selfadjoint operator which induces an orthogonal decomposition E = E^ ^E~ ^E^, where E^ := {u e E : u{x, t) =
^
Ujk sin jxe'^^},
±(fe2-j4+6)>0
E^ := {u e E : u{x, t) =
^
Ujk sin jxe'^^}.
Lemma 10.50. Assume that (C3) and {Cf) hold. Then hminf
G{x,t,v)( ± fo^ G(x, t, v)dxdt ^^ AUM) / ; , / „ : — > 0.
Proof. Note that dim^^ < 00. By the definition of ttoo, we have that
I
Ac.{\v\) dxdt nAoo(lbll) JQ
f
(Prrfdxdt
^C
f
{Prr)'^dxdt
< c.
Note that E^ has the unique continuation property. It is easily seen, for any £ > 0, that there exists a 5{£) G (0,1) such that meas {fl\fl{v,£)) < £ for any v G E^, where Q{v,£) := {{x,t) G O : |'y(x,t)| > (5(e)H^;!!}. Since / ^ a^(x, t)dxdt > 0, we may choose an e > 0 so small that (10.94)
/ a^{x,t)dxdt jQ{v,e)
> a^{x,t)dxdt 2 JQ
>0
for any v G E^.
By (Cf), for any £1 > 0, there exists a T(£i) > 0 such that (10.95)
±^(^lM) >a±(x,i)-£i
forany(x,i)efi, |ei>T(£i).
Set 0(1;, £1) Oi(i;,£) ^2(^,£)
{{x,t)en:\v{x,t)\>T{£i)}, {{x,t)en{v,£):\v{x,t)\> {{x,t)en{v,£):\v{x,t)\<
\\v\\}, \\v\\}.
274
CHAPTER 10. COHOMOLOGY
GROUPS
Then for fixed £ and Si (hence T{si) is fixed ), Q{v, s) C ft{v, Si) for H^;!! large enough. By (10.94)-(10.95), we have ±G{x,t,v) Q{v,s)
^oodbll
-dxdt
(P\r)'^dxdt^
/
a^(x,t)(p\rfdxdt
- £ic
ni{v,s)
a^{x,t)dxdt > /
-\- /
a^{x,t){6{£)fdxdt
>c{6{£)f
a^(x, - £ic
-£lC
and ±G{x,t,v) —.—,,. ,., dxdt
I
, ±G(x,t,v) ^^ > -£i + / dxdt J(Q\Q(v, lin\niv,e))nniv,ei) ^oo[\\v\\) > —c£i + / a^(x,t) ^ dxdt Jin\niv,e))nniv,ei) ^oo(|p||) > -c£i + / >-c£i-
a^{x,t){p\7)^dxdt
/
> -csi -
a^{x,t){5{£))^dxdt {8{s)fcs.
It fohows that ±G{x,t,v)
-dxdt > {S{s)fc{l
-s)-csi>0
in ^oodblL as \\v\\ ^ oo.
n
Lemma 10.51. Assume (Ci) and (C2). (1) For each fixed v G M^, inf
/ F(x, t, z + v)dxdt
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
275
is attained at a unique z := z{v). (2) z{vj) -^ z{v) in E whenever Vj -^ v in 1/^(0). Proof. (1) By condition (C2), (10.96)
/
F{x,t,z^v)dxdt>^\\z^v\\l.
Further, we note that ||i^|| = ||i^||2 in A/Q. Therefore, it means that the lefthand side of (10.96) is a coercive function of z. It is also strictly convex. The conclusion (1) follows. (2) Let Vj ^ V in L'^{Q) and let {vm) be any subsequence of {vj). Then by conclusion (1) and (10.96), we see that -^\\z(Vm) ^ VmWl <
I F{x,t,z{Vm)
< /
^Vm)dxdt
F{x,t,Vm)dxdt.
JQ
Then, {z{vm)) is bounded in both the || • || and || • II2 norms. For a subsequence, we may assume that z{vm) -^ z* weakly. By conclusion (1), we have (10.97)
/ / ( x , t, z{v) + v)wdxdt,
Mw e Afo.
Combine (C2) with (10.97), we have /
f{x,t,Z*
-^Vm){z*
-
z{Vm))dxdt
[J{X, t, Z* + Vm) - f{x,
t, z(Vm)
+ ^ m ) J (^* -
z(Vm))dxdt
>£o||^* - z{Vm)\\l' It implies that z{vr^) ^ z* strongly. Using conclusion (1) again, we have F{x,t, z{vm)-\-Vm)dxdt
< / F{x,t,
z{v)-\-Vm)dxdt.
It follows that
/ F{x,t,z* ^v)dxdt< Consequently, z* = z{v).
L^{n).
/
F{x,t,z{v)^v)dxdt.
Hence, z{vj) -^ z{v) in E whenever Vj ^
v in
n
276
CHAPTER 10. COHOMOLOGY
GROUPS
Lemma 10.52. || J'(i^)|| < c(l + aoo(||t^||)) for any ue E. Proof. Note that aoo{t) ~
a t
for any s^t > 0. Therefore, \(J'{u),v)\
Jn
{l + aooi\u\))\v\dxdt
j^al{\u\)dxdty
< c\\v\\ + c\\v\\ ( f aUluDdxdt W|«|>||„||
f
+ / al{\\u\\){l >/|u|<||u||
+ f ^l«l
al,{\u\)dxdt)" ^
^al{\\u\\){^f^-''>dxdt + a ll^ll
^{j^y-rdxdty ^
D
Lemma 10.53. Assume that (Ci), (C2) and (C3) hold. If b ^ ^{B) or b e (j{B) and (C4 ) hold, then I satisfies the {PSy condition with respect to Proof. We only consider the case when b G o-{B) and (C^) holds. Let (m^) be a {PSy- sequence, i.e., m^ G Ejj^.^I{mi) is bounded and P^./^(m^) -^ 0 as j ^ 00. Since E^ C E^^ similar to the proof of Theorem 10.34, E^ splits as En = E^ 0 E~ 0 E^^ and there exists a c > 0 independent of n such that ^{Lu^^u^) > c||i^^|p for u^ G E^. We write nii = m'^ -\- m~ + m^ with mf e E^,m^ e E^. By Lemma 10.52, {PnJ\mi),mj -m~) = {Pn,L{mf -^ m~) ^ Pn^J'{mi),mf >c||m+ + m - f - | | J V O I I l K + ^ n i >c||m+ + m - | | 2 - c f l + ||m+||^-i |m-f-i+aoo(||m^||))||m++m, Thus, c||m+|p + c | | m - f < c a L ( | | m ? | | ) + c .
-m~)
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
277
Therefore, I{mi) =
-(Lmi,mi) + J{mi)
>c\\mif- IcWm-f -c(l + ||m+f-i + || rrii >c||m+f- •c\\m-f-
r
.(KID
"^ +«oc
ca^dh<\\) + J K ) .
Consequently, /(mi)
«^(l|m?|
> _^, -^K)
aUWm^l
^cJ^G{x,t,m'l)dxdt
^^^_0
^
By Lemma 10.50, we have I{mi) 00
as
m?
l|-^,
and we have a contradiction. Then ||m^|| is bounded. Let rrii = Vi-\- Zi with Vi G A/'Q^ , Z^ G A/Q . We assume that Vi ^ v weakly in Af^- and Vi ^ v strongly in 1/^(0). Noting that PnJ'{mi) -^ 0 and A/Q C E^.^ we have / f{x,t,mi){zi
— z{vi))dxdt^O
as z ^ oo,
where z('y^) G A/Q is determined by Lemma 10.51 and satisfies / F{x,t,z{vi)-\-Vi)dxdt
= inf
/ F{x,t, z -\-Vi)dxdt.
Further, we note that z{vi) is a critical point of the functional z^
F(x,t,
z-\-Vi)dxdt.
278
CHAPTER 10. COHOMOLOGY
GROUPS
Then (10.98)
/ f{x,t,z{vi)^Vi)cpdxdt
=0
for all cp G Afo- Thus, by condition (C2) and (10.98), c\\zi - z{vi)\\l < / U{x,t,mi)
- f{x,t,Vi^
f{x,t,mi)(zi
-
z{vi)\{zi-
z{vi))dxdt
z{vi))dxdt
^0. It follows that Zi — z{vi) ^ 0 in E and hence Zi -^ z{v) in E. Consequently, nii = Zi -\- Vi ^ z{v) -\- V in 1/^(0),
J'(rrii) -^ J\z{v)
-\- v).
Therefore, nii -^ z{v) -\- v in E, since L is invertible on E~^ 0 E~.
D
Lemma 10.54. Assume that (Ci), (C2) and (C3) hold. If b ^ ^{B) or b e (j{B) and {Cf) hold, then I\E^ satisfies the {PS) condition for each n. Proof. This is similar to that of Lemma 10.53.
D
Lemma 10.55. Assume that (Ci) and (C2) hold. Letiro andito be the number of eigenvalues of B in the interval (0, 60] ^^^ (O^^o) (counting multiplicities) respectively. Then (1)
bo ^ a{B) implies that C^^°(/,0) ^ [0];
(2)
bo e a{B) and (C^) imply that C^''°(/,0) ^ [0];
(3)
bo e a{B) and (C+) imply that C-^°(/,0) ^ [0].
Proof. For (1) and (2), we let Vo = E^QEl
Wo=E^;
Vo = E^,
E^^®E^.
for (3) we let Wo =
Then / satisfies the local linking condition. Furthermore, it is easy to check that dim(Vb H En) = n — TTQ for cases (1) and (3), dim(Vb H En) = n — TTQ for case (2), and by Theorem 10.33, we get the conclusions of this lemma. D
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
279
Let —Mg{L) denote the number of eigenvalues of the operator B in the interval (0, b] (counting multiplicities) and M^(L) = dimker(L) = d i m ^ ^
Lemma 10.56. Assume that (Ci)-(Cj) hold. (1)
(2)
Ifb(^a{B),
then [
[T]
iovq =
y
[0]
otherwise;
If be a{B) and (Q"), then Cl{I,)C{I))
= l [
(3)
M^{L),
[0]
otherwise;
If be a{B) and ( C / ) , then
cwMi))
[^]
iovq =
M^{L),
[0]
otherwise.
:
Proof. Since E~ HE^ is spanned by the eigenfunctions e^ with b < A^ < A^, it follows that dim(^~ n En) = n — TTOO
when n is large, where TTOO is the number of eigenvalues of B in (0, b]. Since ker(I/) C En and L{En) C E^, by the definition of Mg (L), we therefore have that M^(L) = -TToo. If ^ ^ cr{B), let h{u) := ^{Lu,u)
+ (1 - X)J{u),
A G [0,1].
By the proof of Lemma 10.53, J\u) = o(||i^||) as ||i^|| -^ oo. By the continuity property of C | ( / , /C), we have that
ci{i,ic{i)) - ciihMh)) = ^,M^(L)\n Conclusions (2) and (3) follow immediately from Theorem 10.34.
D
280
CHAPTER
10.
COHOMOLOGY
GROUPS
Let —Mg{Lo) denote the number of eigenvalues of B in the interval (0, bo] counting multiplicities.
L e m m a 1 0 . 5 7 . Assume 0. Then (1)
(2)
(C^)
implies
that ( C i ) , (C2), (C's), (Cf)
and (CQ) hold, and bo >
that (
[^]
ioTq = M^{Lo)
[
[0]
otherwise;
(
[^]
iovq =
[
[0]
otherwise.
+ MO{Lo),
( C ^ ) impl ies that
In particular,
if bo ^ cr(5), conclusions
M^{Lo),
(1) and (2) still hold with M^ {Lo) = 0.
P r o o f . (1) We first consider t h e case of bo G CF{B). W i t h o u t loss of generality, we may assume in {CQ) t h a t a < 0 < 6. Write u = u^ ^ u~ ^ u^ with u^ e E^.vP e E^. Recalling (10.91), we observe t h a t (10.99)
{I'{u),u-^
-u-
= {Lou'^jU'^)
-u^) — {Lou~,u~)-\-
/ go{x^t^u){u^
— u~ — u^)dxdt
JQ
> / {K.'^{u'^f
^ K.~{u~f
^ go{x,t,u){u'^
- u~ -
u^))dxdt.
JQ
Now we estimate t h e integrand in (10.99). Define (10.100)
Qi
{{x,t)
e n :u{x,t)
= 0};
(10.101)
O2
{{x,t)
G n :u{x,t)
j^O,\u-
(10.102)
O3
{{x,t)
G O : 0 < \u{x,t)\
(10.103)
O4
{{x,t)
en-.ao
-^u^\ < |i^+|};
< \u{x,t)\,\u~
^u^\
-^u^\ >
> |^+|}; \u^\}.
If {x,t) G Oi in (10.100), then (10.104)
K.-^{u-^f^K.-{u-f^go{x,t,u){u-^-u-
- u^) > 0.
If ( x , t ) G O2 in (10.101), then (10.105)
-u{u-^
-u-
-u^)
<0
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
281
and by (Ce), (10.106)
go{x, t, u){u-^ -u>a{u-^f -a{u-
- u^) ^u^f
Hence, (10.107)
/^+(^+)2 + K.-{u-f
+ go{x, t, u){u-^ - u' - u^)
>0. If {x,t) G O3 in (10.102), then -u{u~^ -u~
- u^) > 0.
By (Cf), (10.108)
gQ{x,t,u){u'^ -u-
-u^)>{)
>d{u'^f.
Also we have that K.'^{u'^f ^ K.- {u-f
(10.109)
^ gQ{x,t,u){u^
- u' - u^) > 0.
If (x,t) G O4 in (10.103), by (CQ) we have (10.110)
go{x, t, u){u'^ -u-
- u^)
>-6(^-+^0)^ consequently, (10.111)
/^+(^+)^ + K-{u-f + go{x,t,u){u'^ -u> /^+(^+)2 + K-{u-f - b{u- + u^f > K^{u^f
+ {K- - b){u-f
- b{uy
- u^)
- 26|^%-|.
Note that dim^Q < ^^- Hence there exists a /3o > 0 such that sup{|^0(x,t)| : {x,t) G 0 } < /3o||^^||, Choose
1
. rmin{/^+,/^
for ah u^ e E^.
- 6}crg(/^ - 6 )
CTQ 1
282
CHAPTER 10. COHOMOLOGY
GROUPS
Then for ||i^|| < po, we have that |i^^|
<
sup{\u^{x,t)\
: (x,t) G n}
< Mu'W < Mu\\ < (3oPo and \u^\
< ^.
> ^ ^ + 1, then ( K - - 6 ) | U - | 2 > 6 | U O | 2 + 26|UOU-|.
Hence (10.112)
K+iu+f + (K- - b)iu-f
- biu^
- 2b\u%-\ > 0.
If W~I ^ ; F ^ + 1 . then for {x, t) G ^^4, \u\ > OQ and |u°| < ^ . It follows that IM"*" +M~| > ^ , and max{|u+|, |w~|} > ^ . Consequently, K+{u+f +
{K--b){u-f 2
> min{/T:^, n~ — ^ } T ^ > /3o/3ofo + 26/3o/9o(
= + 1) K
— b
>b\u'^\'^ + 2b\u'^u-\. Thus (10.113)
K+{u+f + ( K - - b){u-f
- b{uy
- 2b\u^u-\ > 0.
Combining (10.99)-(10.113), {I'iu),u+ -u-
-u^)>0
for||u||
By Theorem 10.35, we have f
[J']
[
[0]
forg
= M^-(Lo) + MO(io),
otherwise.
(2) The proof for (2) is similar; we omit the details.
D
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
283
Lemma 10.58. Suppose that (Ci), (C2), (C's), (Cf) and (CQ) hold, and bo = 0. Then (C^) implies that (
[^]
for q = 0,
[
[0]
otherwise.
Proof. Let s > 0 such that 0 < s < ^So{so comes from (^2)) and that maxj/c^ - f
:k^ -f
<0,keZJ
eN}^s
= maxjA:^ - j ^ + £ : k'^ - j ^ ^ s < 0,k e ZJ G N } , minjA:^ -f :k'^ -f >0,k e ZJ G N } + e = min{A:^ - j ^ ^ s : k'^ - j ^ ^ s > 0,k e ZJ G N } . Let / ( x , t, ^) = ^o(^, ^, 0 = ^^ + ^o(^, ^, 0 - ^^ •= ^^ + ^o(^, t, C). By Lemma 10.57,
where L^ is defined as {LsU, u) = I {u^ — u^^ + su )dxdt. Ji Then the negative space of L^ is E~ := {u e E : u{x, t) =
^
Ujk sin jxe'^^}.
Hence dim(^~ H^^) = ^ when £ is small enough. Consequently, Mg{Ls) = 0 and C | ( / , 0 ) ^ (^g,o[^], yqeZ. D
Proof of Theorem 10.47. In fact, if X:(/) = {0}, then
c|(/,o) = c|(/,x:(/)). Then Theorem 10.47 follows immediately from Lemma 10.55 and Lemma 10.56. D
284
CHAPTER 10. COHOMOLOGY
GROUPS
Proof of Theorem 10.48. From Lemma 10.56, Lemma 10.57 and Lemma 10.58 we get the existence of a nontrivial solution ui ^ 0 from the fact that C|(/,0)^C|(J,/C)
for some q. Moreover, suppose that ui is nondegenerate, i.e..
In order to deal with all cases simultaneously, we assume that
If there is no other critical point, then by the Morse inequalities, we have
(-ir[i] + (-ir[i] = (-ir~[i], a contradiction.
D
Proof of Theorem 10.49. For any £ > 0, there exists co^ > 0 such that ±{u;g{x,t,u;) -2G{x,t,uo))
< (a^(x,t) + £)|cj|^+^
for \uo\ >uo^.
Therefore, da /±G{x,t,u;)\ / duo \
<
(a±(x,t)+£)|cj|i+^
for \uj\ > ujs'
Integrating the above inequality over the interval [c(;,C(;i] C [cj^, oo) yields the estimate
fG{x,t,ui) ±
G{x,t,u)\
2
9
(a^{x,t)^£) —
^
^_^ V^l
^_^ ~^
'
Therefore, cj^oo
CJ "^^
1 — cr
Similarly, the above limits are also true if cj -^ — oo. We have shown that [Cf) implies [Cf) with a^(t) = |t|^ and a G (0,1). D ^4
Notes and Comments. When / is superlinear, (B) was studied in F. C. Chang-L. Sanchez [92] and G. Feireisl [151]. Several papers have dealt with (B) in case that / is asymptotically linear, see for examples: T. Bartsch-Y.
10.7. ASYMPTOTICALLY
LINEAR BEAM EQUATIONS
285
Ding [36] and D. Lupo-A. M. Micheletti [246, 247]. These papers were done under the assumption that g is bounded globahy and satisfies the condition of Ahmad-Lazer-Paul type (i.e., G{x,t,^) -^ oo (or —oo) as \^\ -^ oo uniformly in (x, t)). Their main tool was a minimax argument (linking and limit relative category). Lemma 10.51 was originally proved in W. Kryszewski-A. Szulkin [200] for wave equations (see also K. Tanaka [365]). We refer the readers to A. C. Lazer-P. J. Mckenna [206, 207], J. Liu [231, 232], A. M. Micheletti-C. Saccon [257] and A. M. Micheletti-A. A. Pistoia-C. Saccon [256] for beam equations via linking type arguments; H. Brezis-L. Nirenberg [68], H. Brezis-J. M. Coron-L. Nirenberg [69], P. Rabinowitz [291], W. KryszewskiA. Szulkin[200] and S. Li-A. Szulkin [215] and the references cited therein for wave equations. The main results of this section were established in W. Zou [389]. Assumptions (Cg ) are a generalization of the so-called nonquadratic conditions considered by D. G. Costa, C. A. Magalhaes, E. A. B. Silva, etc. (see [107, 111, 112, 113, 339]), which were used in T, Bartsch-M. Clapp [32] to deal with superlinear noncooperative elliptic systems. By linking arguments, some results were established for the study of strongly indefinite functionals. One of the most important parts of the theory was developed by V. Benci-P. H. Rabinowitz [55]. In E. A. B. Silva [339], the framework introduced in [55] was used to prove the existence of subharmonic periodic solutions for a class of asymptotically quadratic first order Hamiltonian systems satisfying the generalized version of the LandesmanLazer condition introduced in E. A. B. Silva [336] (see also D. G. Costa[107] and D. G. de Figueiredo-L Massab6[161]). In [336, 340], E. A. B. Silva has also established some abstract critical point theorems to study the existence and the multiplicity of critical points for strongly indefinite functionals of the form ^(Lu^u) -\- J{u), with J{u) unbounded and satisfying the (PS)* condition. An earlier result on strongly indefinite functionals with applications can also be found in H. Hofer [187]. The result of [340] is used to establish the existence of nonzero solutions for noncooperative elliptic systems (cf. E. A. B. Silva [341]). We also refer readers to N. Ghoussoub's duality and perturbation methods in critical point theory (see [170, 171]) which involve some minmax principles with relaxed boundary conditions and to A. C. Lazer-S. Solimini [208], M. Ramos-L. Sanchez [299], K. Perera-M. Schechter [287] and S. Solimini [345] for Morse index estimates in minimax theorems.
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Index {PS)*, 217, 218
H^m, 3 H-\n), 3 LP{n), 2 Lfo^m, 2 W^'\n), 3 PF'^'J'(f2), 3 ^"(f)), 3 C~(f)), 2
Conley-Zehnder index , 254 Continuity Theorem, 224, 226 Contractions, 117 Contravariance, 216 Critical Sobolev Exponents, 89 Double linking, 118 Ekeland's variational principle, 31 Exact sequence of a triple, 217 Exactness, 216 Filtration, 216 Filtration-preserving, 216 Finsler structure, 30 Fountain theorems, 37, 43 Frechet derivative, 7 Fucik spectrum, 131, 139, 195, 203 Gagliardo-Nirenberg's inequality, 6 Gateaux derivative, 7 Globally admissible pair, 225 Gradient-like vector field, 218 Gronwall inequality, 60 Holder continuous, 4 Hamiltonian System, 253 Heteroclinic orbits, 167 Homoclinic, 64, 66 Homotopy invariance, 217 Imbedding, 4, 5 Jumping nonlinear it ies, 203, 204
INDEX
318 Kronecker Theorem, 15 Linking, 117, 179 Local linking, 242, 244, 278 Locally finite, 1 Locally Lipschitz continuous, 16 Maslov-type index, 254, 255 Monotonicity method, 43 Morse index, 215, 232, 254, 265 Morse inequalities, 228, 267, 268, 284 Mountain pass theorem, 116 Mountain pass type, 43 Naturality, 216 Nondegenerate, 234 Nonresonant, 139 Nonvanishing, 68, 70, 72, 84, 100 Oscihating, 204 Paracompact, 1 Partition of unity, 1 Pseudo-gradients, 25 Resonant, 139, 204 Sard's theorem, 13 Schrodinger equations, 37, 55, 74 Schrodinger system, 101 Shifting theorem, 239 Sign-changing solutions, 195 Sobolev exponent, 94 Sobolev space, 3, 66, 73 Strong excision, 217 Strongly indefinite functional, 215 Superquadraticity condition, 75, 89, 116 Symplectic, 255 Symplectic matrix, 64, 253 Talenti's result, 5 Tangent space, 30
Topological degree, 13 Twin critical points, 120 Unitary isomorphism, 105 Vanishing, 68-70 Weak linking theorem, 56 Weak solution, 20 Weak topology, 55
