"\
,
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Conference Board of the Mathematical Sciences REGIONAL CONFERENCE SERIES IN MATHEMA TICS supported by the National Science Foundation 1
I
i
,
Iv
Number 65
MINIMAX METHODS IN CRITICAL POINT THEORY WITH APPLICATIONS TO DIFFERENTIAL EQUATIONS Paul H. Rabinowitz
Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island
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Expository Lectures from the CBMS Regional Conference held at the University of Miami January 9-13,1984 Research supported in part by National Science Foundation Grant DMS-8303355. 1980 Mathematics Subject Oassifications (1985 Revision). Primary 34C25, 35J60, 47H15, 58E05, 58E07, 58F05, 70H05, 70H30.
Library of Congress Cataloging-in-Publication Data Rabinowitz. Paul H. Minimax methods in critical point theory with applications to differential equations. (Regional conference series in mathematics, ISSN 0160-7642; no. 65) "Expository lectures from the CBMS Regional Conference held at the University of Miami. January 9-13. 1984"--T.p. verso. "Supported by the National Science Foundation." Bibliography: p. 1. Critical point theory (Mathematical analysis)-Congresses. 2. Maxima and minimaCongresses. 3. Differential equations, Elliptic-Congresses. 1. Conference Board of the Mathematical Sciences. II. Title. III. Series. QAl.R33
no. 65
510s
[515.3'3)
86-7847
[QA614.7J ISBN 0-8218-0715-3 (alk. paper)
Copying and reprinting. IndiVIdual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940. The owner consents to copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that a fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc" 21 Congress Street, Salem, Massachusetts 01970. When paying this fee please use the code 0160-7642/86 to refer to this publication. This consent does not extend to other kinds of copying sllch as copying for general distribution, for advertising or promotion purposes) for creating new collective works, or for resale. 1
Copyright @1986 by the American ~athematical Society. All rights reserved. Printed in the United States of America The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in t.his book is acid-free and falls within the guidelines established to ensure permanence and durability.
Contents
Preface l. An Overview
2. The Mountain Pass Theorem and Some Applications 3. Some Variants of the Mountain Pass Theorem 4. The Saddle Point Theorem 5. Some Generalizations of the Mountain Pass Theorem 6. Applications to Hamiltonian Systems 7. Functionals with Symmetries and Index Theories 8. Multiple Critical Points of Symmetric Functionals: Problems with Constraints 9. Multiple Critical Points of Symmetric Functionals: The Unconstrained Case 10. Perturbations from Symmetry 11. Variational Methods in Bifurcation Theory Appendix A Appendix B References
v
Preface
This monograph is an expanded version of a CBMS series of lectures ered in Miami in January, 1984. As in the lectures, our goal is to provi introduction to minimax methods in critical point theory and their applic to problems in differential equations. The presentation of the abstract mil theory is essentially self-contained. Most of the applications are to semi elliptic partial differential equations and a basic knowledge of linear ellipti, ory is required for this material. An overview is given of the subject mat Chapter 1 and a detailed study is carried out in the chapters that follow. Many friends have contributed to my study and organization of this J rial. I thank in particular Antonio Ambrosetti, Abbas Bahri, Vieri Benci, Berestycki, Halm Brezis, Michael Crandall, Edward Fadell, Suffian Hm Jiirgen Moser, and Louis Nirenberg for their inspiration, encouragement advice. The CBMS conference was hosted by the Mathematics Departmc the University of Miami. Further thanks are due to the members of the dE ment, especially to Shair Ahmad and Alan Lazer for their efficient handli the meeting and their kind hospitality.
vii
1. An Overview
A focus of these lectures is the existence of critical points of real valli< tionals. The most familiar example occurs when we have a continuo usl: clltiable map g: Rn ~ R. A critical point of 9 is a point ~ at which g' Frechet derivative of g, vanishes. The simplest sort of critical points of f global or local maxima or minima.
g(x)
~-------------------x
The setting in which we will study critical point t.heory is an infnite sional generalization of the above. Let E be a real Banach space. A m I of E to R will be called a functional. To make precise what we mea critical point of I, recall that I is Frechet differentiable at u E E if then a continuous linear map L = L( u): E -> R satisfying: for any c > 0, tht 8 = 8(c,u) > such that II(u + v) - I(u) - Lvi::; cllvll for all Ilvll ::; ( mapping L is usually denoted by J'(u). Note that J'(u) E E*, the dual SJ E. A critical point u of I is a point at which I'(u) 0, i.e.
°
J'(u)
°
for all
(1.1)
- Liu
J(x),
xE
n,
u = 0, ,
/
where here and in future examples n denotes a hounded domain in R n whose boundary. an. is a smooth manifold. Suppose f E C(n). A fUIlction u is a rlassical solution of (11) if 11 E C2(0) n C(n). For such a solution, multiplying 1) by;; E (0) yields
1(\'11 \';;
(12)
dx
()
°
after an integration hy parts. Let H'c;2(n) denote the closure of cO'(n) with respect to
~' == (j~ 1\'1112 dX) 1/2 . If 11 E H'c;·2(n) and satisfies (1.2) for all 'P E CO'(O), then 11 is said to be a weak solution of (1.1). By our ahove remarks, any classical solution of (1.1) is a weak solution. Under slightly stronger hypotheses on f (e.g. f Holder continuous) the converse is also true. Choosing E == w~·2(n), set ( 1. 3)
1 (u) ==
inr (~
~ f u) dx.
It is not difficult to verify that. 1 is Fn?chet differentiable on E and
(l.4)
J'(u)'P
1n(\'u'V'P~f'P)dX
for 'P E E. Thus u is a critical point of 1 if and only if u is a weak solution of (1.1).
As was noted earlier, when E = Rn the most familiar sorts of critical points obtained are maxima or minima. In these lectures we will be dealing mainly with functionals which may not be bounded from above or below even modulo finite dimensional subspaces or submanifolds. Such "indefinite" functionals may not possess any local maxima or minima other than trivial ones. For example and let n = (0, 1T) C R, E = \Yo_
(1.5)
, I
where / == dl dx. It is not difficult to show that 1 is differentiable on E and has u = as a local minimum. For any other u E E and 0' E R,
°
1(au) =
l'r (~2Iu/12 ~ ~\4)
dx
->
~OO
as ex so 1 is not bounded from below. Furthermore for each kEN, sin kx E E, and 1(sin kx) > '2k 2 ~ '2 -> 00 - 4
4
as k -+ 00 so 1 is not bounded from above. Thus it is not obvious that 1 possesses any critical points other than the trivial one u == 0. Nevertheless we will see later as an application of the Mountain Pass Theorem that 1 possesses positive critical values and the same thing is true for higher dimensional versions of (1.5).
As a second example of an indefinite function,,!, consider the Hamiltc system of ordinary differential equations (1.6)
: dp
dt
-Hq(p, q),
where H: R 2n ~ R is smooth, and P and q are n-tllplcs. \Vc are interest, periodic solutions of (1.6). Taking the period to be 27r and choosing E to t appropriate space of 2if functions. solutions of (1.6) are critical p of
(1.7)
I(p, q) = [ " Ip(t) . q(t) - H(p(I.), q(t))] dl..
(This will be made precise in Chapter 6.) To see the indefinite nature of ( suppose n 1. Taking Pk(t) = sin kt and qdt) = -- cos kt shows I(pk, qk) = bounded term ~ ±oo as k --; ±oo. Thus I is not bounded from above or be Despite this, as we shall see later, minimax methods can be applied tc functional (1.7) to obtain periodic solutions of (1.6). There are at least two sets of methods that have been developed to critical points of functionals: (i) Morse theory and its generalizations am minimax theory. For material on "classical" .tI·!orsc theory, see e.g. [Mi, Ch2]. Generalized Morse theories and the so-called Conley index can be fe in the CBMS monograph of Conley ICC] (see also ISm]). Our lectures focus on minimax theory. This subject originated in work of Ljusternik Schnirelman ILLS] although it certainly had antecedents (see e.g. IBill. What are minimax methods? These are methods that characterize a cri value c of a functional I as a minimax over a suitable class of sets S: ( 1.8)
c = inf max J( u 1. AES uEA
• ,
There is no recipe for choosing S. In any situation the choice must r€ some qualitative change in the topological nature of the level sets of 1, i.e sets I-l(s) for s near c. Thus obtaining and characterizing a critical value in (1.8) is something of an ad hoc process. The Mountain Pass Theorem is the first minimax result that we will study statement involves a useful technical assumption--the Palais-Smale conditie that occurs repeatedly in critical point theory. Suppose E is a real Bal space. Let C1(E, R) denote the set of functionals that are Frechet differenti and whose Frechet derivatives are continuous on E. For IE Cl (E, R), we s satisfies the Palais-Smale condition (henceforth denoted by (PS)) if any sequ (urn) C E for which I(u m ) is bounded and I'(uml --; 0 as m ~ 00 posse a convergent subsequence. The (PS) condition is a convenient way to t some "compactness" into the functional I. Indeed observe that. (PS) im that Kc == {u E EII(u) = c and I'(u) = O}, i.e. the set of critical points ha critical value c, is compact for any c E R. We will see many examples lat, when (PS) is satisfied.
';,
A~
OVER\'IEW
Let Br denote the open ball in E of radius r about 0 and let DBr denote its boundary, ~ow the Mountain Pass Theorem can be stated, THEOREY!. Let E be a real Banach space and IE C1(E.R). Suppose satisfies (PS), 1(0) = 0, (1d there exist constants p, 0: > 0 such that IlaB, 2: 0:, and (h) there is an e E E \ DBp such that I(e) S; 0, Then I possesses a critical value c 2: 0: which can be characterized as
c
= gEl inf
11
I(u),
where
r= {gEC([O,l],E)lg(O) =O,g(l)
e}.
This result is due to Ambrosetti and Rabinowitz [AR]. On a heuristic level, the theorem says if a pair of points in the graph of I are separated by a mountain range, there must be a mountain pass containing a critical point between them. Although the statement of the theorem does not require it, in applications it is generally the case that I has a local minimum at O. A second geometrical example of a minimax result is the following Saddle Point Theorem iR4]: THEOREM. Let E be a real Banach space such that E = V 8 X, where V is finite dimensional. Suppose IE C 1 R), satisfies (PS), and (13) there exists a bounded neighborhood, D, 0/0 in V and a constant 0: such that IlelD S; ct, and (14) there is a constant j3 > ct such that Ilx 2: (3. Then I has a critical value c 2: (3. Moreover c can be characterized as
c
= SEf inf max I (u ) . uES .
where
r = {S = h(J5)lh E C(J5, E)
and h = id on DD}.
Here heuristically c is the minimax of lover all surfaces modelled on D and which share the same boundary. Unlike the Mountain Pass Theorem, in applications of the Saddle Point Theorem generally no critical points of I are known initially. \'ote that (13) and (J,J) are satisfied if I is convex on X, concave on V, and appropriately coercive. Indeed the Saddle Point Theorem was motivated by earlier results of that nature due to Ahmad, Lazer, and Paul [ALP] and Castro and Lazer [CLIo Both of the above theorems, generalizations, and applications will be treated in Chapters 2--B, In particular in a somewhat more restrictive setting both the ).fountain Pass Theorem and Saddle Point Theorem can be interpreted as special cases of a more general critical point theorem which is proved in Chapter 5. yl ueh of the remainder of these lectures will be devoted to the study of variational problems in which symmetries playa role. To be more precise, suppose
AC'J OVEH\'lE\V
E is a real Banach space, G is a group of transformations of E into ;;: IE CI(gR), We say I is invariant under G if I(gu) = I(u) for allg EG lL E g As a first example, consider (L5), It is invariant under G == {id where id denotes the identity map on g Note that we can identify (~ Z2, More generally if p(x, 0 is continuous on [0,7r] X R, is odd in P(x,O Jo(, p(x, t) dt, then (1.9) is invariant under G, As another example consider (1. 7), recalling for th that functions in E are 27r periodic, Let 0 E [0, 27r), z (p, q) E E, (ge~t) ) z(t + 0), and G == {gelO E 10,27r)}, Then it is easy to see that r is im under G, YIoreover G can be identified with Sl, The above examples show that functionals invariant under a group of s;', , tries arise in a natural fashion, It is often the case that such functionals r multiple critical points, Indeed results of this type are among the most faJ ing in minimax theory, The first example of such a theorem goes back t( work of Ljusternik and Schnirelman [LLS], They studied a constrained tional problem, Le, I restricted to a manifold (which must be invariant G) and proved THEOREM, If IE CI(Rn,R) and is even, then distinct pairs of critical points,
115n-1
possesses at
Subsequently other researchers extended this result to an infinite dimer setting, Another multiplicity result is provided by the following Z2 symmetric v of the Mountain Pass Theorem [AR, R2]: THEOREM, Let E be a real Banach space and I E CI(E,R) with I Suppose 1(0) = and I satisfies (PS), (1d, and (12) for all finite dimensional subspace8 E c E, there is an R = R(E that I(u) <; for u E E \ BR(El' Then I possesses an unbounded sequence of critical values,
°
°
In order to exploit symmetries of I, one needs a tool to measure the, symmetric sets, i.e. subsets of E invariant under G. Such a tool is provid the notion of an index theory, With the aid of such theories, minimax cl terizations can be given for the critical points obtained in the two theorerr cited, Index theories will be discussed in Chapter 7 and applied to const] and unconstrained variational problems in Chapters 8-9, In particular \I see that generalized versions of (1.5) satisfy the hypotheses of the symr Mountain Pass Theorem and possess an unbounded sequence of critical p' The next question we will study is what happens to a functional wh invariant under a group of symmetries when a perturbation is made whi, stroys the symmetry, No general theory has been developed yet to treat
[)
A" OVEHVrE W
illiltter, and we will confine om attention to an example from partial different ial f'quation:' in Chapter 10. Our finai topic. cOH'red in ChaptN 1 L concerns ,wplicati olls of minimax met hods to bifurcati on problem s. Such problem s ,He of interest since bifurcati on phcnomc nd uccur in a wide vilriet~· of settings in nature. Consider the map F. R x E -, E. where
( 1.10)
F(A. u) = Lv t H(u) - All,
!~ is a rcal Bilnach space, A E R. vEE, L is a continuo us linear map of E into E. ilnd H E Cl E) with H(u) = o(lllllll) as u ~ O. ~ote that F(\O) = 0 for illl A E R. We call these zeros of F trivial solutions of F(A.U) O. A point (11.0) E R x E is cillled a hi/meation point for F if every neighbor hood of (11,0) contains nontrivia l solutions of F()., 11) O. It is well known- see Chapter 11--· that a necessar y conditio n for (11,0) to be a bifurcati on point is that 11 E alL), the spectrum of L. Simple countere xamples show this necessary conditio n is not
sufficient. However in Chapter 11, it will be shown that if (1.10) correspo nds to an equation of the form 1'(11) = 0, 11 E alL) is also a sufficien t conditio n for (11.0) to be a bifurcati on point. Other sharper results will give more informat ion about the nontrivia l solutions of F(A, ll) 0 for ()..u) neaf (11,0) both as a function of and as a function of A. Lastly there are two appendic es. The first, Append ix A, is mainly concerne d with an importa nt tool called the Deforma tion Theorem . It is used to help prove all of our abstract critical point theorem s. Appendi x B contains some technica l results which are useful in verifying abstract conditio ns like IE CI(E, R) or (rS) in a partial differential equation s setting. Some other sources of material on minimax methods and critical point theory in general are [BW, Bg, Ch2, CC, K, LL8, Mi, N2, P2, R2, 82, Va].
2. The Mountain Pass Theorem and Some Applications In this chapter we will prove the llSUilJ version of the' :'Jollntai n Pass Theon""" and give some applicat ions to semiline ar elliptic partial different ial cquatio,-, The ideas involved in the proof of the Mountai n Pass Theorem arc very A key ingredie nt is the so-called Deforma tion Theorem , This latter result pIE an importa nt role in all of our abstract minimax results, Since it is ran, ( lengthy and technica l in n-ature, we have relegated it to Appendi x A (sec Theof(' AA), The following special case is sufficient for the proof of the ylollntai n Pa, Theorem , PROPOSITION 2.1. Let E be a real Banach space, SlLppose I Cl R I and satisfies (PS). For 3. c E R. set Kc 0= {v E[I(lL) =c (' and '= O} OJ :As '= {1L E ElI(l1) <; s}. If C 15 not a critlcal value of 1. IJwen any E > 0, the exists an c E (0. E) and I] E CnO. 1] x E. E) such that 1° 1](1.11)=11 ifI(l1)r t[c-E,c TE]. 2° 1](1,A o+ o ) C A C - E ' Now we can prove THEOREM 2.2 (MOUNTAIN PASS THEOREM [AR]) , Let E be a real B( nach space and 1 E 01(E, R) satisfying (PS). Suppose 1(0) = 0 and (h) there are constants p, a> a sl1ch that IliJBp ~ a, and (h) there is an e E E \ Bp sl1ch that I(e) <; O. Then 1 possesses a critical value c ~ a ' Moreover c can be characterized as
(2.3)
c = inf
max
gEr uE9([O.1J)
1(11).
where
r
= {g E 0([0, 1]. E)/g(O) = 0, g(l) = e}.
PROOF. The definitio n of c shows that c Therefor e max
l1Eg([O,lj)
1(11)
~
< 00.
r,
gUO, 1]) n 8B p oj 0
I(w) ~ a
inf
wEiJB
If g E
p
via (h). Consequ ently c ~ 00. Suppose that c is not a critical value of 1. Then Proposit ion 2.1 with E == 0'/2 yields E E (0, f) and T/ as in that result. Choose
il
THE y!Ol'NTAIN PASS THEORE"!
gE
r
such that
(2.4)
max
uE9([O,1j)
[(u) < c + S -
and consider h(t) == 1')(1. g(t)). Clearly h E C(IO, I], Also g(O) = 0 and 1(0) = 0 < a/2 ::: C E imply h(O) = 0 [0 of Proposition 2.1. Similarly (I( 1) = r: and [(c) ::: 0 imply that h(l) = e. Consequently hEr amI bv (2.3),
c::: But by (2.4), g([O, 1]) i.e.
c
(2.6)
AC-l-£
[(u).
so 2° of Proposition 2.1 implies h([(), 1]) max
uEh([O,lj)
[(u) < c -
A cc ,
S,
-
contrary to (2.5). Thus C is a critical value of [. Although the above proof is very simple, one sees the basic ingredients of a minimax theorem: (il a family of sets which is chosen to exploit the properties of [ and which is invariant under the deformation map 1')(1, .), (ii) a minimax value c obtained from this family of sets, (iii) topological arguments giving some estimates for c (and showing in particular that c is finite), (iv) an indirect argument based on the Deformation Theorem proving that c is a critical value of [. This framework will be used repeatedly in these lectures. As an almost trivial application of some of these ideas we have THEOREM 2.7. Let E be a real Banach space and [ (PS). If [ is bounded from below, then
(2.8)
c
E Cl (E,
R) satisfying
inf [ E
is a critical value of [. PROOF. Clearly c is finite. Set S = {{x} Ix E E}, i.e. S is a collection of sets each consisting of one point. Trivially we have
c = inf max[(u). KES uEK
:\ow for any choice ofE, e,g, E = 1, since 1')(1, ,) as given by Proposition 2,1 maps 8 into 8, the argument of Theorem 2.2 shows c is a critical value of [.:; The remainder of this section consists of several applications of the Mountain Pass Theorem to boundary value problems for semilinear elliptic partial differential equations. Consider (2.9)
- 6u = p(x, u),
u
0,
xE
x EO,
ao,
where OcR" is a bounded domain whose boundary is a smooth manifold, In (2.9) and later applications, ~6 could be replaced by a more general second order divergence structure ulliformly elliptic operator. However we prefer to minimize such technicalities.
THE ,,1OC'lTAIK PASS THEORE"j
The fUllction p will always be assumed to satisfy (pd p(x, 0 E c(n x R, R), and (P2) there are constants a I, a2 :::: 0 such that
01
~
al
-ra21~lj,
where () ~,,< (n +2)/(n-2) ifn > 2. If II ~ 1. (P2) can he dropped while if n= 2, it ,mffices that
01
iP(x.
~ (]1 exp:p(O,
where :p(O~2 ---) () as I~I -, x. The reason for such structural conditi( the following: The functional associated with (2.9) is
10)
flu) =
r (H'Vu!2 - P(x,u))
in
where P(x, 0 = Jo( p(x, t) dt. A natural space in which to treat (2.10) is WJ 2 (O), the closure of Cil(O) with respect to ( [ (I'Vu12
+ u 2)dX) 1/2.
By the Poincare inequality. there is a constant III > 0 such that 2
III [
u dx
~ .~
dx
for all u E E. Hence we can and will take as nom in E.
(2.11)
Ilull
([ !'Vuj2 dX) 1/2 .
In order to apply Theorem 2.2 to the functional f given by (2.10), we ha know that fECI (E, R) and critical points of f are weak solutions of (2.£ Appendix B (Proposition B.lO), we prove that when (pd and (P2) are sati f E CI(E, R) and
(2.12)
J'(u)'P = [('vu 'V:p - p(x, u):p) dx.
thereby verifying these two properties. For our first application, we further assume (P3) p(x, 0 = o(IW as ~ ---) 0, and (P4) there are constants 11 > 2 and r :::: 0 such that for I ~ 1
() < I1 P (X, 0
~ ~p(x,
::::
r,
O·
REMARKS 2.13. (i) Hypothesis (P3) implies that (2.9) possesses the 'It solution" u == O. (ii) Integrating condition (P1) shows that there exist constants (]3,!l4 > 0 that (2.14)
THE \lOUNTAIN PASS THEORE\1
10
for all x E 0 and ~ E R". Thus since Ii > 2. 0 grows at a "superquadratic" rate and by (P4)' P gro\\'s at a "superlillear" rate as I~I ~ 00. (iii) :':ote that if n = 1 and p(x. ~) = i.e. if we are in the setting of example (1.5). (pd(Pl) are satisfied.
e.
THEORE\!
2.15.
1/ p
satisfies (PI) (Pol), (2.9) possesses a nontrivial weak
solutlOn. PROOF. Let E = WJ·2(fl) and 1 be defined by (2.10). The weak solution I of (2.9) will be obtained as a critical point of J with the aid of Theorem 2.2. Proposition B.lO and (pd-(P2) imply 1 E CI(E,R). Clearly 1(0) = O. Thus we must show 1 satisfies (IJ). (I2), and (PS). To verify (I2), note that by (P4) and (2.14), (2.16) for all u E E. where (2.16) implies
J(tu)
(2.17)
Ifll
denotes the measure of
=~
i
fl.
Chosing any u E E \
12 dx ~
In P(x, tu) dx :S ~lluI12~tl'a3 r Ilull'dx-+-04Ifll--> ~OO ~ in 1
'VU
1
as t --> 00. Hence (I2) holds. For (IJ), by (P3), given any E > such that I~I :S 6 implies IP(x, 01 :s: ~EI~12 for all x E
{O}.
O.
there is a 6> 0
0. By (P2) there is a constant A = A(6) > 0 such that IP(x, ~)I :S
AIEls+
I~I
:::: 5 implies
1
for all x EO. Combining these two estimates, for all ~ E R and x EO, (2.18)
IP(x,
01
:S ~IEI2 -+- AI
Consequently,
I) IIul12
(2.19) via the Poincare and Sobolev inequalities. Choosing
Since
osEIlull shows J(u) o(llul1 2 ) IJ(u)! :S
(2.20) E
was arbitrary, (2.20)
J(u)
= !lluil2 ~
J(u)
Ilu!1 :s: (E/2A)1/(s-l)
yields
2
as u
--> O.
Therefore
= ~lIu112 -'-o(lluI!2)
as u -; 0 so (Id holds. Next the verification of (PS) here and in later results is simplified with the aid of the following result whose proof can be found in Appendix B.
PROPOSITIO:\ B 35. Lct.p wt.ls/YiPl) (P2) (wd! hi' 18
is a bounded seqvence m E such thai J'(u m precompllet in E.
) -- () ()8
/)1/ (210
x. then (
m -
By Proposition B.35, to verify (PS), we need only show if(u",)! ) ---+ 0 as m ---+ 00 implies (urn) is a. bounded sequence. For m with 1L Um and T = /clp(x, u)u ~ u) shows (2,21)
J'(u m
'",
iVl -j-Il
_ j ""
Ii 11 II :::
",
l\U) ~11
_1 ' ; ' )
'1111 '1~
:::(~~)
(1 \2
~
l
(2
;' ~ .
1 \ ,. I;) jl1J
-T-, [)i(J:r
TelL,!'
J{xElll ,lI(xl<>T}
I"'
M
, (.rEO
iu(xli
Tc
By (P4), (2- 1 11- 1) > 0 and the second term on the right in (2.21) is posit The third term is bounded by a constant inrlependently of?n, Hence (2 implies (Urn) is bounded in E, Lastly note that 1(0) = 0 while for our critical point u, f(ll) ;::: IX > (], He 11 is a nontrivial weak solution of (2.9), REMARK 2,22, If hypothesis (pd is strengthened to, e,g., (PI) p(x,~) is locally Lipschitz continuous in X R, then (PI )'-(P2) imply any weak solution of (2,9) in E is a classical solutiO! (2,9) (see e.g. [Ag]).
n
COROLLARY 2.23. Under the hypothesis 0/ Theorem 2.15, i/ paisa satis (P'l)' then (2.9) possesses a positive and a ni'9!!tive classical solution. PROOF. Set p(x,
0 = (] for
~ '::: (] and
P(x,
p(x, 0 =
p(x,~) for ~ ;:::
O. Let
~) = fo~ pix, t) dt.
The arguments of Theorem 2.15 show that
(2.24)
7(u) =
~ IIuI12 ~ 10 P(x, 11) dx
satisfy the hypotheses of the Mountain Pass Theorem. Indeed '15 satisfies (p (P3)' Moreover, (P4) holds for ~ > (] while 0 == P = '15 for ~ '::: O. Hypoth( (P4) was required to help verify (PS) and (h). The above weaker version of ( implies P;::: 11- 1'15 for large I~I and this suffices to get (PS). To also satisfy (j note that (2.14) holds for ~ :2: 0. Thus choosing u E E \ {O} in (2.17) to b, nonnegative function, (12) holds. Consequently, by the Mountain Pass Theor (2.25)
~
6.u
=
p(x, u),
u = 0,
x E 0,
x E 80,
has a weak solution, u 'Ie (]. By (pU and Remark 2.22, 11 is a classical soiut of (2.25). Let A == {x E Oiu(x) < O}. Then by the definition of '15,
!:.u = 0,
u =0,
x E A,
x E 8.11.
THE MOUKTAIK PASS THEOREM
12
Consequently the maximum principle shows in n. In fact rewriting (2.25) as -
Ll11 -
11=
- - ' - - - , - ) 11.
(2.26) 11
0 in A so A =:2). Thus
11
xE
0,
xE
11
?: 0
n.
an.
where aT = max(a,O) and a- = min(a,O), and noting that C1p(x, are 0, the strong maximum principle [CHI continuollR if defined to he 0 at ~ implies that l1(X) > 0 for x E nand dl1(X)/aU < 0 for x E dD. u(x) heing the outward pointing normal to aD. The negative solution of (2.9) is produced in a similar fashion. REMARK 2.2i. Theorem 2.15 and Corollary 2.23 can be found in [AR]. Earlier work in this direction was done by Coffman [Col-2] and Hempel [Hel]. RS"lARl':: 2.28. If p = p(O and n > 2, an identity due to Pohozacv [Po] says
f (2nP(11) + (2 -
(2.29)
Jo
n)l1p(l1)) dx =
f
Jao
x· u(x)lvl1I2 dS
for all solutions of (2.9). In (2.29) u(x) is the outward pointing normal to dn. Consequently if n is starshaped with respect to the origin. i.e. x· u(x) ?: 0 for all :r E then
an,
(2.30)
In P(uJdx?: n 2 In 11p(l1)dx.
By imposing additional conditions, e.g. x· u(x) 'Ie 0 and 11 > () in n so Vl1 # 0 on on. the inequality in (2.30) becomes strict. Taking in particular P'(O = (8 + 11 then shows .5 < (n + 2)(n - 2)-1 is a necessary condition for there to exist solutions of (2.9). On the other hand, Brezis and Nirenberg IBN] have established some existence results for P( 0 P' (0 + lower order term when" equals the limit exponent (n + 2)(n 2)--1. In work in progress, Bahri imd Coron have shown if n = 3, and P( 0 P' (0 with s the limit exponent and n noncontractible, then (2.9) has a positive solution. If n is an annulus in Rn it is also known that there exists a positive solution of (2.9) for P( 0 = P'to for all values of s. Just what the relationship between the geometry of nand growth conditions on p need be for there to exist solutions of (2.9) remains an interesting open question. Our next application concerns a nonlinear eigenvalue problem
(2.31)
- 2:,11 11
x En.
= ),p(l1) , = O.
xE
an,
e-e
where), E R. For a class of such problems, being a good model case, we will show (2.31) has at least two positive solutions for all large ),. THEOREM 2.32 [ARj. Suppose p satisfies (p~), (P2) and (Po) there is an r > () such that p(~) > () for ~ E (Oor) and p(r)
O.
THE :"rOUi'\TA li'\ PASS TIlEOTm:,,1
Then there eXIsts a A > claSSIcal solutions wzth are
° such that for all ,\ > III
~. (2.31)
has at lew
ll.
PROOF. As with Corollar y 2.23, the proof begins by redefining p. Set p p( 0 for ~ E [0, rJ and () otherwis e Then 7J satisfies (pi]). (P2), (P:l) ). If 11 is a solution of 6u = AP(U),
(2.33)
U
= 0,
x E 0, x E all,
the argumen t of Corollar y 2.23 shows {x E ll'iU < O} (:3 find U > 0 j Similar reasonin g proves {x E Olu(x) > r} = 0. Hence 0 < nix) :s; r in U satisfies (2.31). By these observat ions and Remark 2.22, to find solutio (2.31), it suffices to produce nontrivi al critical points of (2.34)
dx
Wd
2 on E = (O), £5 being the primitiv e of p. To study the properti es of h" note first that since p satisfies (pd-(p2 )' C 1 (E, R). Moreover (PS) holds for h,: if(u m ) is a sequence in E with I),(u, M, the bounded ness of p implies 1£5(01 :s; a61EI find
(2.35)
M2:I),(um)2:~llumI12
j,\la6!olumldx
Applying the Holder and Poincare inequalities, (2.35) shows (urn) is bour in E. Hence (PS) follows from Proposit ion B,35. ~ote that (2,35) also iill] I A (n) is bounded from below, Hence by Theorem 2.7,
(2.36)
bA == inf h,(u) E
is a critical value of h, for all A E R, It may be the case that b 0 correspo n( A to the trivial critical point u = 0 of fA' Indeed (P2Hp3) and the arguill' centered around (2,18)-(2 .19) show u 0 is a local minimum of fA and satisfies (II!. In fact
(2.37) for 1,\1 :s; AO and Ilull :s; a7 provided that AO and a7 are sufficiently small. On other hand as in (2.35) the linear growth of £5 in E implies
(2.38) for all'\ E Rand u E E. Thus for small A (2,37)-(2,38) show h,(u) 2: 0 h,{ Consequ ently bA = 0, To obtain a nontrivia l solution, let 'P E E \ {O} such that 'Pix) E [0, r) x E O. By (Ps)
,l
P('P(x)) dx > 0,
l4
TilE \!()(::\TAI:\ PASS THEOREM
Thus for A sufficiently large, !),(Y) < 0 ,md bA < 0, Define A =: infp > 01 bA < O}. Then for all A > A, the abo\'(' remarks show IA has a criticial value bA < 0 and a corresponding critical point 1!A which is a posith'c solution of (2,33). \loreover since h(1!AJ < 0, h satisfies (h) of Theorem 2,2, Consequently the \lo11Iltain Pass Theorem yields a second critical point rIA of h such that h(rI A) > 0 > JA(lJ.Al, Clearly rIA is distinct from ]f A and again above remarks is a positive solution of (2.33), The proof is complete, Our final application in this section is to a problem for which one can establish t.he existence of at least three nontrivial solutions, Consider the equation: (2,39)
~ll = II
Aa(x)ll
~
p(X,ll),
xE
= 0,
x
E
n,
an.
The function a is assumed to be positive and e.g. Lipschitz continuous in O. Associated with (2.39) is the Sturm-Liouville eigenvalue problem: ~ ~1J =
(2.40)
IJ
=
/1a(x)1J, 0,
xE
n,
x E an.
As is well known, (2.40) possesses a sequence of eigenvalllefi (A J ) with 0 < Al < A2 ::; ... ::; AJ ::;. . and AJ ---+ 00 as j ---+ 00, (The number of times an eigenvalue appears in the sequence equals its multiplicity,) Concerning solutions of (2,39) we have PROPOSlTIO:" 2,41. Suppose p satisfies (P3) and (pc) (i) () < 0 == p(x, O~-I Jar ~ # 0, (ii) hE C I (0 x R, R) and ~hdx,~) > 0 Jar ~ # 0, (iii) h(x,O ---+ 00 as I~I ---+ 00 llniJormly Jor x E O. Then: (P) Jor A ~ AI, (2,39) possesses only the trivial solution u == 0, (2°) For each A> AI, (2,39) possesses a pair oj solutions with ut (resp, the unique solution oj (2.39) with ut > 0 in n (resp. < 0 in D), In fact the maps A ~ are CI for A> Al and u~ ---+ 0 as A ---> AI·
Since p satisfies (P3), u == 0 is a solution of (2,39) for any value of A, The first assertion of Proposition 2.41 follows simply on multiplying (2,39) by u, integrating by parts, and using (pe)(i) , The second statement can be proved in a variety of different ways~see e,g, [RI, St21~and will be omitted here, We will prove the following result which is essentially due to Struwe !St2], TllEORE!\.{ 2,42. If p satisfies (P3) and (P6), then Jar all A > A2, (2,39) possesses at least three nontrivial solutions,
The proof of Theorem 2.42 will be carried out in a series of steps, First observe that (pe)(iii) implies an a priori bound for solutions of (2,39), Indeed if 11 is a nontrivial solution, either u has a positive (global) maximum or a negative minimum, Assuming the former, if the max is attained at zEn,
o~
~611(Z)
= Aa(z)u(z) -
z)
THE :-'lOUNTAI:\, PASS THEORE:-'l
or
Ii( Z)) :S AO( zj.
(2.43)
The same inequality holds at a negative minimum. Hence by (P6)(iii), the a constant M = M(A) > 0 such that IlulIL= :S 1..,1(,\). In fact since A is fi choosing :\ > A there is an !vi depending on :\ such that Ilnll L= :S M for solution of (2.39) with A :s:\. With this observation, p(x, 0 or actually he can be modified for lei :0: M as in Corollary 2.23 or Theorem 2.32. so hypotheses (p6)(i)-(iii) also hold for the modification 0 c.= 0 a] satisfies
(P2) Ip(x, 01 :S 121
+ 122IEI8-1
for some s as in (P2). Moreover solutions of -
11
p(x, n),
A0.11
611
(2.44)
= 0,
O.
x
ao,
x
for A :S :\ are also solutions of (2.39). It remains to find a third nontrivial solution of (2.44). 2 (rl) and define
Wci·
h.(n) =
il [~lv1112
~an2 -
P(x
11)]
As usual let 1
dx,
where P is the primitive of p. Since (P6)(ii) implies (pd of Proposition B.34 p and (P2) implies (P2), by that result, I), E C2(E.R). Moreover f), satiE (PS) for. suppose
(2.45)
f{
~
:0: h
By (ps)(iii), for any ,8
eE R,
> 0 there
112 is a 1
A
l
= 1(3) :0: 0
P(x, E) :0: l3e
(2.46)
dx -'-
2 ill
l
dx
JIl
such that for all x En
i
1·
Choosing (3 > AllaIIL=/2, (2.45)-(2.46) show (urn) is bounded in E so (I follows from Proposition B.35. We want to find a critical point u), of I), other than the known ones 0, given by Proposition 2.4l. Concerning the latter we have: LEMMA 2.47.
If A> A], then
< 0 and
are strict lowl mini
of h. PROOF. The map A --+
is C 1 for A> Aj. Hence
since I~ (un = O. Moreover by Proposition 2.41. 1;,..(0) = O. Hence I),(n~) < 0 for A> A].
--+
0 as A
--+
Al a
16
THE .\lOl'NTAI:\ PASS THEOhEivl
To show that argument. Let 11.' =
(2.48) [A(11)
are strict local minima of [A' we will lise a comparison or Since h E C'2(£, R), for 11 near w,
= [A (U') +
J~(11')(11 - 10')
+ P;(W)(l1 - W, 11 -tel + 0(1;11
Since J~(w) = 0, to showw is a strict local minimum for Proposition B.34, it suffices to show that
h, by (2.48)
+ (h(]:, w) +
and
dx 2:
for some,) > 0 and all :p E E. The eigenvalue problem
- 6v
(2.50)
h(x,tL')v v
pav, O.
x E \1, xE
an.
U' with v of OIle sign in n. Hence A must be the hiL' iL" a solution p = A ilnd l' "mallcst rigclwalllc to (2.50). ?vloreover A can be characterized as [CHI:
(251)
h(x. w)v 2 ) dx
-r
inf
.\ =
O"'"EE
Comparing (2.51) to
dx shows thilt ! > A. This fact and (2.52) then show
f; (w)
:p)
2: ( 1
(2.53)
2:
(1
~) In ~)
11:p1!2
and the lemma is proved. With the aid of Lemma 2.4 7, we will find another critical point of [A' Consider the larger of h(ll~). (If these numbers are equal either will suffice,) Suppose it occurs at 1l~. Making the change of variables U = and setting 4>A(U) = h(U + ) - JA (l1\), we see that 4>.\ E Cl(£,R), 4>.\(0) = 0, and h satisfies (PS). Moreover Lemma 2.47 and our choice of origin show f.\ satisfies (Jd and (J 2) since h ) 2: h, (Il ~). Thus the Mountain Pass Theorem shows h has a positiH' critical value. Returning to the original coordinate system, it follows that h. hiL'i a critical value CA > fA ), where
1l-1l\
h(g(t)) and
fA = {g E C(lO, Ii. £)lg(O)
= ll~.g(l) =
}.
) by Lemma 2.47 and 0 is a critical point of h., a priori it may be the case that CA = 0 and the corresponding critical point is llA := O. Thus to ensure that 11), is nontrivial it suffices to show that c), < O. To do this,
THE
~10Cj\;TAJ"i
PASS THEOH.EM
the fact that A > A2 will be used. \Ve will construct a curve 9A E l\ sue h is strictly negative on 9A([0.1]). Before doing so, some further obsen about lA are needed. :\Tote first that
l),(tu) S t 2 1),(u)
(2.54)
for all u E E and t E [0, Ii. Indeed by (P6)(ii)
P(x,
==
(2.55)
lZ
0 d~ =
l
p(x, ty)tdy
t 2 r h(x,ty)ydYSt 2 rh(x,y)ydy
./0
./0
for any t E [0, 1J and z E R. Therefore
h,(t11)=t 2
(2.56)
{~(
-A!/(112)d.r+ (P(x,tu)cixSt 2 f:,,(I1)
./n
./n
for all 11 E E. :\Text let V denote t.he span of all eigenfunctions of (2.40) correspond AI and A2 and let fF denote the closure (in E) of the span of the rem eigenvectors of (2.40). Thus E = V Wand it is easy to check that 1/2
(-In
1/2
- Aaw 2 )
cix )
can be taken as equivalent norms on V and W. Abusing notation somewh will write Ilvll, Ilwll for these norms. Thus for u = v + wEE,
(2.57)
h(u)=
~llvI12+~llwI12+ InP(x,v+w)dx.
If h,(u) < 0, (2.57) and (P6)(i) show
(2.58)
+
21n pry, v+ w) dx > IIwl1
Therefore there exists a 0 = <5 (11) >
(2.59)
2
°
such that
Ilvll > (1
+ o)ilwll·
wi-
= v~ + Since h < 0, (2.59) shows II> 0. Set Now the path 9A E r,\ can be constructed. The path consists of five pa (a+) The first part is the line segment {tutlt E ,Ii}, where 7+livt and p > is free for the moment. It is clear from (2.56) that h < 01 segment (b+) Homotopy to 7-'-V;: via T+(1't + 81£'T). (c) Join to T-1';' on aBp n V, where TII = (I. !\ote that aB is a connected set since A > A2 implies the dimension of V is at ICilHt 2. (b-) HomotopYTto as in (b-i-). (a-) Join 7-U), to 11), by a straight line segment. As in (a-i-). here h, <
°
°
THE \10UNTAIN PASS THEOREM
18
To complete the proof we must show sufficiently small. For E [0,1]'
e
h < 0 for steps (b=!:) and (c) if p is
(2.60)
Hence by
(2.59) with 01
(2.61)
(vs:
-I-
'
T
= mint <'i( uI),o( u~)), and some simple estimates.
iJlL'f)) S; p
2
(1-2 +
2(1
g2 + ,sd 2 + 0(1) )
S; canst
<0
as p ~ D. Hence fA < 0 for (b±). A similar estimate shows fA < 0 for (c) and the proof of Theorem 2.42 is complete. RE:vlARK 2.62. For some related applications of the Mountain Pass Theorem see feR, G].
3. Some Variants of the Mountain Pass Theorem
In this brief section we will examine some other versions of the Moun', Pass Theorem as well as some small generalizations. In Theorem 2.2, the crit value c was obtained by minimaxing lover all curves which join and e. '] choice of sets is rather arbitrary. There are several other natural classes of with which one can work. E.g. set
°
ro = {g([O, l])lg is 1-1,9(0) = 0,9(1) = e}, r1=
{K
c ElK
is compact, connected. and 0, e E K}.
and
r2 =
{K
c EiK
is closed, connected. and 0, e E K}.
Define c, = inf sup I(u),
i
= 0, 1, 2.
KEr, uEK
Since ro c {g([O,l])lg E r} C r 1 c r 2 , it follows that Co 2: C 2: Cl 2: Moreover the proof of Theorem 2.2 shows that Cl is a critical value of I. Not that by Theorem A.4, 1)(1,.) is homeomorphism of E onto E, it follows t 1)(1,.): ri - t r" i = 0,2. With this additional observation the proof of Theor 2.2 also shows Co and C2 are critical values of I. What is the relationship between these numbers? It is not difficult to see t if K E r 1 and [ > 0, there is a g([O, 1]) E ro such that K lies in a unifc [-neighborhood of g([O, 1]). Thus Co = c = el. We do not know if C2 can di from this common value. Remaining in the setting of Theorem 2.2 there are other ways in which criti values of I can be characterized that are "dual" in a sense to those that we h, just seen. E.g. let
(3.1)
W == {B
THEOREM 3.2
(3.3)
c
E!B is open,
°
E
B, and e ric B}.
[AR]. Let I satisfy the hypotheses of Theorem 2.2. Then b == sup inf I(u) BEW uE8B
is a critical value of I with a ::; b ::; c. 19
20
SOME VARIANTS OF THE MOUKTAIN PASS THEOREM
PROOF. By (h), Bp E W. Therefore (3.3) implies b 2: BE W, there is awE g([O, 1]) aB. Hence
Ct.
If g E rand
inf 1::; I(u') ::; max I(9(t)).
(3A)
aB
tEIO.I]
Since 9 and B are arbitrary, (3.4) implies that b ::; c. Lastly suppose that b is not a critical value of 1. Then by (the Deformation) Theorem A.4 (with f 0'/2) and Remark A.7(ii), there is an E E (0, f) and ~ E CliO, 11 x E, E) such that
(3.5) where
As
=
{u E EII(u) 2: s}. Choose BE W such that inf 1> b -
(3.6)
BB
-
f.
By 3° of Theorem AA. ~(l..) is a homeomorphism of E onto E. Therefore ~(1, B) is an open set. Our choice of f and 2° of Theorem AA imply ~(1, 0) = and ~(l,e) = e. Hence E ~(1,B) and e t,t ~(I,B) = i)(l,EJ). Consequently ~(1. B) E W with ai)(l, B) = i)(1, aB). Thus by (3.5)-(3.6) and the definition of b.
°
°
(3.7)
b+ f
::;
inf
a(ry(I,B))
1 ::; b,
a contradiction. RE)'1ARK 3.8. Simple examples for E = Rn or even R show that b may be less than c. REMARK 3.9. The class of sets W of Theorem 3.2 is "dual" to classes introduced earlier in that B E W implies that B n K # 0 for all K E foTl' etc. Define WI = {h(aBp)lh is a homeomorphism of E onto E and h(O)
= O,h(e) = e}
and
W 2 = {K E ElK
n 9([0, 1]) # 0
for each 9 E f}.
Thus W2 :J {aBIB E W} :J WI. Therefore if
b, = sup inf flu),
i = 1,2,
KEW I uEK
then c 2: b2 2: b 2: bl and it is not difficult to show that bl and b2 are critical values of I. In fact h = c since for each 9 E L there is a ( = E(g) E E such that max 1 = I(E). g(IO,I:)
Set K = UgEf E(g)· Then K and infK 1 = c.
n g([O, 1]) #
(7) for each
9 E r. Therefore K E W2
SOME VARIANTS OF THE :>.lOUNTAlf\ PASS TBEORE:V! The next result generalizes Theorem 2.2 by weakening (II!. W refers to ( THEOREM 3.10. Let I E CI(E,R) and satisfy (PS). SlLppose that (Ij) 1(0) = and there is (j B E W SlLch that IiaB :::: 0, and
(I2l
°
there is an e E E \ E slLch that I(e) SO hold. Then b defined In (3.3) IS a cntical vallLe of 1 with b :::: 0 [f b = 0, exists a critical vallLe of I on CJ B.
I
PROOF. If b > 0, the proof of Theorem 3.2 with e.g. E = b/2 carries oV( the present situation. Thus suppose b = O. Since 0 E Band e f,i E. without of generality we can assume (3.11)
min(dist(e,E),dist(O,DB)) > 1,
where dist(X, Y) refers to the distance between sets X and Y. Suppose th has no critical value on BB, i.e. l' of 0 on DB. Since the set of critical pc Ko is compact, there exists a neighborhood, 0, of Ko such that 0 tl DB = By Theorem A.4 with e.g. E = 1 and Remark A.17(iii), there is an E. E (0. 1) r; E C([O, 1] x E, E) such that (3.12) In particular by (Ii! and our choice of 0, BB C A_f: \ O. Therefore by (3 and 3° of Theorem A.4, il(I, DB) = Bil(l, B) c Ai: and (3.13)
inf
i)(1,8B)
I> -
E.
> O.
We claim il(1, B) E Wand therefore (3.13) contradicts b O. By 3° of Theo: A.4, there is an x E E such that il(I, x) = 0 and by 4° of Theorem A.4, Ilil(I,x)-xll = Ilxll S 1. Hence by (3.11), x E Band il(I, B) is a neighborhood of O. Lastly Indeed, again by 3° and 4° of Theorem A.4, there is ayE E such = e and Ilil(I, y) - yll = lie - yll 1 while if y E E, Ily - ell> 1 by proof is complete. REMARK 3.14. Theorem 3.10 was motivated by work of Pucci [PSI. The following corollary can also be found in [PSI.
s
e f,i il(1. that il( (3.11). and Sel
COROLLARY 3.15. Let I E CI(E,R) and satisfy (PS). If I has a pail local minima (or maxima), then I possesses a third critical point. PROOF. Let Ul,U2 be the two critical points and suppose I(lLIl :::: l(v Translating variables so that Ul becomes the origin and replacing I( 1L) by 1* (11 I(u) - I(uIl, we see that l' satisfies the hypotheses of Theorem 3.10 with ( 1L2 - Ul and B the boundary of any small neighborhood of O. Hence I posses a critical value larger than I( uIl or equal to I( lLd in which case I has a criti point on BB.
22
SO:V!E
VAR[A~TS
OF THE MOUNTAIN PASS THEOREM
RE:V1ARK :U6. If the third critical value equals I( ud, then I has a critical point on aB for all small neighborhoods B of Uj. An argument from point set topolog~' then shows Kb contains a component of solutions which meets aB for all such small B rWh]. H EMARK 3.17. In roughly the setting of Theorem 2.2 or 3.10, several authors haw studied when any of the corresponding critical points is a "saddle point" of ~ !lul!4 one sees that there need some sort. Taking e.g. E = R" and l(u) = not be any saddle point. Howe\"er several positive results have been obtained. Sec c.g, [Ni. HoZ, PSi.
4. The Saddle Point Theorem The main goal of this section is to prove the Saddle Point Theorem stated in Chapter 1. The proof requires a simple applicat ion of the theory of topological degree in Rn, i.e. Brouwer degree. Since degree theory will also be needed in Chapter 5, we will make a brief digression to discuss this subject. A detailed treatme nt in the spirit of our exposition can be found in [821. Let 0 eRn be bounded and open, 'P E CleO, Rn), and bERn \ 'PUJO). The theory of degree is concerned with the existence and mUltiplicity of solutions of the equation
(4.1)
'P(x)=b .
Suppose 'PI(X) is nonsingular whenever 'P(x) = b. Then the Inverse Function Theorem shows the solutions of (4.1) are isolated. By hypothesis, b t/: 'P(80) and 0 is compact. Hence (4.1) can have only a finite number of solution s in O. For this "nice" case, the (Brouwer) degree of 'P with respect to 0 and b, denoted by drip, 0, b), is defined to be
(4.2)
d('P,O,b) =
sign det 'PI (x), XE",-l
(b)nO
where det A denotes the determin ant of a square matrix A. From this definition it is immedia te that for such "nice" 'P:
1 d('d 0 b) = {I0 0
,1,
(4.3)
2° 3° 4°
5°
,
if b EO. if b t/: 0,
d( 'P, 0, b) 1= 0 implies there is an x E 0 such that 'P( x) = b, d('P,O,b) =0 ifbt/:'P( O), (Additivity-excision property): If 0 1 , O2 C 0 are open with 0 1 n 02 = 0 and b 7': 'P(O \ (0 1 U O2 )), then drip, O,b) = drip, Ol,b) d('P,02,b), (Continuity in 'P): d( 'P, 0, b) = d( 1/),0. b) for all 1}; near 'P
Property 5° follows from a straightf orward application of the Inverse Function Theorem , In 5°, 1/J near 'P refers to the Cl norm. The notion of degree given above and properti es 1°-5 0 extend to all 'P E C(O, R n) and b ERn \ 'P( 80) with "near" in 5° now referring to the C (0, R n) 2:1
24
THE
SADDLE POINT THEOREM
topology. Of course the formula (4.2) is lost in making this extension. A useful and important consequence of 5° is PROPOSITIOC'{ 4.4. If HE C([O,I] x u,Rn) and b t/: H([CJ,l] x aO). then d(H(t, .), O,b) == constant for t E [0,1]. PROOF. Our above remarks imply that d(H(t, .), O,b) is defined for all t E [0, I]. By 5° and the definition of degree, the mapping t - d(H(t, -), 0, b) is continuous and integer valued. Since [0, I] is connected. the mapping must. be constant. COROLLARY 4.5. Ifrp,1;JEC(u,Rn), rp=1iJ onaO, andbERn\rp(30), then d(p, O.b) = d(1/!, O,b). PROOF. Set H(t, xl = trp(x) + (1 - t)1jJ(x) and invoke Proposition 4.4. The definition of degree given above and its properties extend to an infinite dimensional setting as follows: E is a real Banach space, C E is a bounded and open set, <J> E C(u, E) with <J>(u) u-T(u) and T compact, and b E E\ <J>(30). The resulting degree~still denoted by d( <J>, 0, b)~--is called the Leray-Schauder degree (see [LJ8 or 82]). With these preliminaries in hand, we turn to the Saddle Point Theorem [R4]:
°
Tm;OREM 4.6. Let E = V EBX, where E IS a real Banach space and V of {O} and is finite dimensional. Suppose IE CI(E, R), satisfies (PS), and (I3) there is a constant Q and a bounded neighborhood D of 0 in V such that llaD :S Q, and (14) there IS a constant (J > Q such that llx 2: .3. Then 1 possesses a critical value c 2: (J . Moreover c can be characterized as
(4.7)
c = inf maxI(h(u)). hEr uED
where
r = {h
E
C(15,E)lh
id on aD}.
PROOF. Let P denote the projector of E onto V obtained from the given splitting of E. If hEr. Ph E C(D, V). Moreover u E 3D implies Ph(u) = Pu = u of O. Identifying V with Rn for some n, d(Ph, D, 0) is defined and by COfo]I'lfY 45 and 10 of (4.3).
d(Ph, D, 0)
= d(id,
D, 0)
= 1.
Hence by 2° of (4.3), there exists an xED such that Ph(x) = O. Consequently for each hEr, there is an x = x(h) E D such that
(4.8)
h(x)
(id- P)h(x) E X
Hypothesis (1 4 ) now implies
ma:>Sl(h(u)) 2: I(h(x)) 2: (J. uED
THE SADDLE POINT THEOHEM
Thus by (4,7), c 2: ,3, To show that c is a critical value of 1 requires a fami argument, If Kc = Z, W8 set E = ~(p n:) and invoke Theorem A4 obtainin and 1] as usual. Choose hEr such that
1(h(u)) <:; ('
(4,9)
s,
If 11(1, h) E r, by (4,9) and 7° of Theorem A.4, we get our usual contradicti Since 1](1, h) E C(T5, E), it belongs to r if 1](1, h(v)) v for 11 E 3D, But such 11, h( u) = u and 1] (1, u) 11, This last equality follows via 2° of Theor A.4 since 1(u) <:; Cl < Cl + € <:; (3 ~ € <:; c + E via our choice of E, The prooj complete, RE~ARKS 4,10, (i) If V = {O}, (13) makes no sense but (14) and Theor, 2,7 imply 1 has a critical value, (ii) Hypotheses (13 )-( 14) will be satisfied if e,g, 1 is concave on V, convex X, and appropriately coercive as one approaches infinity along these subspac Theorem 4,6 was motivated by a PDE existence result due to Ahmad, Laz and Paul [ALP] (see also [Am, AZ]), As an application of Theorem 4,6 a versi of a result in [ALP] will be given next. Consider ~
(4.11 )
t.u = Aa(x)u + p(x, u), u = 0,
x E (1
an,
xE
where p satisfies (ptl and (P2) with s = 0, If A is not an eigenvalue of (2,3! then (P2) with s = and standard elliptic estimates imply an a priori bou for solutions of (4,11) in W5,p(n) for e,g, (3 > n, Then one can use the line elliptic theory for such spaces and the Schauder Fixed Point Theorem to get solution of (4,11), Thus an interesting question to study is what happens wh A is an eigenvalue of (2.40),
°
THEOREM 4, 12. Suppose A = Ak
s
< Ak-l- I and p satisfies (PI), (P2) Wi
= 0,
and (P7) P(x,O = Jo~ p(x, t) dt -+ 00 as I~I Then (4.11) possesses a weak solution.
-+ 00
uniformly for x E
n.
PROOF, Let =
Jr \(1,~ -2
,)
iVU !-
(]
A
"
~2av-
.\ u)) dx
\
for u E E WJ,2(O). Since (pJ) and (P2) hold, I E (;1 R), Let V span{vj" where v) is an eigenfunction of (2.39) corresponding to A] ar normalized so that
i
dx = 1 = A]
l
dv.
Let X = span{v]iJ 2: k + I} so X = V.L, Therefore E = V X. We will sho that I satisfies (h), (14), and (PS). Then Theorem 4,12 follows from TheoreJ 4.6,
26
THE SADDLE POI;\T THEOREM
(4.13) Let ,VI
/
in sUPY(Il<E=R
I
(4.14)
Ip(x.
~)I. Then
r P(x.u) dxi <:: M inr !ul dx <:: MJllul1
,in
I
for iilll1 E E Viii the Holder and Poincare inequalities. Combining (4.13)-(4.14) shows 1 is bounded from below on X, i.e. (14) holds. Next if u E V, then u 1I()+1l . where llllE:: E():::Sjxlrl{vJI),j ),dandu EE-=span{vJI),J<),d· Then ( 4.15)
l(u) =
~ I>; j
(1 - ~k) - JP(x, uo) dx + J(P(x, uO + u" n
J
0
Estimating the last term as in (4.14),
(4.16)
l(u) <::
-M2 1Iu-11 2
-
10 P(x,uo)dx+MJllu-ll·
:\ow (4.16) and (P7) show l(u) -+ -00 as 11 -+ co in V. Hence I satisfies (h). Lastly to verify (PS), by Proposition B.35 it suffices to show that 11(u m )1 <:: K and 1'(u m ) -+ 0 implies (urn) is bounded. Writing U m = u~ + u;;; + u;;" where u~ E EO, u;;, E E-, E X, for large m: (4.17)±
i!'(um)u;,!
= I[
r[vu m . vU;' -
,}o
Consequently since X (4.14), (4.18)
=
- p(x, um)u;,1 dxi <::
II.
Vl., by (4.17)+, (4.13), (P2 ), and an estimate like
',i> II -
(1-~) )'k+J
which shows that {llu;;'il} is bounded. Similarly (4.17)- shows { II} is bounded. Finally we claim that {lfu~i!} is also bounded. Then (urn) is bounded in E and we are through. To verify the claim, observe that (4.19)
K::: II(u m )! = i10 nUvu;;;1 2 + Ivu;;;1 2 - ),ka((u;;;)2 + (u;;;)2)]_ (P(x,u m )
-
P(x,u~))}dx
- 10 P(x, u~) dxl·
THE SADDLE POIlI:T THEORE M
By what has already been shown, the first term on the right is bounded il pendent ly of m. Therefor e
(4.20) so
Un P(x, u~) dx) is bounded. LEMMA
4.21. If P satisfies (pd, (P2), and (P7), then
f P(x,v)dx---> 1n
(4.22) as v
-t
This implies (u~) is bounded . Indeed
(X)
co uniformly for v E EO.
PROOF. By (pd-(p2)' the function al in (4.22) belongs to C I (E, R). By ( for any k > 0, there is a dk such that P(x, 0 ;:: k if I~I ;:: d for all x E k v E EO and write v = t'P where 'P E BBl. Then
n.
(4.23) where Ok
r P(x,t'P)dx;:: 1nk r P(x,t'P)dx-Mo,
1n
nk(t'P)) = {x
E
OIP(x, t'P(x)) ;:: k} and
Mo ;:: (meas 0)
IxEn,cER .inf p(x,OI·
Since 'P =Ie 0, there is an Xo = xO('P) E nand > 0 such that 'P f 0 in B 2r (, (the hall of radius 2r about xo). Therefor e by (P7), for all large t, 11k :J Br() and P( x, t'P) - t (X) on Br (xo) as t - t 00. Hence the right-ha nd side of (4.: - t co as t - t 00. Since BBI n EO is compact , (4.22) holds uniforml y for v E" and the lemma is proved. REMARK 4.24. If (P7) is replaced by P(x, 0 -+ -(X) as I~I -+ 00, the abo argumen ts can easily be modified to handle this case.
5. Some Generalizations of the Mountain Pass Theorem
This section begins with a generalization of the Mountain Pass Theorem in which (Id is weakened. This enables us to treat (2.9) where the nonlinearity also contains a linear part at O. Then an abstract critical point theorem is proved which contains earlier results both of mountain pass and saddle point type. To describe and state it, we will introduce an infinite dimensional notion of linking. To motivate the first abstract result of this section, consider
- L>1L
J)
1L
=
'\a(x)lL + p(x.lL),
0,
x E 0,
x Eon,
where p satisfies (pd-(P4)' This differs from the case treated in Theorem 2.15 due to the presence of the linear term in (5.1). The corresponding functional on E is
(5.2)
1(1L)
=
10 [~iVlLI2
~a1L2 -
P(x, 1L)] dx.
If ,\ < '\1, with '\1 as in (2.39), then (fn(lVlL\2 - '\a1L2) dX)I/2 can be taken as a norm on E. 1 satisfies (h), and one can use the Mountain Pass Theorem to establish the existence of a weak solution to (5.1) (and even a positive solution). However if ), > ), I, (11) no longer holds so our previous existence mechanism fails. The next result from [R5] gives us a tool to treat problems where a milder ver8ion of (Id is satisfied. THEOREM 5.3. Let E be a real Banach space with E = V 9 X, where V IS finite dimensional. Suppose 1 E CI(E, R), satisfies (PS), and (Ii) there (Ire constants p, Ct > 0 slLch that llaB p r-rX ;:: Ct, (lnd (h) there is (In e E oBI n X (lnd R > p such that if Q == (IJ R n V) EEl {relO < r < R}, then llaQ s: O. Then 1 possesses a critical value c ;:: Ct which can be characterized as
(5.4)
c
inf maxl(h(u)),
hEr uEQ
where
r REMARKS 5.5. (il
=
{h E CCQ. E)lh
id on aQ}.
3Q refers to t.he boundary of Q reiative to V@span{e}. 28
SOME GE:'-1ERALIZATIO);S OF THE
~10U"'TAI]\;
PASS THEORE l\l
(ii) If j/ = {O}, then X = E and (I;) reduces to (h). If further 1(0) ) becomes (h). Thus Theorem 5.3 is a generali zation of the Mounta in] Theorem . (iii) Suppose Ilv <:; 0 and there is an e E 8B l n X and an R > p such I( Ii) <:; 0 for u E F ES span{ e} and Ilull 2: R. Then for any large R, Q as deli ill (Is) satisfies IIQ <:; 0. PROOF OF THEORlcM 5.3. The only novelty in the proof is to show thai
(5J))
c
2: o.
Once that has been done, a familiar argumen t shows that c is a critical v, of I. Indeed if not, set E = 0./2 and invoke Theorem A.4 to get E E (0, E) '1 E C(iO, 11 x E. such that
Choose hEr such that (5.8)
maxI(h( u)\) < c + [. uEQ . , -
Then since I(u) <:; 0 on 8Q, 2° of Theorem A.4 and our choice of E imply t 1)(1, h(u)) = u for u E 8Q. Hence 1)(1, h(u)) E r. But (5.8) and (5.4) then im c
<:; maxI(1)(l, h(u))) <:; (.LEQ
C - E,
a contradic t.ion. To verify (5.6), it suffices to establish the following interscc twn theorem: PROPOSITION 5.9. If hEr, then
(5.10)
h(Q) n 8Bp n Xi- 12.
Given Proposit ion 5.9, let hEr and w E h(Q) n 8B n X. Then p
(5.11)
maxI(h( u)) 2: I(w) 2: uEQ
inf
vEBBpnx
I(v) 2:
0,
the last inequali ty following via (Ii). Since h was arbitrary , (5.4) and (5.1 imply (5.6). PROOF OF PROPOSITION 5.9. Let P denote the projecto r of E onto V I the given splitting . Then (5.10) is equivale nt to
(5.12)
Ph(u) 0, { II(id P)h(u)11 = p
for some u E Q (depend ing on h). Expressi ng u E n V and 0 <:; r <:; R, define
v E BR
(r, v) = !l1(id - P)h(v + re)ii, Ph(v Then E C(R x V, R x V). Since hlaQ (5.13)
(r,v)
id, if
1L
Q as u = v + re,
+ re)).
E 8Q,
= (Ilreli,v) = (r,v),
whE
:lO
SO\lE U;;NEInLlZATIONS OF THE \lO,:NTAIN PAS:; THEORF:\l
ao
i.e.
d(
(5.14)
Hence by 2" of (4.3), there exists u Q such that
;d(h, Q, (p, 0)) f O}.
{h E C(Q,
We now turn to an application of Theorem 5.3 to
1).
THEORE\l 5.16. Suppose p sa.tisfics (pJ! (p,d and 02'Ojorall~ER. (ps) Then jor all A E R, (5.l) possesses a nontnvial weak solution.
PROOF. If ), < AI, the smallest eigenvalue of (2.40), the result follows from fCmarks made at the beginning of this chapter. Thus suppose A 2' ),1, e.g. ), E I),b ),k~l]' We will show 1 as defined by (5,2) satisfies the hypotheses of Theorem 5.3. Clearly 1 E C 1 R) via (p!l(P2). Set V == span{vl,,,.,vd and X = F -"-, where the v) 's arc as following (2.40). For u X,
I)
(Ivul 2
),u 2 ) dx 2'
(1 ),
IIul1 2
as in (4.1:3) while hy (P.,) (and the proof of Theorem 2.15),
10 P(x, u) dx = 0(liuI1
2
)
as
(L - .
O.
Hence 1 satisfies (1;). To verify (15), by Remark 5,5(iii), it suffices to show (a) Ilv 0 and (b) there is an e E oBI n X and R > p such that l(u) 0 for 11 E V 8 span{e} == E and Ilull > R. But (a) follows from (ps) and (b) with e.g. e = Vk+l on noting that the argument of (2,16)-(2.17) is uniform for finite dimensional subspaces of E. It only remains to check that 1 satisfies (PS) or (via Proposition B,35) that )1 },,f and 1'(u m) -.0 implies (Urn) is bounded. Choose f3 E (p,-l, 2- 1 ). Then for m large and 11 = Urn
s:
s:
s:
.'vf +
2' l(u) - 31'(11)U =
17)
r i( ~ - 3)I,vu\2 - ),( ~ - fJ)au
io
2
+ fJp(x, u)u -
2' (~ - 3)llu11 2 ),( ~ - 3)llaIIL=(o)!luII12(o) + (13J.i - 1)
10 P(x, u) dx
a5
-),q - J3)llaIiL=(o)!luII12(o) + (3J.i - l)a6Iiulli"(o) - a7,
2' (~-3)lluI12
P(x, u)] dx
SOME
GEI\ERALIZATIO,\;'; OF THE \1()li'\T.\I\ PASS THEOHE\!
where (P.l) and (2.14) hilH' iWf'1l used. Ih ,[illlCl;lfCj ilH''I'Uliil i"s.sillC(' 11 >
,,> O. , Ii
(5.18)
U(O)
(18
lJ : V'
Un'
where K(E) ~ x as E ~ O. Choosing" SO Slllilil 111<11 the Ui term ill ( absorbs (with the aid of (5.18)) the L2 terlll. we sec (lIm) is bounded in 1 the proof is complete. REMARK 5.19. In Corollary 2.2:3, we shows that (2.9) possesses a po solution. The same arguments apply to the current setting if A < A1. HOI if A 2': AI, we cannot expect a positive solution of (,5.1). Indeed if 1)1 is an ( function corresponding to Al in (2.40), we can assume VI > 0 in rl. Theref 11 is a solution of (5.1),
(5.20)
r(A011 + p(x, 11))111 dx ior(-6u)vI dx = r(-6vdudx io =
in
=
Al
r oVludx.
in
Consequently (5.21 )
(AI - A)
r 01LVI dx = .!0p(x, u)vJ dx.
.10
If u is positive in 0, the left-hand side of (5.21) is nonposit.ive while by (ps) right-hand side is nonnegative. Thus there can only be a positive solution if A = Al and p(X,11(X)) == o. An examination of our abstract critical point theorems shows that an im tant ingredient in their proofs is an "intersection theorem" which allows L show that the minimax values defined in these results are indeed critical va E.g. in Theorem 5.3, this is carried out in Proposition 5.9 while in Theor·err it is done in (4.8). Our next goal is to introduce a topological notion of lin which modulo assumptions on the form of the functional will enable us to p a result which contains both Theorems 4.6 and 5.3. It also enables us to we, the splitting assumptions of these theorems. Thus let E be a real Banach space with E = E1 9 E z, where both EI E2 may be infinite dimensionaL Let PI, Pz be the projectors of E onto EI associated with the given splitting of E. Set
S == {1> E G([O, 1) x E, E)[1>(O, u) = u and P21>(t, u) = P2u - K(t, 11), where K: [0,1] x E
~
E2 is compact
Recall K compact means it is continuous and maps bounded sets to relati· compact sets. Let S, Q c E with Q c E, a given subspace of E. Then BQ refer to the boundary of Q in that subspace. We say Sand BQ link if when! 1> E Sand 1>(t,BQ) n S = 0 for all t E [0,1]' then 1>(t, Q) n S of 0 for t E [0, 1). This notion and the examples below are due to Benci and the aut IBR]. For heuristic purposes. one can think of the sets Sand BQ as linkin
32
SOME GE:"ERALIZATlO:"S OF THE :vIOG:"TAl:" PASS THEOREM
every manifold modelled on q and ,hiuing the same boundary intersects S, \\'e will give two examples of such linking corresponding to what occurs in Theorems 4,6 and 5,3, EXAMPLE 5,22, Let q B n E 2 , where B is a neighborhood of 0 in E 2 , E E 2 , q E q, and S q + E 1 , Suppose
E Sand
=
(5,23)
=
(t,aq)n(q+Ed
:2
for all tE [0,11,
\Yo claim (t, q) n S # 2 for all t E [0,1]' i,e, there is a w = w(t) E q such that P2(t,w(t)) = q, For u E E 2 , oct W(t,u) P2 (t,u) = u - K(t,u), By (5,23), W(t, u) # q for u E aq, Therefore d(W(t, ,), Q, q) is defined for t E [0,11 and by the infinite dimensional versions of Propositions 4.4 and 4,3, for any t E [0,11,
=
(,'),24)
d(W(O,), q, q) = d(id, q, q) = 1.
d(W(t,), Q, q)
Hence there is a 11'( t) E q as desired and Sand aq link, II EMARK ,5.25, Setting q 0, S = X, and q = D, we are in the setting of the Saddle Point Theorem, EXA:vlPLE 5,26, Let p > 0, S \".E], e E E1 noB 1 , 1'1> p, 1'2> 0, span{e} E 2 , Suppose E Sand Q E iO.rJ]}
=
(,),27)
(t.aQ)
S=2
for all tE [0,1],
\,\'c claim for each t E [0,1]' there is a w(t) E Q such that (t, w(t)) E S, i,e, P 2 ¢(t, (] and ::P1 (t, p, For u E E2 and r E R, set
'11 (t,(r, By (,5,27), d(W(/,),
q, (p, 0))
is well defined and as in (5,24),
d(W(t,(r,u)),q,(p,O)) = 1. Consequently there is a wit) as claimed and Sand aQ link, RE".IARK 5,28, Setting E1 = X, E2 = V, and T1 = R = T2, the relationship between Example 5,26 and Theorem 5,3 becomes clear. .'\ow we are ready to state a critical point theorem which unifies Theorems 4,6 and 5,3, THEORE"1 5,29, Let E be a real Hilbert space with E = EJ ttl E2 and E2 = Et Suppose 1 E C 1 (E,R), satisfies (PS), and (Is) l(u) = HLu, 1l) + btu), where Lu = L 1 Pj u + L 2 P2 U and L,: E, ~ £, is hounded and selfadjoint, i 1. 2, (h) b' is compact, and (17) there exist5 a subspace E c E and sets SeE, q c E and constants o > w such that (i) S c El and lis:::: (x, q is bounded and lidQ <:: w, (iii) Sand DQ link. Then 1 possesses a critzcai calue c :::: Q,
SOME GENERALIZATIONS OF THE MOUl\TAI" PASS THEOREM REMARK 5.30. Roughly speaking, c will be obtained as the minimax over all surfaces modelled on Q. Although the hypotheses of Theorem 5.2~ more restrictive than in our earlier results because of (1.0 ), (h), £'2 == Et, it is easy to check that the PDE applications presented earlier do indeed sa these conditions. PROOF OF THEOREM 5.29. The proof follows the same lines as our ea results. An appropriate class of sets will be introduced and c defined as minimax of lover this class. Let
f == {h E C([O, IJ x E, E)lh satisfies (f J)-(f 3)}, where
(ftl h(O,u) = u, (f2) h(t,u) = u for u E 8Q, and h(t, u) = eB(t,U)Lu+K(t, u), where e E C([O, IJ xE, R) and K is comp; Note that id E f and if g, h E f so is go h with egoh = eg(t,h(t,u)) +Oh(t,U) and
Kgoh(t, u) = e8g (t,h(t,u))L K h(t, u)
+ Kg(t, h(t, u) J.
Also by (f3) and (h),
(5.31) where u, property.
= Piu, i = 1,2.
The mappings in f have an important intersecti
PROPOSITION 5 32. If hE f, then
h(t, Q)
(5.33)
n S Ie
0
for each t E [0,1).
PROOF. Equation (5.33) is equivalent to solving
(5.34) and via (5.31) the latter equation is equivalent to
(5.35)
U2
+ e- BL2 K 2 (t, u) = 0,
where K2 = P2 K. Define
+ U2 + e- BL2 K 2 (t, u)
so by (5.34)-(5.35), (5.33) is equivalent to
(5.36)
By its definition,
(5.37)
n S Ie 0
for some t E [0,1].
S. Thus to prove (5.36), by (17 )(iii), it suffices to shm
=0
for each t E
10,11.
:34
SO\JE CE\ERALlZATIO\S OF THE \lOC\TAI\ PASS THEOREM
But
=c
7 by (i) and (ii) of (/7). Hence
(5.37) holds and the proposition is proved. :\ow we can define
(5.38)
c = inf sup I(h(1. u)) hEr uEQ
By hypothesis (Ic), b is weakly continuous (i.e. if (u m ) converges weakly to u, b( 11 m) -- b( 11)) [K] The boundedncss of Q and (10) then imply SUPuEQ I (u) < x. Since id E r. it follows that,~ < x. By Proposition 5.32 and (i) of (h), (5.39)
c :::
Ct.
To complete the proof. we must show that c is a critical value of 1. Let 5' = ~ (0 - ...'J. By our usual argument, if cis 1101, a critical value of 1, there is an S E (0,5') and II E CliO, 1] x E, E) given by Theorem AA such that 'I( 1. c Choose h E f such that sup I(h(l, u)) :<:: c -;-
S.
uEQ
TheIl our usual argument leads to a contradiction provided that '10 h E f. By all above remark, th s will be the case if 'I E f. By I O of Theorem AA, 'I satisfies (r Il and by 2° of that theorem, (5.39), (h )(ii), and the choice of E, (f2) holds. Finally by Proposition A.I8. fl also satisfies (f3)' RE\!ARK 5AO. See [Ni, BR, and HoI] for other such results. Au applicatiou of Theorem 5.29 will be given in the next chapter.
6. Applications to Hamiltonian Systems This chapter contains applicati ons of Theorem 5.29 to problem s in which both EI and E2 are infinite dimensional. Such situation s arise in studying the existence of periodic solutions of Hamilto nian systems. For p, q E Rn and H E e l (R2n, R), the system of ordinary differential equation s (6.1) is called a Hamilto nian system. Setting z = (p, q) and J = (~ -;}), where] is the n-dimen sional identity matrix, (6.1) can be written more concisely as (6.2) Onc of the basic propertie s of such systems
IS
H(z(t)) := const, i.e. "cnergy" is conserved. Indeed
that if z( t) is a solution,
(6.3) via (6.1) or (6.2). It will be shown that (6.2) possesses periodic solution s under certain conditions on H. Before giving a variation al formulat ion of (6.2), some prelimin ary material on function spaces and norms is needed. Let L2(SI,R m) denote the set of m-tuples of 27f periodic function s which are square integrable. If z E L2(SI, Rm), it has a Fourier expansio n z LJEZ ajei)t, where a) E a_ J = CiJ ' and LjEZ [2 < 00. Set
em,
:=
(2:::(1 + JEZ
and let
W e,2(SI, Rm) := {z E L2(SI, Rm)llIzllw8.2 < oo}. We will mainly be intereste d in () = ~, m = 2n, and E := W Ij2,2 (S 1 , R 2n). For e.g. smooth z (p, q) E E where p and q are each n-tuples , set
(6.4)
A(z)
2"
1 o
35
p·qdt.
36
APPLICATIO:--JS TO HA.\IILT O:\IA:\ SYSTEI-I S
Then it is easy to check that
I :; const
(I:, :Ji )ccZ
Therefor e A extends to all of E as a continuo us quadrati c form. This extensio n will still be denotd by A. Let Cl . . . . . e2n denote the usual bases in R2n and set
EO '=
Sp8.n{el, ... e2n},
Et '= span{(si n]t)ek
(cos
, (cosJt)ek -+- (sinJt)ek+nl JEN.I :;k:;n} .
and
E- '= span{(s in]t)c, + (cos
Jtlek - (sin
JEN.I :;k:;n} . Then E = EO E- E-. In fact it is not difficult to verify that E+, E-. EO are respectiv ely the subspace s of E on which A is positive definite, negative definite, cmd null. and these spaces are orthogon al with respect to the bilinear form
10r
2r
I"i
(p~'-+-pq)dt
il$sociated \vith A. Here z = q) and I" = (p, E.g. if z E E+ and I" E E-, 1": = 0 and -+- 1") = A(z) -+- .'1(11. It is also easy to check th8.t Eo,E+, and E- 8.re mutually orthogon al in U(Sl.R 2n). These remarks show that if z = zO + z-'- T z- E E.
(6.5) 8.S an equivale nt norm on E. Hencefo rth we use the norm defined in (6.5) as the norm for E. The spaces EO, E-'-, E- are mutually orthogon al with respect to the associate d inner product via our above remarks . One further analytica l fact about E is needed.
SNVP,
PROPOSITIO'-:
6.6. For each
3
there is an
E [1,(0), E D's
13
compactly embedded in
> 0 such that
16.i)
for all z E E. PROOF. See iFr or R7]. To obtain periodic solution s of (6.2). we are led to study critical points of an associate d function al OIl E. Suppose H E Cl(R2n .R) and there is an s E (1,00) :iueh that
\6.8)
APPLICATIONS TO HAMILTONIAN SYSTEMS for all z E R 2n. Then by Proposit ion B .39,
)I(z) == (2rr H(z)dt E CI(E,R ) .fo as does !(z) defined by (6.9)
!(z)
A(z) - )I(z).
If Z C I (SlR2n ) is a critical point of 1 and <; = (p, shows that
E E, a computa tic
(2Jr
I'(z)<; = 0 = .fo
which implies that z satisfies (6.2). More generally a critical point z of 1 in E wi be a weak solution of (6.2). However a simple regularit y argumen t then sho\\ Z E CI(SI R2n) (see the proof of Theorem 6.10 below). Thus we are intereste in critical points of (6.9). THEOREM G. 10 [R6]. Suppo.se H E CI (R2n, R) and satisfies (HdH? O, (H 2) = o(lzj2) as Izi -+ 0, and (H3J there are constants il > 2 and r > 0 such that 0 < ilH(Z) 'S z . Hz(z for all ? r. Then for (my T > 0, (6.2) possesses a nonconstant T periodic solution. PROOF. Making the change of variables
(6.11)
T
= 2'rrtT- 1 ,
(6.2) becomes
dz/dT = ).,JHAz),
where)., = (27T)-IT and z is 27T periodic in T. Since this has the same form as (6.2) with H replaced by )"H, without loss of gcncmlit y w(' can takc T = 27T and work with (6.2). Basically the proof reduces to verifying that. Theorem 5.29 is applicab le here. However there are some technica l complica tions. Since the growth conditio n (6.8) has not been assumed for H, the correspo nding term in (6.9) need not belong to C 1 (E, R). Thus to get a C I function al on E, H wi!! be modified so that it grows like a power of Izl as [z! -+ 00. Let K > 0 and X E COO(R,R) such that x(y) == 1 if y::; K, X(y) == 0 if y ? K + 1, and X'(y) < 0 if y E (K, K + 1), where K is free for now. Set (6.12) where R = R(K) satisfies
R? Then HK E C I (R2n,R) , satisfies (HJ)-(H2) and (6.8) with" = 4. ~loreo\'er a straightf orward computa tion shows (H3) holds with II replaced by v = min(![, 4). Integrati ng this inequali ty then yields
(6.13)
HK(Z) ? a3lzi"
a4
APPLICATIO:-;S TO HA\llLTOI\IAI\
for illl z
c R 2'1.
where
SYSTEMS
> 0 and ilre independent of K.
(lJ .11.j
set
(G.14)
We will ,;how II{ satisfies the hypotheses of Theorem 5.29. This willieild Ilonconstant 27i periodic (weak) solution 21{ (I) of
to
a
(6.15) It will then be shown that ZK is a classical solution of (6.15). Further estimates then prove there is a Ko such that for all K > KO, I ZK I L= ::; Ko. Therefore Jh(ZK) = H(ZK). Hence for such K, ZK satisfies (6.2). Turning to a study of (6.14), by Proposition B.39. IK c CI(E,R). Choosing E 1 == E~ ami E2 == EO ~ E-. we see that I K satisfies (I5) of Theorem 5.29 with L, clefined by for
Z
E E, aIlCI
b(z) =
_12K HK(z) dt.
Proposition B.39 implies that b'(z) is compact. Hence (h) holds. The next three lemmas establish (17).
LE\!\1A 6.16. If H satisfies (H 2 ), (Ii) (i) holds for h· PROOF. Dy (H2), for any E > 0, there is a 6 > 0 such that HK(Z) ::; slzl2 6. Since H K (z)lz!-4 is uniformly bounded as Izl ~ 00, there is an M = M(s. K) such that HK(z) ::; Mlzl4 for Izi 2: 6. Hence
if
Izi ::;
HK(Z)::;
(6.17)
+
for all z E R2n. Therefore by (6.17) and Proposition 6.6,
fTC HK(z) dt::; [llzI11, + Mllzll1, ::; (m2 + Ma411z112)llz112
(6.18)
Consequently for z E El = E+,
h(z) 2: Choose
E
1
!lzil 2 -
(m2 +
Ma411zi12)llz112
and p so that 3M a4p2 = 1. Then for z E fJBp
h(z) 2: 1p2
n E1 ,
a.
Hence IK satisfies (h)(i) with S fJBpnEl' REMARK 6.19. p and a depend on K through M.
LEMMA 6.20. If H satisfies (Hd and (H3J, then PROOF. Let e E BBI
n El
h satisfies
and set
Q == {relO ::; r ::; rl} EB (Be2
n E 2 ),
(I7)(ii).
APPLICATIONS TO HA\llLTO\,IA\, SYSTE'dS
where r1 and r2 are free for the moment. Define Let Z = zO + z~ E Br2 Ii E 2 . Then
Jf{(Z
(6.21)
+ re)
3'
E == span{e}
E2 so Q c
E
= r2 -
'\Tow by simple inequalities and orthogonalit.ies:
(" Hf{(z + re) dt 2:'
a3
"0
2:'
a5
(6.22)
~2" iz + rei" dt -
.10
(1
2 "
27fa4
Iz + rej2 dt) ,,/2
-
a6 v/2
a5
2:'
(1h(lzOI2 + Izj2 +
a7(lzOI" + rV) -
dt )
- at
a6
Combining (6.21)·(6.22) shows
19(z
(6.23)
+ rel
::; r2 .~
Choose r1 so that (6.24) for all r ;::: rj. Set
M = max 'P(r). rE[O,rd
Since
1/J(z) == uniformly as Ilzll -+ 00, in E 2 , 1/J(z) 2:' M if Therefore by (6.23), if Ilzlj ;::: r2,
Ilzll
is large enough, e.g.
19(z + re) ::; M -1/J(z) ::; 0.
(6.25)
Ilzlj ;:::
r2
°
Combining (6.24) and (6.25) with the fact that 19 ::; on E2 via (Hd, we see == won DQ and (h )(ii) holds. REMARK 6.26. Both rj and T2 and therefore Q are independent of K.
19 ::;
°
LEMMA 6.27. If Sand Q are defined as in Lemmas 6.16 and 6.20, then S and DQ link, i.e. If{ satisfies (h)(iii). PROOF. Immediate from the definitions of Sand Q and Example 5.26. The above three lemmas show If{ satisfies (h). '\Tow to be able to use Theorem 5.28, it only remains to verify that 19 satisfies (PS). Thus suppose Ilg(Zm)'1 ::; A1 and Ij.;(zm) --> 0 as m --> 00. Then for large m and z = Zm:
M (6.28)
+ IIzl!
2:lg(z) -1Ij.;(z)z = J02"[~z. Hf{z(z) - Hf{(z)] dt
2:
(2~j - l/~j)
l"
z· Hf{z(z) dt - Mj ;::: M 2 11zlli,
-
M3
10
APPLICATIOi\S TO HAMILTONIAi\ SYSTEMS
via (H3) and the form of H K . In (6.28). both M2 and M3 depend on K. \",'riting zO + z-'- ~ z', (H3) and simple estimates show
z=
"\1
(6.29)
+llzll :: (2- 1
v- 1 )v
12rr
Hg(z) dt - M4
"ote that .\16 and ;\h are independent of K via (6.13). Inequality (6.29) can also be written as
(6.30) \ext taking
Z
= Zm
and
<;
in
=
(6.31) and using the Holder inequality and (6.7) yields
21Iz+I\2 = A'(z)z'" ::; ::; "\lg (llzlli,
1
r2rr
lio
H gz
+ 1)
II.
Hence
(6.32) via (6.28). Similarly choosing <; -z- in (6.31) yields (6 ..32) with z+ replaced by Z-. Combining these inequalities with (6.30) shows
Ilzll ::; M12 (1 + Ilz1l3/4 + Ilzll"l)
(6.33)
which implies that (zm) is bounded in E. To show that (zm) is precompact in E. note that Since EO is finite dimensional. (z~) is precompae!. By the argument of Propo· sition B ..35. (6.34)
where pi: is the orthogonal projector of E onto E'" and T'"' (z) is compact. Hence ((U4) and IK (zm) ~ 0 as m ~ x implies has a convergent subsequence. Theorem 5.29 is now applicable and from it we conclude that IK possesses a critical value eK :: a = arK) > 0 with a corresponding critical point ZK = (PI{. qK). It remains to show that Zg is a classicalnonconstant solution of (6.2). Since IK )\. = 0 for all \ E K choosing \ = (',0,0) E W1.2(SI,R2n) shows (6.35)
f2rr[PK'
io
D-
qg . c; -lhz(zl{)' c;J dt = O.
In particular if \' == 1. (6.35) implies
HA~IILTO:;IA:\ ~\,::;TE:,l::;
APPLICATIOt\S TO
Fourier expansion shows that if y E L 2(SI,R 2n), [y] = 0, and ~ E R 2n, ther there exists a unique x E W I ,2(SI,R 2n) such that [x] = E and:i; = y. Th( choice of HK and Proposition 6.6 show that HKz(ZK) E L2(SI,R2n). Conse quently choosing y = lHKz(ZK) and ~ = [ZK], there is a unique Z = (p,q) E W I .2(SI,R 2n) such that !z] = and (6.36) Taking the inner product of (6.36) with and integrating by parts yields
le;. where e;
1
2 1<[p.,J; - q
(6.37)
E
=
. R2n'
I] dt = O.
Comparing (6.35) and (6.37) which hold for all s' E W1.2(SIR2n) showE Z ZK E W1,2(SI,R2n) and ZK satisfies (6.15) in an a.e. s{'nse. l3ut ZK E W1.2(SI,R211) easily implies ZK is Holder continuous of order~. Therefore HKz(Zk) E C(SI,R 2n) and (6.15) tells us ZK E CI(SI,R2n) and is a classical solution of (6.15). If Zf{ == const,
h(Zf{)
=
_121< HK(ZK)dt:s; 0
via (Hd. But h(ZK):::: arK) > 0 so ZK is a nonconstant solution of (6.15). The last step in the proof is to show that there is a Ko > 0 such that for all K:::: Ko, IlzKIIL= < K. Then HKAzK) = Hz(ZK) and we have the desired solution of (6.2). By Theorem 5.29 and in particular (5.38)
h(ZK) Since h(t, z)
==
Z
E
= CK
= inf suph(h(l,z)). hEr zEQ
r, Cf{ :s; suph(z)
(6.38)
zEQ
with Q as defined in Lemma 6.20. By Remark 6.26, of K. Then for Z = re + zO + z- E Q,
ior
rl
and
h
(6.39)
112 -
via (H1)' Consequently by (6.38)--(6.39)' CK (6.28), this implies
ri : : h(ZK) (6.40)
Hf{(z)dt:s;
:s;
T2
are independent
ri
rr independently of K.
As in
Vk-(ZK )ZK
:::: (Tl _v-I)
121r ZK' HKz(ZK) dt -
MI.
Thus (6.40) provides a K independent upper bound for J~" Zf{ . H f{ AZf{) dt. By
(H3 ), (6.41)
HK(d :s; v-II' HKz(r;) + lvh
,PI'L1( ATJO:\S TO lL\\IILTO:\L\;\ ",.':;TDlS
·j2
for all;; E R2n. Choosing \ = ::K. integrating (641) over [0, 27f), and rccalling that HI\ ) == rDnst SillCecl\ satisfies a Hamiltonian system, (641) yields
27f H K ( Z 1\)
(6.42)
-:: /.1- 1 /' h ( Z 1\ . .0
H 1\ z ( Z K )) dt
-j-
27f Ah
The right-hand side of (6.42) is hounded from aboY(' independently of K via om above remarks. Thu., (6.42) and (6.13) yield a K independent LX bound for ZK imel the proof of Theorem 6.10 is complete. R E\L\ H h C.4:l. J11 int Theore]]1 6.10, a caveat is ill order. In its setting. given an~' T > 0, tlwrc l'xins a IlOllconstcmt solution of (6.2) having period T. However T may not he the minimal period of the solution which may Ill'Tk 1 for :;OIlle k E: N, k > 1. Under further hypotheses on II, Ambrosetti and Mancini [AM2], Deng [De], and Ekeland and Hofer [EH] have proved there exists a solution having minimal period T for any T > O. An easy consequence of Theorem 6.10 is COHOLLAH.Y 6.44. Under the hypotheses of Theorem 6.10, there exist inJimtely many (hsUnd T periodic .solutions of (6.2). PROOF. Theorem 6.10 provides one such solution, ZI(t). Suppose its minimal period is Tkjl Apply Theorem 6.10 again with T replaced by T(2ktl- 1 to get a nonconstant T(2ktl- l periodic solution Z2(t). Certainly Z2 is T periodic and it is distinct from ZI since its minimal period is less than tkjl. Repeating this process produces a sequence of distinct nonconstant T periodic solutions of (6.2). RD1AHK 6.45. There is a much stronger version of Theorem 6,10 [R9]: If HE C l (R 2 n,R) and satisfies (H 3 ), then for any R,T > 0, there exists aT periodic solution Z of (6.2) with > R (sec also [Bel]). Simple examples show that T may not be the minimal period of the solution. E.g. suppose n = 1 and H(z) = G(!z:2), where G is smooth and monotone increasing. Then (6,2) hecomes
and
== const for a solution. Setting
¢ = 2iG'(ki
~ 2
= p + iq, , satisfies )\
,0 exp(2iG'
Therefore if T is the minimal period, T = In particular if G' :2 1, T -:: 7f, We conclude this chapter with an application of a seemingly different nature, Consider (6,2) where instead of fixing the period, the energy is prescribed, e,g, H(z) == L 1.
THEORE~1 6.46 [R61. Suppose H E C l (R2n,R) and H-l(l) is the boundary of a compact starshaped neighborhood of 0 with z· Hz # 0 on H-l (1). Then (6.2) has a periodic solution on H-l(l).
This theorem will be obtained with the aid of Theorem 6,10, First a technical result.
APPLICATIONS TO HAMILTO\iL\\i
SYSTI-:\l~
1 PROPOSrTIOT\ 6.47. Suppose H.H E Cl(R2n.R) wl.th H 1(1)= H- (1) and Hz,H z lOon H-l(1). If dt) satisfies ~ = JHz(r;)
(6.48)
and 1(0) E H-l(1), then there exists a reparametrization z(t) of Itt) such that z satisfies (6.2) and z(t) E H-l(l). In particular if \ is periodic, so 18 PROOF. Since H- 1(1) is a level set for Hand Hand Hz,Hz lOon this set, these gradients must be proportional on H-1(1). i.e. there is i1 1/ E C(H-l(l),R \ {O}) such that Hz(z) l/(z)Hz(z) Now dO) E H-1(l) and (6.48) is a Hamiltonian system so ((t) E H-l(1). Set z(t) == \(r(t)) where r(O) = 0 and r is a solution of (6.49) Therefore z satisfies z(O)
i
\(0) E H-l(l) and
= ~~r = 1/(\"(r(t)))JHz(\"(r(t))) = JHz(z(t)),
i.e. (6.2). To justify the assertion about the periodic case, more care must be taken since the solution of (6.49) with r(O) 0 may not be unique if 1/ is merely continuous. Suppose \" is T periodic and without loss of generality assume 1/ > O. Then 1/ > 0 on d[O, Til and (6.49) implies there is a first positive value I of t snch that r(l) = T. Set 3(t) r(t) for t E [0, II, and for J E N, set 3(t) = JT + r(t - JI) for t E [JI, U + I)I]. Then S E C 1 . Indeed we need only check what happens at
t
=
I. lirrlS'(t) tit
= lirrH'(t) = lir!.w(dr(t))) tTt
tTt
= l/(dT)) = v(\(O)) = liJ'!.lI/(\"(r(t
I)))
tll
= liJ'!.lr'(t - I) = liJ'!.l3'(t). tll
ilt
It follows that z(t) = d3(t)) is I periodic. PROOF OF THEOREM 6.46. By Proposition 6.47, it suffices to find an H 1 such that H- (1) = H-l (1) and for which the existence of a periodic solution of (6.48) can be established. The geometrical assumption on H-l(l) leads to the construction of such a function. By the starshaped assumption for each z E R2n, there is a unique w(z) E H-l(l) and a(z) > 0 such that z = a(z)w(z). Indeed a(z) = izi Iw(z)I- 1 Note that a is homogeneous of degree one and is C 1 for z I O. Define H(O) = 0 and H(z) = a(z)4 for z 10. Then H E C 1 (R2n,R), H-1(l) = H-l(I), H 2: 0, H(z) = 0(lzI2) as Izl--. 0, and z·Hz(z) = 4H(z) since H is homogeneous of degree 4. Thus H satisfies all of the hypotheses of Theorem 6.10. Hence for e.g. T = 27f, there exists a nonconstant 27f periodic solution \" of (6.48). It need not be the case that H(<;(t)) == 1. However H(,,(t)) == f31 0 since <; 1= O. Set! = f3- 1/ 4 Therefore Hhl) == 1. Moreover (6.50)
h~) = ~IJHz(\)
=
44
APPLICATlO;';S TO HAMILTOKIAN SYSTEMS
via the homogeneity of Hz. Finally let t 271,-2 periodic and by (6.50) satisfies
dz dT
= ,2T
; - (~ ) dt _ Hz II dT -
and z( T)
= ,s-( t).
Then z is
r- ( ) oj
Hz z .
Thus z is the desired solution and the proof is complete. REMARK 6.51. It is possible to make a more direct study of the existence and multiplicity of periodic solutions of (6.2) having prescribed energy using a direct variational approach and ideas from Chapters 7-9. See e.g. [AMI, BLMR, EL, VG, WI·
7. Functionals with Symmetries and Index Theories
Probably the most striking applications of minimax methods are results which establish the existence of multiple critical points of functionals which are invariant under a group of symmetries. In this chapter some machinary to treat such questions will be introduced. To be a bit more precise, let E be a real Banach space, 9 a group of mappings of E into E, and I E C1 (E, R). We say I is invariant under 9 if I(gu) = I(u) for all g E 9 and u E E. As a simple example, wppose I is even, i.e. I(u) = I( -u) for all u E E. Then I is invariant under 9 {id, --id} =:: Z2. As was noted in Chapter 1, a second example is provided by (1. 7), a functional associated with periodic solutions of Hamiltonian systems where 9 =:: 51 The earliest multiplicity result for symmetric functionals is due to Ljusternik and Schnirelmann wo proved [LL8]: THEOREM 7.1. If IE Cl(E,R) and is even, then Iism-l has at least m distinct pairs of critical points.
1)
In order to prove Theorem 7.1 or other such results, we need a tool to measure the "size" of a symmetric set. (By a symmetric set, we mean one which is invariant under the symmetry group.) The so-called Ljusternik-Schnirelmann category [LL8] was introduced for this purpose. A simpler notion, that of genus, is easier to deal with and will be used here. Genus is due to Krasnoselski [K] although we will use an equivalent definition due to Coffman [Col] (see also [CF]). Let E be a real Banach space and let E denote the family of sets AcE \ {O} such that A is closed in E and symmetric with respect to 0, i.e. x E A implies -x EA. For A E E, define the genus of A to be n (denoted by ,(A) = n) if there is a map 'P E C(A, Rn \ {O}) and n is the smallest integer with this property . . When there does not exist a finite such n, set ,(A) = 00. Finally set ,(0) = O. EXAMPLE 7.2. Suppose BeE is closed and B n (-B) 0. Let A = BU(-B). Then ,(A) = 1 since the function 'P(x) 1 for x E Band 'P(x) =-1 for x E -B is odd and lies in C(A,R \ {O}). REMARK 7.3. If A E E and ,(A) > 1, then A contains infinitely many distinct points, for if A were finite we could write A = B U (- B) with B as in Example 7.2. But then ,(A) = 1. 45
IT\'CTI()\'.\L:; \\ITll Sy\!:-IETHIES A\,D I\,DEX THEOHlES
46
.,'n
EXA\1PLE ,.j If n ? I iind A is hOlll('oillorphic to • by an odd map. then > 1. Otlwrwisr therr i, a mapping; E ('IA. R {Of) 'xitil ; odd. Choose 'ln~' E A sllch that Plr) > O. Then -.ri < 0 and hy th(' Interlllcdiate \'alll(, Theorelll. ; lllust \'i\nish ,olTll'\\'hrre on ,1m' path in A joining .r and ".1'. i\ contradiction. ~.(A)
The main proper! ies of grnus wrll he listed in thl' nC'xt pl'Oposit ion. For A (- (; and ~ > O. let So(A) denote a uniform 6-neighborhoorJ of A. i.e. X,,(A) ~ {r E £1 IT - A <:;,s}. PHOI'OSITIO\' 7 ..'J.
Let A. B
E [.
Then
Ie Normal1zn.tion: If:r f O. !({:r} 0 { -.7}) '" 1. 2c .Hopping property: If there PI?"t.' an odd mop f
(=
C( A. Il). then') (A)
:S
-.( Il). :]0 MonotOrt1Clty property: If A c n. !(A) < :(n) . .12 Sl1badditivity: :(A U B) <:; -((A) ~ :(13). 5° Continuity property: If A 18 comport. -,(A) < x ond there is !!!l > 0 s11ch that Nh(A) E (; ond (A)) == -,(A).
PROOF. lOis a special case of Example 7.2. To prove 2°5° we assulTle that :(A).';(B) <x: the remaining ('Clses are trivial. For 2°, suppose -I(B) = n. Then there exists a function 'P belonging to C(B. R" \ {O}). Consequently 'P 0 J is odd and in C(A.Rn \ {O}). Therefore -/(..1) :S n '" ,(B). Choosing f = id in 2° !'ields 30. For 4°. suppose IrA) = III and '/(B) '" n. Then there exist ll1appiniis p E CiA. Rrn \. {O}) and L' t= G(B. R" \ {O}). both odd. By the Tietze Extension Theorem. there are mappings'; t= Rm) and 0 E C(E. Rn) such that';I.\ = P and
A. A c I T x , for some finite set of points Xl,"" Xk· Therefore i(A) < 00 via 40. If '" n. there is a mapping P E C( A R n \ {O}) with 'P odd. Extend 'P to an odd function'; as in 4°. Since A is compact, there is a {; > 0 such that .; f 0 OIl.Yo Therefore -f(.Y6 (A)) <:; n ,(A). But by 3°, <:; (A)l so we have equality. RDIARK i.B. For later arguments it is useful to observe that if < x, 2> -1(..1) - :(B). Indeed A C B so the inequality follows from 3°-4° of Proposition 7.5. :"iext we will caiculate the genus of an important dass of sets. PROPOSITION 7.7. If AcE. 0 i., a bounded neighborhood of 0 in R k, and there exists a mapping h E CIA, aO) with h an odd homeomorphism, then ,(A) '" k.
FCc.:CTIONALS WITH
SY~njETR.IES
Ac.:l) INDEX THEOhII'>17
PROOF. Plainl), :S k. If = } < k. there is a,:; E Ri {O}) with,:; odd. Then 'P 0 h 1 is odd and belongs 1.0 C(ao. RI \ {O}). nut this is contra,'y to the Dorsuk-Glam Theorelll [82] since k > j. Therefore co {~. The next result illustrates how s:i1ll111etry provides us with a tool for obtaining int.ersection theorems.
PROPOSfTfOC'l 7.8 [el!. Let X /;e (L A E [ with ,(A) > k. Then A n X # 0.
ol1bsJ!ILU:
01 1'; 01
wrizlII.c7I.mJrl
k and
PROOF. Writing E V 1) X with V a k rlimensional complcll]('nt of X, let P denote the projector of E onto V. If AnX = 0, ]' E C(A. V \ {O}). Moreover P is odd. Hence by 2° of Proposition 7.0, ,(A) :S ,(PAl. The radial projection of P A into aE 1 n V is another continuous odd map. Hence ,( A) :<: ,(a B J n V) k via Proposition 7.7, contrary to hypot.hesis. As was mentioned earlier, there arc other ways in which to measure the size of symmet.ric sets, not. only in a Z'2 setting but for morc general group actions. Suppose E is a real J3anach space with a group 9 acting on it. Set.
Fix 9 == {u E EI!lU = u for all q
9}·
E.g. for 9 {id, -id}, Fix9 = {O} while for the Hamiltonian (1.7), Fix 9 consists of the set of 2n tuples of const.ant vectors. Let [ ocnotc the family of invariant subsets of E \ {O}, i.e. A E [ if fJX E A for all (I E 9 ano x E A. We say we have an index theory for 9) if there is a mapping i: [ -+ N U {oo} such that for all A, B E [, 1° Normalization: If x ri Fixy, i(U gE9 !11') = I. 2° Mapping property: If I erA, D) and I is cCjuivariant. i.e. 19 =!il for all 9 E 9, thcn irA) :S i(E). 3° Monotonicity property: If A c B,i(A) :S 7(D). 4° 8ubadditlvity: itA U E) :Si(A) -I- i(D). 5° Continuity property: If A is compact and A II Fix 9 = 2, then < 00 and there is as> 0 such that No(A) E [ and i(No(A)) = itA). REMARK 7.9. If A E [and AnFix9 # 2, then irA) = sUPBEE l(B). Indeed let x E A n Fix::; and define I: A -+ {x} via I (u) = x for all 17 E A. This map is continuous and eCjuivariant. Hence i( A) ::; i ({x}) ::; i( A) via the mapping and monotonicity properties of i. But A can be replaced by for e.g. r < ~ I!xll and any B E [ lies in such a set. The genus, ,. provides us with a simple indcx theory where::; {id. As to other index theories, Benc; has int.roduced an 8 1 versioll of genus [Be]. Cohomological index theories can be found e.g. in iY, FRI-2, FRRI and the references cited there. E.g. an especially general such situation is contained in Fadell-Husseini [FRj. One can also find analogues of Propositions 7.7-7.8 for these theories. In t.he next two chapters we shall see how these tools can be used to obt.ain multiplicity results for symmetric functionals.
8. Multiple Critical Points of Symmetric Functionals: Problems with Constraints
The following two chapters study the existence of multiple critical points of fllnctionals possessing a Z2 symmetry. The same ideas together with an appropriate index theory can be used to treat other kinds of symmetries. SeC' e.g. :AZ, Be2-3, BF, Bg, Brl--2. Pal-2, Sl-2]. Theorem 1.10, the classical ]'('sult of Ljusternik and Schnirelmann ILLS] is for a constmined funct.ional, i.e. i! functional on a manifold. In this chapter we will prove that result. as well as an infinite climensional generalization and give an application to (2.31). Chapter 9 treats unconstrained functionals. For either type of problem, the main clifficuJty is to find an appropriate class of sets with respect to which one can minimax the functional. We will see how this can be done in the constrained setting first, following Ljusternik and Schnirelmann. For convenience we restate their result: THEOREM 8.1. Suppose I E C 1(RTI. R) and is even. Then
1
possesses
at least n distinct pairs of critical points. PROOF. For E = Rn and 1 ~ )
(8.2)
"iJ
~
n, define
= {A E ciA c 5 n -
1
and ,(A) :-,> j}.
This family of sets possesses the following properties: 1c
2° (8.3)
3° 4°
¥ 0. 1 ~ ] ~ n. Alonotonicdy property: ~!l :) ~!2 :) ... :) In' Invariance property: Suppose p E C( 5 n - 1 , 5 n - 1 ) and is odd. Then 'P' -, IJ' i.e. E IJ wheneyer A E Excision property: If A. E ~;J and BE C with I(B) ~ s < ], I]
then Indeed ro follows from Proposition 7.7 with D = sn-l. 2° is trivial, 3° is a consequence of the mapping property of Proposition 7.5, and 4° follows from Remark 7.6. Define
(8.4)
c]
=
inf maxI(u),
AE~I)
uEA
48
1~
J ~ n.
PROBLEMS WITH COl\'STRAINTS From the monotonicity property of the 'J' it is clear that Cl :s: C2 :S:. .:S: Cn . 1 will show that cJ is a critical value of Ilsn-l. This fact in itself is not sufficient prove Theorem S.l since some of the minimax values may coincide with only c corresponding critical point. The following proposition together with Rem, 7.3 shows the Cj'S are critical values and we get enough corresponding criti points. :\ote that 1 (lL) = 1'(lL) AlL, where A = (1'(11),1/). PROPOSITION S. 5. If CJ = ... = CJ + P == c, and and = O}, then ,(ke ) 2: p T 1.
kc
=
{,y E
sn-l
PROOF. Suppose that ,(ke ) :s: p, Then by 5° of Proposition 7,5, then a [; > 0 such that ,(Nt(kc )) :s: p, Hence if .N == No(k e ) n sn-l, by 3° Proposition 7.5, ,U-:V) :s: p. Invoking Theorem A.4 and Remark A.17(iv) w 0== intJV and E = 1, there is all c: E (0,1) and TI E C([O, 1] x sn-l,sn-l) W II(t, u) odd in 11 and satisfying (S.C))
TI(l.
c
\
Chom;c A E
By 4° 01' (S.3), A \ /./ E' ') imel E I)' Therefore by (S.C)) and the definition of c,
I :s: c
c:s:
a contradiction. REMARKS S,7. (i) The minimax values rical characterization, namely (S.S)
CJ
c)
c,
can be given another more geon
= inf{r E Rh(k) 2:]}.
Thus the c 7 's are just those numbers at which the sets A. r change genus. Ind an denote the right-hand side of (S.S) by cJ . If r > c]' ,(k) 2:.7 so Ar E
'J
(S.9)
CJ
:s: max 1= r. A,
~ Then there is an A E Thus (S.9) shows cJ :s: cJ . If c] < cJ ' let c = such that maxA I :s: c. Therefore > 2: j by 3° of (8.3) but c < (' contradiction. (ii) There are other ways to obtain critical values of lis define
bk = sup min I(u),
1 :s: k:S: n,
AE'Ik uEA
Clearly b1 2: b2 2: .. 2: bn and using Theorem A.4 and Remark A.17(iv) s11 the bk's are also critical values of Ilsn-l. Note that Cl = mins"-' I sill! x E sn-l, {x} U { -x} E ,1, :V10reover c 1 bn . To prove this, it suffices to s that = {sn-1}. If not, there is a set A E In such that A # S'1-l Then there is a point y E sn-l \ A. Without loss of generality we can assume y = ( with 6 the origin in R n-1 The projection map P( 11) = (11 1,. ., Un-l, 0) bell to C(A, Rn-1 \ {O}) and is odd. Therefore by the argument of Proposition
'71
PEOBLE\lS \\TfH CO';STHAI';TS
50
~J(A) 'S 11 -- I. a contradiction. Thus Cj = hn · Similarly en = III = maXSn-l f. \Ve do not kno\\' if r J = Ii" -).1 if.J ~ 1. n. However if one used the cohomological index theory of iFRlj instead of genus and defilled corresponding minima:.:: and maximin values it call be shown that = b;, I' Thus ill this sense the cohomological index is a nicer tool to deal with. There are manv infinite dimensional generalizatiolls of Theorem 8.1. E.g.
r;. b;,
r;
THEOHE\l 8,10. Let E be an dImensional Hilbert space and lei f E Cl(E,R) be even Suppose r > 0, f\iJJJ, satisjies (P8), and fliw, 18 bounded from below. Then f,)B, possesses lnjimtely many distmet pairs of entieal points. PHOOF. Define the sets J) iL'i in (8.2) for J E N with 8,,--1 replaced by aE T • These sets still satisfy properties 1CAe (again with 8,,--1 replaced by anT)' !\ow define
(8.11 )
= inf sup
CJ
[(11),
.JE N,
AET] l1EA
Since JjiJlJ, is hounded frolllblll()w, 1:1 / - J(:'. :Vlorccwer (PS) illlplies fe· {11 C anrlf(u) = c and f!~3B,(U) = O} is a compact set for any c E R. With these observatiom3 and Remark A,17(iv), the argument of Proposition 8.5 proves the theorem as earlier. REl\IAHK 8.12. The requirement that fia1J e satisfies (PS) is too stringent a condition for applications. Consider e.g. -
~11
Ap(X.11).
(813) =
11
O.
X
:r E
n.
an,
n as usual. Suppose that p satisfies (pd--(p2) and (ps) ~p(x. 0> 0 if Ef 0, and (pg) p(x,~) is odd in E 2 Let E == \Vci (O) and 11 E E, Set
with
(8,14) Then f E C 1 flaB, we have
flu) == - f P(x, 11) dx. In R) by Proposition B.10 and f is even, At a critical point
(8.
=
for all y E E. Choosing:p =
11
0=
-
f p(x,u)'Pdx
io
11
of
pfYll'Y'Pdx
in
and using (Ps) shows
p=I'(ll)U=- (p(x,u)udx
io
Therefore 11 is a weak solution of (8.13) with A = _ p - I This suggests using Theorem 8.10 to get weak solutions of (8.13) on aB 1 (or aE T ). To apply Theorem 8,8, observe first that by Proposition B,lO, f is weakly continuous, i.e. Urn ~ u implies f(u m ) ~ f(u), This implies that fl aB, is
PH013LE'vl;; WITH
CO:-;;-;THAI\lT~
a
bounded from below for otherwise t here is it sequence (IL", ) C 13 1 such tha l(u",) < -nl. But, being bounded, (u.mJ has a subsequence com'erging weakl ) ~ l(u) = -oc contrary to 1 E el(E, R) in E to U B~ Hence To ycrify (PS), let (11m) be any sequcnce in aB l such that (l(u m )) is bounde· and (11m) -~ 0 as m ~ x, i.e. ) = 1'(11",)- (T'(l1rn)lI m
(8.16)
Since
.
4
0
is bounded in E and I' is COlnpact (Proposit.ioll 13.10), along sortl
subsequence
Uut
converges weakly to
SOIlH; II
E [~ a]](1
1'(l1 m ) ~ I'(11) - (I'(l1)U.)l1 =
o.
By the weak continuity of f, 1(11m) -~ 1(11). If 1(11) f 0, by (Ps), 11 f rrL. Consequently (8,16) shows
()
Henc
1'(11)11 f () and 1'(l1m )l1 m f 0 for large
)) possesses it conycrgcnt subseqnence. However if [(v.) 0, II", Ileed not hav a convergent subsequence, In fact by the above argument any sequence which is weakly convergent to 0 has f(l1m) ~ 0 and fi~Bl ., 0 but does not necessarily have a convergent subsequence. What we have actually shown above is that flaB I satisfies (PS)lor for all c 'F where (PS)loc is defined in Remark A,17(i), Moreover (PS)loc at c is all we nee for the proof of the Deformation Theorem, Thus Theorem 8,10 is also valid wit (PS) replaced by (PS)loc for each c] defined by (8,11), To apply this result t (8.13), by the remarks just made, we need only verify that cJ < 0 for all j. Bu this is immediate from (p~) and (8,11), Thus we have proved THEOREM 8,17. 1fp satisfies (pd,(P2),(PS), and (pg), (8,13) possesses sequence of distinct pairs of weak solutions (Ak' on R x aB r . where Ak '
-(I'(l1k)l1k)-l COROLLARY 8,18, ck=f(uk)->Oask->x, PROOF, Let VI", . ,Vk be as in (2.40) and set Ek span{ v], .. , ,vd an Et its orthogonal complement, By Proposition 7,8, if A E ;k, A If;; Therefore
Consequently (8.19)
(8,20)
52
PROBLEMS WITH CO!,;STRAINTS
By the Gagliardo-Nirenberg inequality (8.21)
Ilull~(A) :::;
(}8
f ( in
[NI], or [Fr],
Ivul 2
)a/2(f
in u2
)(I-a)/2
where a E (0,1) is defined by
1
s+ If u E Et-l'
(8.22)
i
u 2 dx :::; Ak-1
i
Ivul 2 dx
as in (4.13). Combining (8.20)-(8.22) shows
(8.23)
II(u)1 :::; a4(A;(I-a)(s+l)
+ Ak 1/ 2)
for u E Et_l (laB 1 . As is well known [CHI, Ak ~ 00 as k (8.19), (8.20), and (8.23) show I(Uk) -40 as k -400,
-4
00. Hence (8.11),
9. Multiple Critical Points of Symmetric Functionals: The Unconstrained Case
This section contains two abstract results: Clark's Theorem and a symmetric version of the Mountain Pass Theorem as well as some applications of these results. We begin with the following version of Clark's Theorem. THEOREM 9.1 [C). Let E be a real Banach space, IE Cl(E,R) with 1 even, bounded from below, and satisfying (PS). Suppose 1(0) = 0, there is a se: K c E such that K is homeomorphic to SJ~l by an odd map, and SUPK 1< 0 Then I possesses at least J distinct pairs of critical points. PROOF. We argue almost exactly as in Theorem 8.1. Let
and define (9.2)
Ck
=
inf sup I(u),
lSkS).
AE,k uEA
The sets Ik satisfy properties 1°~4° of (8.3) with sn~l replaced by E. HencE C2 S ... :s; cJ . Moreover Cl > -00 since I is bounded from below and cJ < 0 since ,(K) = J via Proposition 7.7 and 11K < O. The result now follow, from:
CIS
PROPOSITION 9.3. If Ck = ... = J'(u) = O}, then I(K c)?> p+ 1.
Ck+J
C
and Kc == {u E EII(u) =
C
anc
PROOF. The proof follows almost the same lines as that of Proposition 8.3 and will be omitted. (Use must be made of the fact that 1(0) = 0 and c, < 0 fOJ 1 SiS k. Therefore 0 rf- KCi so K c, E [ and I(Kc,} < 00 via (PS).) REMARK 9.4. Actually in [C), Clark does not assume that I is bounded frorr below or that there is a K as above but merely that if Ck is defined as in (9.2) then -(X) < Ck < O. The above proof then shows Ck is a critical value of I The form of the special case given in Theorem 9.2 is perhaps more useful fOJ applications. Two such applications will be given next (see e.g. [AR, He2, and R2]). The) are related to problems treated in Chapter 2, namely (2.31) and (2.39) .. QC(nsidm 53
THE (':\(,O:\STH.-'c!:\ED CASE
54
first -
(9.5)
~11 = 11
=
A(a(.r)lJ - plIo u)).
0.
e!fl.
.r
where \2 is a.s usual and p satisfies (P'1)' (P3). (p~) there i" a (l > 0 such that a(.r)(j - p(x. 6) ::; 0 for all x E \2. and (P9).
9. G. Suppose p (p'j). (P:3)' (Ps), (pn), and A> Ak, the kth of (2.40). Then (fl5) possesses at lenst k lilstmet of nontrivial
THEORE,-,!
solutwns. PHOOF. We begin by modifying the problem in a familiar fashion,. For x E \1. set q(I, 0 = a(x)( p(x.O if (E [0. (11. q(x. 0 = q(I. Ed if ( > (l, and let If be odd in (. Consider -
(9.7)
~11
u
= Aq(X, u), x E \2. x E ao. = o.
Arguing ,~s in the proof of Corollary 2.23 shows any solution of (9.7) is a solution of (9.5). Hence to prove Theorem 9.G. it suffices to produce at least k distinct pairs of critical points of
1n[~I\}uI2 -
I(u)
AQ(x.u)ldx,
where Q is the primitive of q and E = WOI2(O) as usual. Since q satisfies (p~) and is a bounded function, I E C l (E, R) via Proposition B.lO. Likewise I is bounded from below and (PS) holds as in the proof of Theorem 2.32. Clearly 1(0) = 0 and I is even. Thus the proof of Theorem 9.G follows from Theorem 9.1 once the existence of a set K as in that theorem has been established. Let Vl, ...• Vk denote the eigenfunctions of (2.40) corresponding to AI, . . , Ak normalized so that
I = 1=
A,
In v~ dx,
1 ::; i ::; k.
Set
(9.8) It is clear that K is homeomorphic to sn-1 by an odd map for any r > O. We claim < 0 if r is sufficiently small. Indeed for small T and u E K, 11[( x) i ::; E1 so Q(u) au 2/2 P(x,u). Therefore by (P3), 1
I(u) =
r - 2'A L }o1J'fdx
k
k
2' L 1=1
1 =-L 2 k
1=1
o(r2)
i=l (
A)
1-~ A,
for small r. Since A > Ab I(u) < 0 and the proof is complete.
THE C"ICONSTHAI"IED CASE
As a second application of Theorem 9.6. we CXiillliIlC agclill the proiJlcIn t real in (2.33) and Theorem 2.32: c,u
(9.9)
= Ap(U),
u = 0,
x EO,
xE
a~.
THEOREM 9.10. Suppose p satisfies (pit), (P3), (P5), and (pg). Then for a J E N, there exists a >:J > 0 such that if A > >:J' (9,lD) possesses at leasl
distinct pairs of solutions. PROOF. As in the proof of Theorem 2.32. it suffices to show that the modifi functional
(9.11) has the appropriate number of critical points. This in turn will follow frc Theorem 9.l. In the proof of Theorem 2.32, it was already established th hE CI(E,R), is bounded from below, satisfies (PS), and IA(O) = O. CleaJ (pg) implies h is even. Thus we need only verify that for any J E N, there a >:) > 0 such that for each A > >:), there is a set K as in Theorem 9,1. Let be as in (9.8). For sufficiently small r, F(u(x)) = P(u(x)) for all u E K and (ps), P(u(x)) > 0 if u(x) f 0, Therefore inf "EK
inr P(u(x)) dx =a
> O.
Choose>: = a- 1r2 Then IAIK::; ~r2 + o(r2) < 0 for small r. The proof complete. Now we turn to a Z2 version of the Mountain Pass Theorem. Such a rest was stated in Chapter l. A more general version will be given here. THEOREM 9.12. Let E be an infinite dimensional Banach space and, IE CI(E, R) be even, satisfy (PS), and 1(0) = O. If E = V Ell X, where V finite dimensional, and I satisfies (I;J there are constants p, a > 0 such that IIDB p 0x ? a, and (12) for each finite dimensional subspace E c E, there is an R = R(E) su that I::; 0 on E \ BR(E)' then I possesses an unbounded sequence of critical values. REMARK 9.13, All of our previous symmetric results have used roughly tJ same class of sets: ik = {A E [h( A) ? k} (or the corresponding class on B to construct critical values. These sets do not suffice for the setting of Theore 9.13. To see why, as a model case consider a functional of the form
a
(9.14)
s
+1
J) dx,
THE lJ;\CO,-;STRAI"ED CASE
56
where 1 < s < (n + 2) (n ). It will be seen later in this chapter that Theorem 9.12 applies to such a functional. Since A E can be chosen to be an arbitrarily large sphere in span{v).. with Vi as in (2.40). (9.14) shows
'k
I(u)
inf
(9.15)
=
-x.
AEik
On the other hand choosing
where] = J(r) is large enough. and using arguments as in Corollary 8.18 we sec minuEA I(u) :::: ~r2 Hence sup minI(u) =
00.
AE""1k uEA
Thus a new family of sets must be produced t.o prove Theorem 9.12. PROOF OF THEOREM 9.12. A sequence of families of sets f m will be introduced and a corresponding sequence (em) of critical values of I will be obtained by taking a minimax of I over each I'm. A separate argument then shows is unbounded. Suppose V' is k dimensional and l' = span{ e) .. .. ,ek}. For m :::: k, inductively choose em+ 1 rt span{e), .... em} == Em· Set Rm == R(Eml and Dm == BRm ",Em· Let
E)ih is odd ar:d h = id on BB Rm
(9.16) ;';Ot8 that id E
em
for all mEN so
Ih
(9.17)
E
em of 2.
Set
em, m :::: J, Y
E
The following proposition shows that the sets f
E, and fry) :; m - J}. J
satisfy conditions like (8.3).
The sets f) possess the following properties: Z for all} E N. 2° (Monotonicity) rJ~l C f J . 3° (Invarwnce) If p E G(E.E) is odd, andp = id on BBRm " Em for all :::: J, then p: r J ---> r J . 4° If BE rj , Z E E, and ,(Z) :; s < J, then B \ Z E r J - 8 · PHOPOSITIO:O: 9.18.
1"
m
r J of
PROOF. Since id E em for all mEN, it follows that r J f/: 0 for all J' E N. If B = h(Dm\Y) E rJ~l. then m :.:: j + 1 :::: J, hE Y E E, and -1(Y) :; m + 1) :; m - J. Therefore B E f J . !'Iext to prove 3°, suppose B =, E f J and p is as above. Then po h is odd, belongs to C( Dm, E), and yO h = id on BBR~ n Em. Therefore yO h E Gm and po h(Dm \ Y) p(E) ~ r). Lastly to get 4°, let B = h(Dm \ Y) E r J and Z E E with ,( Z) :; .) < J \Yc claim
em,
(9.19)
B\Z
THE UNCONSTRAINED CASE Assuming (9.19), note that since h is odd and continuous and Z E [, h- 1 (Z) E [ Therefore Y U h- 1 (Z) E [ and by 4° and 2° of Proposition 7.5,
,(Y U h- 1 (Z)) ::: ,(Y) + ,(h-1(Z)) ::: ,(Y)
+ ,(Z)
:::m-J+s=m-(J-s). Hence B \ Z E r)-s' To prove (9.19), suppose b E h(Dm \(YUh-1(Z))). Then b E h(Dm \Y)\Z B \ Z c B \ Z. Therefore
c
h(Dm \ (Y U h-l(Z))) c B \ Z.
(9.20)
On the other hand if b E B \ Z, then b = h( w) where wE Dm \ Y \ h-1(Z)
c Dm \ (Y U h-l(Z)).
Thus (9.21)
B \ Z
c h(Dm \ (Y U h-l(Z))).
Comparing (9.20)-(9.21) yields (9.19) since h is continuous. Now a consequence of minimax values of I can be defined. Set (9.22)
JE N.
c) = inf maxI(u), BEf, uEB
It will soon be seen that if J > k = dim V, Cj is a critical value of I. The following intersection theorem is needed to provide a key estimate. PROPOSITION 9.23. IJJ>k andBEr J , then (9.24)
BnaBp nX
i= 0.
PROOF. Set B = h(Dm \ Y) where m ~ j and ,(Y) ::: m - J. Let 6 = {x E Dmlh(x) E Bp}. Since h is odd, 0 E 6. Let 0 denote the component of 6 containing O. Since Dm is bounded, 0 is a symmetric (with respect to 0) bounded neighborhood of 0 in Em. Therefore by Proposition 7.7, ,taO) m. We claim
h(aO)
(9.25)
c aB p .
Assuming (9.25) for the moment, set W == {x E Dmlh(x) E aBp}. ThereforE (9.25) implies W :::J ao. Hence by 3° of Proposition 7.5, ,(W) m and by Remark 7.6, ,(W \ Y) ~ m (m - J) J' > k. Thus by 2° of Proposition 7.5, ,(h(W \ Y)) > k. Since co dim X k, h(W \ Y) n X i= (2) by Proposition 7.8. But h(W \ Y) c (B n aBp). Consequently (9.24) holds. It remains to prove (9.25). Note first that by the choice of Hm, (9.26)
Since m > k, aBp (9.27)
n X n Em i=
2. Hence by (1;),
THE UNCO:\STRAI:".'ED CASE
,'i8
Comparing (9,26) and (9,27) shows
Rm >
p, :"ow to verify (9,25), suppose
x E 00 and E B p ' If x E Dm there is a neighborhood /If of x such that hiS) c B p ' But then x (j. 00, Thus x E oDm (with 0 relative to Em), But on oDm' h = id, Consequently if x E oDm and h(x) E B p , ilh(x)l! = = Rm < p contrary' to what we just proved, Thus (9,25) must hold, R E\lARK 9.28. A closer inspection of the above proof shows that
2:) - k, COHOLLIllY CJ
29
JI»k'('l
rx>O
PROOF, If) > k and BE f l , by (9,24) and (1;), maxuEB I(u) 2: 0:, Therefore by (9,22). c) 2: 0:, The next proposition both shows cJ is a critical value of I for J > k and makes an appropriate multipicity statement about degenerate critical values. PROPOSITION 9,30. II)
> k, and c) = ' . , = cJ+p ==
c, then ,(Ke)
2: p + 1.
PROOF, Since 1(0) = 0 while c 2: 0 > 0 via Corollary 9.29, 0 (j. Ke. Therefore Kc E [ and by (PS), Kc is compact. If ,(Ke) <::; p, by 5° of Proposition 7.5, there is a b > 0 such that ,(NIi(Ke )) <::; p, Invoking the Deformation Theorem with 0 = .YE(Ke) and '[ = 0:/2, there is an E E (0, '[) and T) E G([O, 1] x E, E) such that T)(1..) is odd and (9.31) Choose BE
such that max J (u) <::; uEB
(9,32)
C
+E
By 4° of Proposition 9,18, B \ () E f), The definition of Rm shows J(u) <::; 0 for u E OBRm n Em for any mEN. Hence 2° of Theorem A.4 and our choice of E' imply rl(l, ,) id on OBRm nEm for each mEN, Consequentlyr) (1, B \ ()) E r J by 3° of Proposition 9,18, The definition of c) and (9,31)-(9,32) then imply max J <::; ry(I,B\Oj
C-
E,
a contradiction" The next proposition completes the proof of Theorem 9,12. PROPOSITIO?\ 9,33. c)
~ 00
as
J -+
00,
PROOF, By 2° of Proposition 9.18 and (9.22), C)+I 2: c). Suppose the sequence (c)) is bounded. Then c) ~ C < 00 as J -+ 00. If c) = c for all large j, Proposition 9,30 implies ,(Ke) = 00, But by (PS), Ke is compact so ,(Ke) < 00 via 5° of Proposition 7,5, Thus c > c) for all j E N, Set
K == {u E Eh+1
<::;
J(u)
<::; c and
I'(u)
O},
By (PS) again, K is compact and 5° of Proposition 7.5 implies ,(K) < 00 and there is a 0 > 0 such that ,(N8(K)) = ,(K) == q, Let s max(q, k + 1). The
THE
U~CONSTRAIl\ED
CASE
Deformation Theorem with c = C, E = C ~ cs , and 0 = N5(K) yields an as usual such that
1/(1, Ac_€
(9.34) Choose
JEN
such that c] > C ~
E
\
0)
C
and B E
,59 E
and 1/
Ac- £ '
r
such that
maxl
(9,35)
Arguing as in the proof of Proposition 9.30 shows is in f) as is 1/( 1, B \ 0) provided that 1/(1,,) = id on DB R ", r.Em for all m ,7 But J SOon aB n", nErn for all mEN while c ~ E C s 2: Ck+l 2: a> 0 via Corollary 9.29 Consequently 1/(1, B \ 0) E f J and by (9.34) (9.35) and the choice of c]'
CJ:c:;
m~
I :c:;
c ~ [ < c],
~(l,B\O)
a contradiction, The proof is complete. REMARKS 9.36. (i) If E is finite dimensional, the result of Theorem 9.12 also obtains with the conclusion being that I possesses at least dim X critica points, (ii) There are analogues of Theorem 9.12 when V is infinite dimensional and when we have an SI rather than a Z2 action (see e.g. !FHRj) , Such a result leads to the generalization of Theorem 6,10 mentioned in Remark 6.45. As an application of Theorem 9.12, consider
c.u = p(x, u),
(9,37)
u = 0,
x E {1
x E 30.
THEOREM 9,38. Suppose p satisfies (pd, (P2), (P4), and (pg). Then (9.37)
possesses an unbounded sequence of weak solutions. PROOF, With E = W6,2(0) as usual and
l(u) =
fo[~lvuI2 ~ P(x,u)] dx,
the proof of Theorem 2,15 shows I E Cl(E, R), satisfies (PS), and 1(0) = 0, Moreover the argument in Theorem 2,15 that showed I satisfied (h) equally well yields (I2)' Clearly (pg) implies I is even. Assume for now that I also satisfies (Ii), Then Theorem 9,12 implies that I possesses an unbounded sequence of critical values Ck = I(Uk), where Uk is a weak solution of (9.37). Since I'(Uk)Uk = 0,
(9,39) and it follows that
(9.40)
Ck
=
fo[~P(X,Uk)Uk ~ P(x,uk)]dx -> 00
and k -> 00. Hence by (9,39)·-(9.40) and (P4), (Uk) must be unbounded in E and in LOO(O).
THE C:\CONSTRAINED CASE
60
To verify (I;). choose V = span {Vj, ... ,vd where the functions v) are as in (2.40) and X = V ~. By (P2). for all u EO E.
(9.41)
flu) ::::
By (8.21)(8.22), if u
EO
I, ~
dx
as
in 111ls~l
dx -
a6
BEp n X,
1) - as.
(9.42)
Choose p = p(k) so that the coefficient of p2 in (9.42) is
±
Therefore
(9.43) for 11 E BEp X. Since Ak ~x as k that p2 > 2as. Consequently
i
~
x. p(k)
~
x as k
~ 00.
Choose k so
(9.44) and (I;) holds. As a final example of the ideas used in the proof of Theorem 9.12, consider (9.9) again. In Theorem 9.lD, we proved under appropriate hypotheses on P if A > ),k, (9.9) possesses at least k distinct pairs of weak solutions and these functions correspond to negative critical values of (9.11). Let K be as in (9.8), i.e. K = BET n E k · By our choice of rand ),k, I < 0 on K. Setting Rm = r and Dm = Er n Em, define G m for 1 ::; m ::; k as in (9.16) and define r) as in (9.17) with the further proviso that m ::; k. Proposition 9.18 then holds for j EO [1, k] and m ::; k as does Proposition 9.23. Defining Cj via (9.22), observing that 7)(1..J = id on BET n E k, and noting that Corollary 9.29 holds with k O. the proof of Proposition 9.30 works equally well in this setting. Thus we have proved:
THEORE:,I 9.45 [AR]. Under the hypotheses of Theorem 9.10, for all A> ),k, (9.9) possesses at least 2k distinct pairs of nontrivial solutions, k pairs cor-
responding to negative and k pairs to positive critical values of (9.11).
10. Perturbations from Symmetry
In the last two chapters several examples have been given of the existence of multiple critical points for functionals invariant under a group of symmetries. A natural question to ask is: What happens when such a functional is subjected to a perturbation which destroys the symmetry? Some special cases of this question have been studied and while progress has been made, there are not yet satisfactory general answers. This chapters treats a perturbation question in the setting of Theorem 9.38. Thus consider
- 611. = p(x, 11.),
(10.1)
11. = 0,
xE
x EO,
ao,
where p satisfies (Pl), (P2), (P4), and (pg). By Theorem 9.38 the corresponding functional has an unbounded sequence of critical values and (10.1) has an unbounded sequence of weak solutions. Suppose f E L2(0) and (10.1) is replaced by
611. = p(x, 11.)
(10.2)
11. = 0,
+ f(x), ao.
x EO,
xE
The corresponding functional is
(10.3)
J(11.)
and J is not even if
f
fo[~IV11.12-P(x'11.)-f(X)11.]dX
=f= 0. Nevertheless we have:
L 2(0), then (10.2) possesses an unbounded seq11.ence of weak solutions provided that s in (P2) is further restricted by THEOREM 10.4
(10.5) REMARKS
(10.7)
[RB]. lfp satisfies (pd, (P2), (P4), and (pg) and f
= (n+2)-(n-2)s
/3 _
( ) ns-1
>
E
_/1_
/1-1'
10.6. (il Inequality (10.5) is equivalent to /1n s < 1m
+ (/1 - l)(n + 2) + (/1 - l)(n - 2)'
It is easily checked that if s satisfies (10.7), then 8 < (n + 2)(n 2)--] Also, observing that s = 1 and /1 = 2 satisfies (10.7) shows that (10.5) is nonVaC\lOllS. 61
PERTT:RBATIO:\S FH()\l SY\I\IETRY
62
Iii) Slightly Ie,s general versions of Theorem 10.4 were proved by Bahri and Berestycki :BB. and b:.' Struwc lSI. See also Dong and Li :DL]. Their arguments differ fWIIl the one giv'cn herf'. which is somewhat in the spirit of the s:;nHnetric ~lountain Pil.SS Theorem. For the proof we requirE' an estimate on the deviation from symmetry of I of the form
( 10.8) for 11 E E. Cnfortunately I does not satisty (10.8): however it can be modified in snch a [,t'ihioll that the new functional, J. satisfies (10.8) and critical values and points of J are critical value'S and points of I. To motivate the modified problem, a priori bounds for critical points of I in terms of the corresponding critical values will be obtained. ;'\ote first that by (P4) there are constants a4, a.o > 0 such that
o 2: a51~!!1 ~ a4
(10.9)
for all ~ E R. Therefore there is a constant 1 (10.10) ~(~p(x. 0 + (/3) 2: PIx. 11. for all ~ E R. PROPOSITJO:\ 1 (). 11.
wnstant A depending on
> 0 such that
0 + 01 2:
(151~!!1
Under the hypotheses 0/ Theorem 10.4, there eXlsts a ::/1:1'(0) such that i/u is a critical point 0/ I,
r (P(x, u) +
110.12)
a3
j[]
0.4)
ciJ:
:S A(I(v)2 + 1)1/2
REMARK 10.13. If u is a critical point of (10.3), then (10.10) and (10.12) easily imply a bound for v in E in terms of I (v). PROOF OF PROPOSITION 10.11. Suppose u is a critical point of I. Then by (10.10) and simple estimates,
I(v)
= I(v) 2:
(10.14)
~ ~I'(v)u =
G~ t) l
l [~UP(X'lL) ~
(up(x, u) + (3) dx
2:
a71
2:
°2711(p(x,v)+a4)dx~a9
(P(x, u)
~/v]
P(x,u)
dx
~ ~ IlfIIU(Il) IlvIIL2(O) ~ 0.6
+ (4) dx ~ aslluIIL"([]) ~ ae
and (10.12) follows immediately from (10.14). To introduce the modified problem, let X E C=(R, R) such that X(O E:S 1, X(O := 0 for E 2: 2, and x'(El E (~2,0) for EE (1,2). Set
Q(v) := 2A(J(vJ2 + 1)1/2 and
'I/;(u) := X ( Q(v)-l
l
(P(x, u)
+ 0.4) dX)
.
1 for
PERTURBAT10'iS
FJW~1
SY\I\IETllY
:';ote that by (10.12), if u is a critical point of 1. the arg1ll11(,llt of \ Ii", ill and therefore 11)(11) = 1. Finally set
-1
[1. 2 J(u)= 0.21vul
(10.15)
ro.
~
•
( IJ dx. Px,u)-v(u)fl1
Then J(u) = 1(11) if u is a critical point of I. The following result contains the main technical properties of .1 which we need. PROPOSITION 10.16. Under the hypotheses of Theorem 10.4: 1° JECl(E,R). 2° There exists a constant (jl depending on I f Ii L' (0) such that
(10.17)
!oralluEE. 3° There is a constant Mo > 0 such that if .1(11) ?: Mo and .1'(11) =,0, then J(u) = I(u) and 1'(u) = o. 40 There is a constant M 1 ?: M 0 such that for any c > M I, .1 satisfies (P S) lac at c. PROOF. Hypotheses (Ptl-(p2) imply I E Cl(E,R). Since X is smooth, the same is true for 1/; and therefore J. To prove 2°, note first that if 11 E su pp 10 (the support of 1/;), then
(10.18) where Ctl depends on and (10.10),
lin
II!IIL'(D). Indeed by the Schwarz and Holder inequalities
!UdXI ::; IlfllL'(o)llullL'(D) ::; Ct21Iulb(0)
(10.19)
::; Ct3
(In
(P(x, u) + a4) dX)
III'
If further u E supp 1/;, (10.20)
In
(P(x, 11) + a4) dx ::; 4A(I(u)2 + 1)1/2::; Ct4(II(u)1 + 1)
so (10.18) follows from (10.19)-(10.20). Now to get (10.17), by (10.15) and (P9),
(10.21)
IJ( u) - J( -u)1 ::; (1/J( 11) + 11)( -u))
lin
fu dxl·
To estimate the right-hand side of (10.21), by (10.18),
(10.22)
1/J(u)
lin
fudxl ::; Ctl1/J(u)(II(uWlfL
By (10.3) and (10.15), (10.23)
II(u)l::; IJ(u)j
lin
fUdxl·
+ 1).
PERTCRBATJO:\S FROy! SYW,lETIW
64
Therefore ii/II
(10.24)
\) ~
1
.
Using Young's inequality, the f term on the right-hand side can be absorbed into the left-hand side yielding
(10.25) Combining (10.25) with a similar estimate for the term gives (10.17). To prove 3°. it suffices to show that if A10 is large and 11 is a critical point of J with J(u) 2' ,\10' then
. \ Q(ui
(10.26)
Ifrp. . \
\.1
(X,llj-a4}CLX
n
The definition of v then implies 1,)(1:) == 1 for l' near 11. Hence 1/(11.) J(u) = J(u), F(u) = J'(u), and 3° follows. We will show that (10.26) holds. l3y the definition of J,
u)
(10.27)
(0(u)
0 so
+ 1/J'(u)u)fu] dx,
where' (llJ.LS)
=
, X, (8(UJ)Q(11)- 2 rlQ(u). /'n up(x, u) dx-- (2A) 2 8(u)J(u)J (u)u]1
and
8(u) = Q- i (u)
J
(P{x.u)
T
(L4)
dx.
Regrouping terms in (1O.27)-(1O.28) yields
dx (10.29)
where
(10.30)(i) and
( 1O.:30)(ii) Consider
(10.31 )
J(u) - ~-----::=--C-7CJ'(u)U.
If = 1 and TI{u) = 0 = 12(11). (10.31) reduces to the left-hand side of (10.14) so (10.26) follows from (10.12). Since 0 'S 1/)(11) 'S 1, if TI(u) and T2(U)
PERTCIWATIO'iS FROM S\';-'!i\!ETHY
65
are both small enough, the calculation made in (10, when carried out for (10,31) leads to (10.12) with A replaced bv a constant which if smaller than 2A, But then (10,26) holds, It therefore suffices to show that T J (11,), T2 (u) ,~lJ as ,\10 ~ x, If u supp I), Tl(U) = lJ = T2(U), Thus we assume 11 E SUppe}, By (1O.3lJ) and (lO,IS),
IT1 (u)l:::
(10,32)
4al(ll(u)11/1l + 1)1[(u)I- 1
We need an estimate relating l(u) and J(u) for u E supp
(10,15),
l(u)
~ J(u)
-Ii
By (10,3) and
Ij),
Judxl,
Thus by (1O.1S),
l(u) -;- a1
(10.33)
I/1
L
~
J(u) -
01
~
1Y10 /2
for Mo large enough, If flu) ::: 0, (10.3:3) implies
a1 /1/ + 11(u)l/p ~
(10.34)
+ IL ~< 1
Mo/2 + 11(11,)1,
(10,34) is not possible if Mo is large enough, e,g" Mo ~ 2a11/-1, which we can assume to be the case, Therefore l(u) > 0, Hence (10.33) implies I(u) ~ Mo/4 or l(u) ~ (Mo/4ad l1 , In any event, l(u)-> 00 as Mo -+ x, which together with (10,32) shows T1(U) -+ 0 as Mo -+ 00, Analogous estimates yield T2(U) -+ 0 as Mo ~ 00 and 3° holds, where
1/-1
= L But
The verification of 4° follows similar lines to 3°, It suffices to show there is an Ml > Mo such that if (1im) C E, M1 ::: J(u m ) ::: J{, and <~ 0, then (u m ) is bounded, For large m and any p > 0,
pllumil + J{
~ J(uml - pJ'(um)u m
=(~
p(1+T1(u m )))llum
+p(1+T2 (u m
(10.35)
))
2 I1
r u"p(x,um)dx- Inr P(x,um)cix
Jll
+ [p(1)0(u m ) + T 1(u m
))
w(u m )]
i
fUm
cix,
For Ml sufficiently large and therefore T 1 , T2 small, we can choose p E (p-1, 2- 1 ) and E > 0 such that
(10,36) Hence by (10.35), (10,36), and (P4), (lO.37)
il,,'+
J{
> ~ - 2
Using the Holder and Young inequalities, (10,37) implies (u m As in Proposition B.35,
D- 1 JI(u m) = (1 + Tdum))u m - Plum),
)
is bounded in E-
PEETL"EBATIOi\S FRO\! SYMMETRY
66
where D: E - E' is the duality map, .D is compact. and !TJ(llm)! S ~ (for ;1,[] large). Hence (PS) holds. On the basis of 3" of Proposition 10.16 and the argument of Theorem 9.38, to prove Theorem 10.4 it suffices to show that J has an unbounded sequence of critical points, This we shall do via a series of steps. The first is to introduce a ) be as in (2.40) (with e,g, sequence of minimax values of J. Let the functions =: 1), let =: span{ t']" , . ,I'}}, and let be the orthogonal complrment of E) in E, Comparing J to the functional in Theorem 9,38, the 1:' term in J does not affect the \'crification of (J~) for J, Therefore there is all R) > 0 such that J(u) S 0 ifll E E) \B R), Set D) =: BR) nE) and G) =' {h
C(D 1 , E)lh is odd and h = id on DB R ,
},
Define
J(h(u)),
(10,38)
j E N,
These minim;tx \'/\lu('s will llot in general be critical v;tlues of J unless f =, 0, However we will WiC them a.s part of a comparison argument to prove that .J has an unbounded sequence of critical values, First we will obtain lower bounds for h} .
P EOPOSITiO\, 10 ':l9, There all k 2' k,
IS
a constant 32 > 0 and kEN such that
fOT
(1040)
where ,3 was defined in (10,5),
1J.'
PROOF. Let h E G k and P < Rk, E h(Dk)·~ cJBp n J' Therefore
(1041)
max J(h(u)) 2' J(w) 2' uEDk
By Proposition 9,23, there exists a
inf.
J(u),
uEBBpnEt_l
To obtain a lower bound for the right-hand side of (10.41), we argue along the same lines as (9.41)-(9.44) of Theorem 9,38 to choose P = Pk satisfying 1
= const
and therefore (10.42) for all u E cJB Pk n Et-l' Since Ak 2' a6k2/n for large k [CHI, (10.42) and our choice of Pk yield (10.40) for large k, To get critical values of J from the sequence (b k ), another set of minimax values must be introduced, Define
I'"
PERTURBATIONS FHO'.1 SY:V!METRY
and
= {H
Ak
E C(Uk,E)IH!D k E fk and H
u E Qk
= id
for
dU
(3BRk~1
"
' BIlk)"
Ek]}'
Set
J(H(u)). Comparing the definition of
Ck
to (10.38) shows
PnOPOSITIO:\ 10.43. As,mme
Ck
:>
bk ·
Ck
> bk :> hi i . For Ii
inf
max J(H(u)).
E (0,
Ck -
od,
define
and
Ck(O)
( 10.44) Then
cd 0)
HEAdo) UEUk
is a critical value of J,
.h(o) implies this set is nonempty, Since Ado) c SUpposeCk(.5) is not a critical vnJucofJ. Self i(Ck--iJk-5)so
PROOF, The definition of
Ak ,c,{.5) 1:'
:>q.
> () and invoke the Dcformat ion Thc:or('1ll obtaining
H E
,'cd b)
and 1; as llslla.l. Choose
such that
J(H(u)) S
(10.45)
Consider T}(l,H(,)). Clearly this function belongs to C(lh, Moreover if u E Qk, H(u) = u since H E ih and therefore J(H(u)) SO via the definitions of Rk and Rk+l, Since we can assume bk :> AI] > 0 and ck(5) :> Ck > bk , the choice of f implies TI(1, H(u)) = u on Qk, Therefore r;(l, H()) E Ak , Moreover since H E Ak (5), if u E Dk ,
J(H(u))Sb k +ESck-E'Scd5)
E'
by the choice of f. Therefore by 2° of Theorem A.4, T}(l, H()) E AdS), Consequently by (9.45) and T of Theorem A.4 ma?C J(T}(l, H(u))) S q(6)
UEVk
E,
contrary to (10.44). Now we are nearly through with the proof of Theorem 10.4. If Ck > bk for a sequence of k's --> 00, by Proposition 10.43 and 10,39, J has an unbounded sequence of critical values and the proof is complete. It remains to show that Ck = bk for all large k is impossible. PROPOSITION 10,46, If Ck = bk for all k
and
k~
:>
k*, there exists a constant 0J > 0
k * such that
(lO.47) for all k ~
k.
Comparing (lO.47) to (10.40) and (10,5) yields a contradiction and the proof of Theorem lOA is complete,
PERTURBATIONS FROM SYMMETRY
68
PROOF OF PROPOSITIO'."
10.46. Let c > 0 and k 2: k'. Choose H E '\.k such
that max J(H(u))
( 10.48)
uE[h
s: bk + f
Since D k + 1 = Uk U (-Uk), H can be continuously extended to D k_ 1 as an odd function. Therefore by (10.38),
(10.49)
bk + 1
s: uED max + k
for some
11'
E Dk+ l . If
11'
J(H(u)) = J(H(w)) 1
E Uk, by (10.48)-(10.49),
J(H(w))
( 10.50)
s: bk -1- E.
~uppose
1)' E -[h. Then since bk -> CXl as k -> (Xl via (10.40), (10.49) and (10.17) imply J(-H(w)) > 0 if k is large, e.g. k 2: k. By (10.17), the oddness of H. and (10.48),
J(H(11')) = J( -H(-w))
( 10.51)
s: J(H(-w)) + PI ((J(H( -W)))I/I' -1- 1)
s: bk + E -1- 01((b k + ElI/1' + 1).
Combining (10.49)-( 10.51) yields
bk _
( 10.52) Since
f
1
S:b k
€+3d(b k -+-
-rl).
(10.52) implies
is
( 10.53)
bk+!
s: bk -i- 31(b~/1' -1- 1)
for all k 2: k. It remains to show that (10.53) implies (10.47). This will be done by induction. Suppose (10.47) holds for all k E [k,]] n N. We claim it also holds for j + 1. \Vithout loss of generality, we can assume j 2: 2k and ~
> max, bk+ I - o<;l<;k
(k
+ 1)1'11'+1
By (10.53), bJ + I
s: b
J
+;31 ((wjl'll'-1 )1/1' + 1) J
s: bi; + ,3 1 '2.)w l/ l'll/l'-1 + 1) (10.54)
l=k
s: bi: +8
J
1 (j
-
k -r 1) + Plw l/11 L
11/ 1'-1
l=k
We must prove that the right-hand side of (10.54) does not exceed w(j + 1)1'/1'-1 I\'ow J
(10.55 )
L
Is: t .Ii:
- 1 fL
PERTURBATIONS FROM SYMMETRY
Comparing (10.54) and (10.55), we see that to get (10.47) for] that w satisfies (i) bk ::; w(l 20)(J + 1)1'//1 - 1, (ii) (31 ::; wo, (10.56) ::; wO (iii) (31 w1/ 1'
69
+ 1,
it suffices
/1
for some 8 E (0,1). Since
J 2: 2k,
(1O.56)(i) holds if
1 ::; (1 - 20)21'/1'-1,
(10.57)
which is certainly satisfied for 0 near 0. With 0 so chosen, (1O.57)(ii) and (iii) also hold if OJ is large enough. Thus we have (10.47) for all k 2: k and the proof is complete. REMARKS 10.58. (i) If f(x) in (10.2) is replaced by f(x,lL), where (10.59)
If(x,OI::;03+04IElo,
0::;0'
and (10.5) is strengthened to (10.60) a variant of our above arguments gives a stronger version of Theorem 10.4 for this case. (ii) It is an interesting open question as to whether Theorem 10.4 holds without (10.5). Bahri [Ba] has given a partial answer. He proved for -
6:.1L 1L
= = 0,
+ f(x),
x EO,
x E 80,
where s < (n + 2)(n - 2)-1, that there is an open dense set of f in W- 1 ,2(0) for which (10.1) possesses an infinite number of distinct weak solutions. (iii) A recent announcement of Bahri and P. L. Lions [BLI improves the value of sin (10.5) to s < n(n - 2)-1
11. Variational Methods in Bifurcation Theory
In this final chapter, the usc of variational methods in bifurcation theory will be studied. Recall that if X and Yare Banach spaces, 1: X -+ Y, and 1-1(0) contains a curve Z, we sayan interior point z E Z is a bifurcation point for 1 with respect to Z if every neighborhood of z contains zeros of 1 not on Z. After some preliminary transformations a situation often encountered in applications is X = R x E with E a real Banach space. Y = E. and (11.1 )
l(A.u)=Lu+H(u)
A11.
where A E R. 11 E E. L E [(E. E). and H E CI E) with H(u) as u. ~ O. Then we can take Z == {(A,O)IA E R} or any subinterval thereof. It is easy to that a necessary condition for 0) E Z to be a bifurcation point is that 11 E alL). Indeed if 11 7'- alL), then for A near 11, L - A id is an isomorphism and l(A, u) = 0 is equivalent to
(11.2) The right-hand side of (11.2) is 0(111111) as 11 -+ O. while 11111111- 1 is of norm 1. Hence {(A,O)IA is near 11} are the only "small" solutions of (11.2) for A near 11 and (11.0) is not a bifurcation point. Simple examples show this necessary condition is not sufficient. E.g. take E = R2 11 (UI,U2), Lid, and H(11) = (-11~,Ull so l(A,11) 0 becomes
(11.3) ~1ultiplying the first component of (11.3) by U2, the second by 111, and subtracting yields 1(/ + 11~ 0 for any solution (A,U). Hence (11.3) has only trivial solutions. Thus 11 = 1 E alL) but (1. (0.0)) is not a bifurcation point. Surprisingly. hO\\'('wr. if variational structure is present. II E alL) is not only necessary blll is also. modulo some additional technicalities, a sufficient condition for bifurcation to occur. To be more precise, suppose E is a Hilbert space,
THEOREM 11.4. Suppose E is a real Hilbert space and I E C2(E,R) with + H(11), where L E [(E, E) is symmetric and H(11) = 0(111111) as
D
70
VARIATIOI\AL METHODS JI\ B1FTI1CAT!OI\ THEOPY
u -+ 0. If J1 E alL) is an Isolated eigenvalue of a bifurcation point for
:().,u)==Dq,'(U)-Al1=Lu+
71
then (11.0)
18
-AlL.
°
Aloreover there is an TO > such that (il for each I' E (0, ro) there exist at least tll.;O IILstmet so!uI7lln., (,I, i = L 2, of: = having 1:11, rand - 111 small. (ii) As r -,0, (A,(r), 11,(1')) ~ 0).
°
If,
(r)).
REMARK 11.5. Actually I need only be defined on a neighborhood of 0 in E. Theorem 11.4 is due independently to Biihme iBii) and .\1arino [Maj. Aside
from smoothness considerations, it generalizes an carlirr result of Krasnosrlski [K]. Minimax methods are not needed to prove Theorem 11.4. A standard construction from bifurcation theory converts the problem of finding zeros of : to a finite dimensional problem in R x N(L - J1 idl, where N(Al denotes the null space of a linear operator A. The latter problem will be solved by obtaining the maximum and minimum of a funct.ional defined OIl a sphcrclikc manifold ill N(L - J1id). The proof of Theorem 11.4 begills with the finite dimensional reduction mentioned above. Let N == N(L J1id) and let NJ. denote its orthogonal comple. ment. Then E = N ttl NJ.. and if 11 E E, 11 == V wEN ttl NJ.. Let P and pJ. denote respectively the orthogonal projectors of E onto N, N 1.. We seek solutions of the equation
Lu + H(u) = AU
(11.6)
near (J1,0). Using P and pJ.., (11.6) is equivalent to the pair of equations
(11.7) (11.8)
Lv+PH(v+W)=AV, Lw+PJ..H(v+w) = AW.
Set
F(\v,w) == (L-Aid)w+PJ..H(v+w). The hypotheses on I imply F E Cl(R x N X NJ..,NJ.). Clearly F(J1,O,O) = 0. Let Fw ().L, v, w) denote the Frechet derivative of F with respect to w. Then Fw (J1,O,O) = L - ).Lid, an isomorphism of N1. onto NJ... Therefore by the Implicit Function Theorem, there is a neighborhood, 0, of ().L, 0) in R x N and a mapping ;p E C 1 (0, N1.) such that ;P().L, 0) = and the zeros of F near (J1, 0) are given by {(A,v,;p(.A,v))I(A,V) EO}, Thus to solve (11.6) near (J1,0), it suffices to solve the finite dimensional problem
°
Lv + PH(v + ;P(A, v)) = AV, or equivalf!ntly since Lv = J1V, ( 11.9) This reduction is called the Method of Lyapunov·Schmidt.
\'AI\IATIO:;AL
72
~!ETHODS
I:; BIFURCATIO:; THEORY
Before continuing further, two useful estimates should be pointed ouL Since for all A near ~I, (IL8) can be rewritten
L - A id is an isomorphism of j\" ~ to X as (1110)
Since H(A,U) =
iL'i
v-"
°
imel
o(llull)
as u·~ 0, (11.10) shm\'s
:PtA, v) =
0(111' + :PtA, 0')11)
therefore
(11.11) ii," V ~
°uniformly for A near
~l. Differentiating
(L - Aid, pi- H'(v
(1112)
T
(11.8) with respect to A gives
:ptA, v))) ~~ (A, v)
= 'P(A, v),
Since H'(O) = 0, (lUI) and (IU2) imply
( 11.13) a.5
°
l' ~ uniformly for A near 11, The next step in the proof of Theorem 11 A is to find A as a function of v near 0). By (11.9), !\
\/1.
, V)l
- il)
or
=0
A -11
( 11.14)
°
for v I and (A, 1') near (11,0), Define G(A, v) to be the left-hand side of (11.14) and G(A,O) == A-/l. The estimates (lUI), (lU3) and properties of H show G is continuous from a neighborhood of (11,0) in R x N to R, G(I1, 0) 0, G is continuously differentiable in A near (11,0), and also with respect to v for v I 0, Since (JC(/I, O)/OA = 1, a stronger version of the Implicit Function Theorem [Di] implies the zeros of C near (11,0) are given by A = 1jJ( v), where 1/J is continuous in a neighborhood of 0, 1.6(0) 11, and 1/J( v) is continuously differentiable in a deleted neighborhood of 0. I"ow set XlV) = 'P(1.6(v),o'), Then X is a continuous map of a neighborhood of in N to lv'"- and is continuously differentiable in a deleted neighborhood of 0, Moreover Xlv) = o(llvll) as v -> 0. For our later purposes, the behavior of X' Ilcar 0 must be studied. By (11.8) for zEN and small t E R,
°
(11.15)
(L -
+ t.z)idJx(v
Diffcn'ntiating (11. 1,5) at t ( 111G)
(L-
+ pi H(l' + tz + Xlv + tz))
0 gives
+ p~ H'(l' ~ X(l')))x'(v)z -pLH'(v't'
= 0,
VARIATIO"AL :-'1ETHODS E\ BIFCRCATIOl\ THEORY
73
Similarly (1114) yields
(?}J'(v). z) = ilvll~21(H'(v
(11.17)
- 2(H(1I
+ X(v))(z T
X'(v)z). v)
+ (H(v + X(v)),z)]
+ X(lI)),V)(v, z)llvll~4
Consequently since H'(O) = 0. by (11.16)~(11.17), for v near 0,
Ilx'(v)zll :s: constll(1jJ'(v), z)i"x(V)11 + IIH'(v
-i-
l'
f 0.
xlv)):!
+ Ilx'(v)zll)
:s: const { ( 11.18)
~'-;,-",;---'-'izi'l"x(v)i' -+-
x(v))11
= o(l)llzll + o(1)llx'(v)zil as v
-+
0, v
(11.19)
f
0, or
Ilx'(v)zll
o(I)llzll
as v -+ 0, v f 0. Equation (11.19) shows X'(v) extends continuously to v = 0 and ;((0) = O. With these preliminaries in hand, the proof of Theorem 11.4 can be reduced to the study of a finite dimensional variational problem. Let 1) be a neighborhood of in N in which 1jJ and X are defined and C 1 Let ]v( := {v -i- x(v)lv E 'V}. Then ]v( is a CI manifold of dimension n dim N in E. Let E > and set D, := ]v( naB, so for small E, D, is a compact CI, n 1 manifold. Consider liD,. This functional possesses at least two distinct critical points corresponding to the maximum and minimum of I on D,. We claim that at any critical point U of II D" I'(u) = AU, where u = v -+- xlv) and A = 1jJ(v). Assuming this for the moment, observe that as f -+ 0, if v€ E D€, then v€ -+ O. Therefore v€ -+- x( v,) --+ 0 and 1jJ(v€) -+ fL. Thus the proof of Theorem 11.4 will be complete once the above claim is verified. If u is a critical point of liD" then
°
°
J'(u)
(11.20)
for all
T(aB,)x
= {
= O}.
Therefore
TD,x = {
(DI'(u) - s-2(DJ'(u),u)u,
\'AHIATJO,\AL \!ETHODS 1,\ BlFURCATJO" THEORY
i4
for all p E span{1!.TDcu}' We will show that (1121) yields £'(11) = AlI, where I' + ilnd A = Since v EM implies 11 = l' + xlv). where l' E V, the constrnction of X shows P"- D1'(lI) c= 1.'(1')X(1'), Therefore 11
(DI'(1J), xlv)) = l.0(1')llx(v)11 2
(11,22) The constrnction of
1;'
further implies
(I,'(v) - 11,)
(11.23)
(11,24) 1L
E
(DI'(V).l1)
=
0(v)iil1I1 2 =
DE' Therefore 1)(v)
for illl v E
=
c- 2(DJ'(1l),11)
A(V)
=
DE' Compilring this to (11.21) shows (DJ'(l1) - 1))(1J)11. 'P)
(11.25) for illl
(DH(v),l').
(11.22)(11.23) yield
Hccillling t hilt
for
2 =
= (]
E span{l1. TDE ,,}, By (11,8), PJ'(DI'(1I)'- Ali) = (] so (11.25) gives
(DI'(l1)
(11 ,26)
=(]
for all 'P E spiln {11, I' DE", ;V J. } == W. We will show W = E and therefore D £' (11) = A11. To prove this, note first that 11 = v + X' (11)1' for some v E V and X' (1J) E N 1.. Therefore W = span{1J, TD Eu , N~}. Let VI,.· .• 1J n -1 be a basis for I'D,,,. To get a basis for I' .M u , we supplement these vectors by a vector 1Jn , Since TMu = {x + x'(1J)xlx EN}, we have n
(1127)
1J+x'(V)1J=L 1=1
\'ote that (3 n f (] in (11.27) for otherwise, v + X'(v)v E TDeu and therefore
(v + X'(v)v, 11)
(11.28)
= (] =
+ (X'(v)v,X(1J))
for v near O. Hence v = 0 if is small, contrary to Thus (3'1 of O. Solving for 1Jn in (11.27) shows that 1Jn E span{v, TEu,;V.l.} =
( 11.29)
+ 0(livI12)
ilv
x(v)11 =
w.
Finally observe that if y E E,
y
(P
+ x'(1'lP)y + (P~ - X'(1')Py)
= span{v n , TD,u, N1.}
E span{T Mu , NJ.}
cW
via (11.29). Therefore W = E and the proof of Theorem 11.4 is complete.
E.
VARIATIONAL METHODS IN BIFURCATION THEORY
75
COROLLARY 11.30. If I is even and dim N ~c n. the equation DI'(I1) AU possesses at least n distinct pairs of solutions (AJ (r) ± iL J (r)) 1 S .7 < n. such that !luJ(r)!l = r. Moreover AJ(r) ~)1 as r --+ O. f
PROOF. If I is even, 'P and X are odd functions, DE is a symmetric set, and 110, is even. A slight modification of the proof of Theorem 8.1 then gives the corollary. We shall not carry out the details. We will briefly sketch an application of Theorem 11.4 to a bifurcation problem for a semilinear elliptic partial differential equation. Consider ~
(11.31)
A(a(x)u + p(x, u)),
6u
xE
n = 0,
x
E
0,
a~,
where p E C 1 (n, R) and satisfies (P3) and a is as in (2.40). Let X(~) E COO(R, R) satisfy X(O = 1 for I~I S 1, xlO = 0 for I~I ;:> 2, and 0 S xW ~ 1 for all ( Define p(x,~) = X(Op(x, 0 + (1 ~ xlO)· Then
p E C 1, and satisfies
(P2) (with s = 0) and (p3). Set
l(u)
r(~an2 + F(x, n)) dx
in
with 11 E E = W c;,2 (0) as usual. The properties of p and PropositiJn B.34 imply I E C 2 (E,R). The argument of Theorem 8.17 shows that a critical point of
11aB, satisfies n))'Pdx
= v/
Vn· v'Pdx
for some v and all 'P E E, i.e. n is a weak solution of (11.31) (for p) on aB T with A = v - I In terms of Theorem 11.4, using (P3) it is not difficult to check that
H(u)'P = L is defined by
In In
p(x, n)'P dx ,
=
{LU'P dx,
and alL) = {,u E Rlau ,u6n 0 for some n 't 0, u E E}, i.e. alL) = p;l is an eigenvalue of (2.40)}. Since these eigenvalues are of finite multiplicity and isolated, by Theorem 11.4, each eigenvalue of (2.40) provides a bifurcation point for (11.31) (for pl. Arguments from elliptic regularity theory show if r is small enough, IluIIL=(n) < 1 and therefore p(x, u) = p(x, u). Thus we have shown THEOREM 11.32. Ifp E C 1 (n,R) and satisfies (P3), every eigenvalue A) of (2.40) gives rise to a bifurcation point (AJ' 0) of (1131). REMARK 11.33. If p is odd, Corollary 11.30 can be used to say more about the number of solutions of (11.31) near (A J' 0).
\'ARIATIO,\AL \lETHODS 1:\ BIFURCATIO,\ THEORY
76
Our final result in this chapter is a variation on Theorem 11.4 where we seek of
80111 tions
(11.34)
Lu
+
= Au
as functions of A rather than of r = Iluli. This is a somewhat morc subtle situation than the case treated in Thcorem 11.4 and involves ideas of a minimax nature. THEOREM 11.35 ]R3]. Under the hypotheses of Theorem 11.4, either (i) (fl,O) is not an isolated solution of (11.34) in {Il} x E, or (ii) there is a one sided neighborhood A of 11 such that for all A E A \ {Il}, ( 11.34) possesses at least two distinct nontrivial solution.5, or (iii) there is a neighborhood ,\ of fl such that for all A E A \ {ILL (11.34)
possesses at least one nontrivial solution. PROOF. By the proof of Theorem 11.4, (11.34) is to (11.9) where 1') E ('1 and ,:;(,\, v) = o(li as v ~ 0 uniformly for A ncar 11. Define
g(A, v) == I(v
+ p(,\, v))
v)
+
Then 9 is ('1 for l' neal' 0 in:V. Letting h denote the primitive of H with h(O) shows (1L36)
vi
0
)0) +h(l'+
=
\Ye claim nitical points of g(A. critical point of 9 we have
ncar v = 0 are solutions of (11.9). Indeed at a
( 11.37)
+
v))D + Iw(v + ptA, v))Pv()', v)iJ
- AI( v, D) + (p(A, u), Pv(A, v )iJ)], Siner by (11.7). for all Ii.' E .\"-, (1
simplifies to
0= (CJ,.(A,
n=
(Dl v
-
AV, D)
for all l' E"Y which is equivalent to (11.9). \'ow ill order to prove Theorem 11.35, we need only analyze the critical points of 9(A,) near 0 for A near 11 The form of 9 implies v = 0 is a critical point of urA,) for all A near /1. If 0 is not an isolated critical point of g(ll, ,), then alternative (i) of Theorem 11.35 holds. Thus for what follows we assume 0 is an isolated critical point of 9(11, .). Consequently 0 is either (a) a strict local maximllTll or minimum for g(p, .], or (b) 9(11,') takes on both positive and negative \'al11es ncar O. The former possibility leads to alternative Oi) and the latter to alt crnative of Theorem 11.35. Before verifying these statements, note that
VARIATIONAL METHODS IN BIFURCATIOK THEORY
g(/1, u)
77
g(A, u)
Case (a)
----~+::__----u
g(A,u) Case (b)
----~~~-----u
----~_+_'''-------u
for the special case of dim N = 1, one can use the shown figures as a basis for a proof. Case (a); 0 is a strict local maximum or minimum for g(/1, .). We will assume that 0 is a strict local maximum for g(/1, .). Replace 9 by -g to handle the case of a strict local minimum. Since g(/1,O) = 0, for r > 0 and small, there exists a (3 > 0 such that g(/1,.) ~ -2(3 on BE r . (Here and for the remainder of this section the sets we are working with lie in N.) Therefore for A near /1, g(A,) ~ -(3 on BE r . By (11.36) and previous remarks,
(11.38)
g(,\,v) =
as v - t O. Hence g(A, u) > 0 if A < Ii and 0 I v is sufficiently small. In particular there exists apE (O,r) such that g(A,V) 2: (/1 A)4- 1 p2 == a(A) for v E BEp. Let
(11.39)
e(A) == max g(,\, v). vEB,
Then for A < /1 and A sufficiently close to /1, e(A) 2: a(A) > 0 and is a critical value of g(,\, .). Moreover g(A,.) satisfies hypotheses (Id and (h) of the Mountain Pass Theorem, the latter with e being any point on BE r . Theorem 2.2 cannot be applied directly since g(A,.) is only defined near 0 in N. However since a(A) > 0 > -(3, it is easy (via the proof of the Deformation Theorem) to
VAHIATI00:AL \fETHO[)S 1:\ BlFl:HCATI00: THEORY
78
constmct a ITlap T) C(:O, I' x B r , Br) such that I/(t,·) = id on BBr and the properties of Theorcm A.4 rclati\'c to Dr' Consequently if
r={hEC(iO,l],B r ):h(O)=O andh(l)
(11.40)
lllax
tEio.1j
T)
has
BB r }
UP, h(I)),
the proof of Theorem 2.2 shows <;;(.\) ::: a(.\) > 0 is also a critical value of )1 and near p If <;;(.\) < 7:(.\), it is clear that (11.34) has two distinct nontrivial solutions If <;;(.\) = e(.\), (11.39)-(11.40) show c(.\) equals the maximum of a(.\,·) over ever:, curve in 1'. Therefore the set of points in Br at which a(.\,·) achieves its maximum in Br separates 0 and BB r , Consequently this critical set contains the boundary of a neighborhood of 0 in ;V, In any event we sec Ca'il' (a) implies altcrnative (ii) of Theoem 11.35, Case (b): g(p,,) takes on both positIve and negative values near 0, This case is more subtle than the previous one, Again the idea is to reduce the proof to iUl argumcnt of "mountain pass" type but a more complicated construction is required. To begin, the computation in (11.37) shows
1](.\,) for .\ <
9t,(.\,V) = (fl ~ .\)1'
+ PH(1' + rp(.\,1'))
which is C 1 in (.\, v), Hence the negative gradient flow corresponding to 9(fl., ,):
1,0(0, v)
(11.41)
=
v,
has a unique (local in t) solution for all v near 0 in ;V, Let Br be a small ball in which this flow is defined, Set
={x E Brl1,6(t, x) E Br for all t > O} S- ={x E Brl1,6(t, x) E Br for all t < O}, S+
and LEMMA 11.42,
S, andS- are nanempty,
S+ case, Let (xm) C Br be a sequence such as m -+ 00 and 9(fl, > 0, Consider 1,6 ( t, By assumption, 9v(J.L,V) f in Br for v f 0, If for fixed m, the orbit 1,6(t, remained in Br for all t < 0, it would have to converge to a critical point of 9(fl, ,) with positive critical value, Since this is impossible, there exists a smallest tm < 0 such that 1,0(t m , = Ym E DB r , Since Xm -+ 0 as m -+ 00 and 9v(J.L,O) 0, tm -+ ~OO as m -+ 00, Let y be a limit point of (Ym)' Then 1,6(t, y) E Br for all t > 0 so yES+ REMARK 11.43. Since H(u) = a(llull) as u -+ 0, the matrix 9vv(J.L, v) vanishes at v = 0, If it were nonsingular, S+ and S- would be respectively the stable and unstable manifolds at of the flow given by 9v (fL, -). PROOF, We will only verify the
that
°
°
Xm -+
°
VARIATIONAL METHODS IN BIFURCATION THEORY
79
PROPOSITION 11.44. There exi"ts a neighborhood Q of 0 in ,V and constants c+
> 0 > c- such that Q c Br and v E aQ implies either (i) g(f.1.,v) = c+, or (ii) g(f.1., v) = C ,or (iii) 1/;(t, v) E aQ for all t near O.
Aosuming Proposition 11.44 for the moment, we can complete the proof of Case (b). Let T- = g(f.1., .)-I(C-) n S-. The construction of S' and Q shows T- 7= C. Set
f:={hEC([O,l],Q)lh(O)
O,h(1)ET-}.
Define
erA)
(11.45)
= hEr inf max g(.\, h(t)). tEla,l]
Letting A < J1 and choosing p as following (11.38) shows c(A) 2: a(A) > O. We claim c(A) is a critical value of g(A, .). Indeed (A.1) shows -gv(f.1., v) is a pseudogradient vector for -gv (.\, v) for all v E aQ. This permits a modification of the proof of the Deformation Theorem to find TI E C([O, 11 x Q, Q) having the usual properties and the use of TI(1,') in the usual fashion to conclude that erA) is a critical value of g( A, .). If A > f.1., the same argument works on replacing 9 by -g. rt remains to construct the set Q. Let Ac:= {v E Nlg(f.1.,v):S e}. LEMMA 11.46. There is an c > 0 such that if c := maxvEB, g(f.1., v) and x E BE \ S-, then as t -+ -00, the orbit 1/;(t, x) leaves Ac n Br via Ac \ aB r . PROOF. If not, there are sequences such that Zm := 1/;(tm, Xm) E aB r and
Cm -+
0,
Xm
E BEm \ S-, and
tm
<0
min g(p, v) :S g(f.1., zm):S max g(f.1., v).
vEBem
vEB<>m
Therefore along a subsequence, Zm -+ Z E aB r and g(p, z) O. Let Tm < 0 denote the time of final exit of 1/;(t, xm) from BEl as t -+ -00 and consider 1/;(t,xm) for t E (tm,Tm). Since Ilgv(f.1.,v)11 is bounded away from 0 in 13 r \ BEll (11.41) shows that
Om:= 9(fl,Zm) - g(fl,1/;(Tm ,X m
))
2: const > 0
om
if Itm - Tml is bounded away from O. But our above remarks show -+ 0 as m -+ 00 so It m - Tmi ---t 0 as m -+ 00, i.e. the time it takes the orbit to go from 1/;(Tm' xm) on aBEl to Zm on aBT tends to O. However this cannot happen since gv(fl, v) is bounded away from infinity. Therefore there exists an e as desired. An obvious consequence of Lemma 11.46 is COROLLARY 11.47. There is an
E
> 0 such that if c- = min g(f.1.,v), vEBi;
80
VARIATIONAL METHODS IN BIFURCATIOCI THEORY
and x E B£ \ 5--, as t -00 the orbit 1jJ(t, x) leaves Ac+ n aB via Ac+ and if x E Be \ ST, as t 00 the orbit 1/J( t, x) leaves Ac ,n BT via Ac . -4
T
-4
;-.Jow let E be as given by Corollary 11.47. For each x E BE \ S-, there is a corresponding t- (x) such that g(l1. (x), x)) = c+ and for each x E Be \ S+. there i8 a tT(x) such that g(I1.1jJ(t+(x),x)) = c. If x E S+, set t-(x) = -00: if x E 5-. set t-(x) = 00. Finally define
Q == {'0(t,x)lx E Be and t-(x) < t < t+(x)}. PROOF OF PROPOSlTIOl\ 11.44. Clearly Be C Q. If Z E Q \ Be, Z = '0(t, x) for some x E Be and t E (t-(x),t+(x)). Therefore for 0 small, 1/J(t,BIi(x)) c Q and is a neighborhood of z so Q is a neighborhood of O. Let v E aQ and suppose g(l1,v) cI c T or C-. Consider Ov == {1/J(t,v)lt E R}. Since v = limm~(X)vm, where Vm E Q, and 0,,= n Be cI 0, there exists an x E Ov n BE' Therefore .T= .1')forsomeTERandv= where-TE (x),t+(x)) since 9(11.,11) E (c-. ). Hence x E aBE for otherwise v E Q. Moreover 9(11, 1/J(s, v)) E ,c T ) for s near O. Since v) = '0(s - T,X), it is clear that 1/J(s, v) E DQ. Indeed 1iJ( s, v) = limm~oo 1jJ(s T, where Xm E Be, Xm - 4 x, and therefore s - T E (t- (x m ), t+ (x m )). The proof is complete. REMARK 11.48. If I is even, a much stronger result than Theorem 11.35 holds (see iFR]).
Appendix A
The goal of Appendixes A and B is to prove some results that were required earlier which are rather technical in nature. In particular in this appendix we will prove a fairly general version of the Deformation Theorem. Thus let E be a real Banach space, U c E, and I E Cl (U, R). Then vEE is called a pseudo-gradient vector for I at u E U [P2] if: (A.l)
(i) (ii)
Ilvll::; 211J'(u)ll, 11J'(u)112
J'(u)v ~
In the future pseudo-gradient will be denoted by p.g. for short. :\ote that a p.g. vector is not unique in general and any convex combination of p.g. vectors for I at u is also a p.g. vector for 1 at u. Let I E Cl (E, R) and E == {u E EII'(u) Ie a}. Then \f: E -~ E is called a p.g. vector field on E if \f is locally Lipschit~ continuous and \fIx) is a p.g. vector for I for all x E E. LEMMA
A. 2. If IE C l (E, R), there exists a p.g. vector field for I on
E.
PROOF. For each u E E, we can find a vector 11J E E such that = 1 and J'(u)11J > ~III'(u)ll. Then z = ~III'(u)ll11J is a p.g. vector for I at u with strict inequality in (i) and (ii) of (A.l). The continuity of I' then shows z is a p.g. vector for all v E N u , an open neighborhood of u. Since {Nulu E E} is an open covering of E, it possesses a locally finite refinement which will be denoted by {MJ }. Let PJ(x) denote the distance from x to the complement of MJ . Then PJ(x) is Lipschitz continuous and PJ(x) = 0 if x f/:. M J. Set /
/3J(x) = PJ(x) / I:>k(X). k
The denominator of /3) is only a finite sum since each x E E belongs to only Let z) == finitely many sets M k . Each of the sets M J lies in some ~liF(uJ)II11JJ' a p.g. vector for I in M) and set \fix) == Li (x). Since o ::; /3J(x) ::; 1 and L J /3J(x) = 1, for each x E E, \fix) is 'a convex combination of p.g. vectors for I at x. Moreover \f is locally Lipschitz continuous. The proof is complete. 81
APPE:\DIX A
82
COROLLARY :\.3. If I(x) by an odd W.
1S
even in x, I has a p.g. vector field on j;; given
PROOF. Suppose I is eVeIl. Let \; be as given Lemma A.2 and set W(x) = . Then 11-' is odd. locally Lipschitz continuous, and since l' (x) is odd in T. (i) ::; + I!. (ii) = V(x)\/(x) ~ 2: Recall that A.., =' {11 E ::: s} and Ie == {11 E EII(u) sand 1'(11) = O}. "ow we will prove the following version of the Deformation Theorem (which in particular contains Proposition 2.1 as a very special case). See e.g. [Brl, C, P1, Sl] for earlier such results. ~ (V (x) ~ V (
1(
THEOREM A.4. Let E be a real Banach space and let I E CI(E,R) and satisfy (P8). If C E R, E > 0, and 0 is any neighborhood of K e , then there exists an c: E (0, E) and T) E 0([0.1] x E, E) such that 10 T)(0.11) = 11 for all 11 E E. 2° T)(I.u) = 11 for all tE [0,1) if 1(11) t/: [c E.e +E]. 3° T)(t.11) is a horneornorphl:srn of E onto E for each t E [0, I]. 4° !1T)(t.u) ~ uli ::: 1 for all t E [0,1] and u E E. 5° I(T)(t. ::: I(u) for nil t E [0,1] nnd u E E. 6° T)(l,A o +£ \ 0) C ..1 0 - £ , 7° If Kc = Z, T)(1, C ..1 0 -£ 8° If I(u) is even in 11, T)(t, u) is odd in u. PROOF. The function T) will be constructed as the solution of a suitably modified negative gradient flow for I. A few preliminaries are needed before setting up this differential equation. By (PS), Kc is compact. Set Ns == {11 E Eiliu ~ Kc II < t5}, where liu ~ Ko II denotes the distance from 11 to Ke. Choosing (j suitably small, Ills cO. Therefore it suffices to prove 6° with 0 replaced by N s . ]\'ote also that if Kc = Ns == Z so we get 7° instead. We claim there are constants b, s > such that
°
(A.5)
1I1'(u)11 2: b
for all u EAcH \ (Ae-i U N o/ 8 )·
If not, there are sequences bn -> 0, sn -> 0, and 11n E AeHn \ (Ae-in UNo/ 8 ) such that Il1'(un)11 < bn . By (PS), a subsequence of Un converges to u E Kc \ N o/ s · But this latter set is empty. Hence there are constants b, s as in (A.5). Since (A.5) still holds if E is decreased, it can further be assumed that
(A.6)
o<s<min(E,~~,~,D.
Choosing any c: E (0, s), define A == {u E EII(u)::: c ~ E} U {u E EII(u)
2: c + E}
and
B == {u E
Elc ~ E :::
I(u) :::
c+ c:}.
APPEN])]X ,\
Therefore All B = 0. Set
°
°
Then 9 = on A, 9 = 1 on B. ~ 9 ~ 1, and 9 is Lipschitz continuous on E. Similarly there is a Lipschitz continuous function f on E such that f = 1 on E \ Nb/4 , f = on No/s , and ~ f ~ 1. Note that if 1 is even, A, D, and Nb will be symmetric sets with respect t.o the origin and f and 9 arc even functions. Nextdefineh(s)=lifsE[O,ll andh(s) = 1!8if.s2 1. Since [ C1(F::,R).by Lemma A.2 and Corollary A.3, there exists a p.g. vector field l/ for [ on E with V odd if 1 is even. Finally set W(x) = -f(x)g(x)h(lIV(x)II)ll(x) for x E E and W (x) otherwise. Then by construction, W is locally Lipschitz continuous on E with ~ IIW(x)11 ~ 1 and W is odd if 1 is even. Now we can define the mapping 1/. Consider the Cauchy problem:
°
°
° °
(A.7)
d1/ dt
W(1/),
1/(O,u)
=
u.
The basic existence-uniqueness theorem for ordinary differential equations implies that for each u E E, (A.7) has a unique solution defined for t in a maximal interval (u), t+(u)). We claim t±(u) ±oo. If not, say t+(u) < 00. Let tn -> t+(u) with tn < t+(u). Integrating (A.7) shows (A.S)
- tn !
since !IW(·) II ~ 1. But then 1/(t n, u) is a Cauchy sequence and hence converges to some u as tn -> t+(u). The solution to (A.7) with u as initial data then furnishes a continuation of 1/( t, u) to values of t > t+ (u) contradicting the maximality of t+(u). Similarly t-(u) = -00, The continuous dependence of solutions of (A.7) on the initial data u implies 1/ E C([O, 11 x E, E) and (A.7) implies 10 holds. Since E > E, g(x) = on A so 20 is satisfied. The semigroup property for solutions of (A. 7) gives 3°. Integrating (A.7) and using IIWC)II ~ 1 and 10 gives 40 . Above remarks on the oddness of W when 1 is even yield So. To verify 50, note first that if W(u) = 0, 1/(t, u) == u is the solution of (A.7) (via uniqueness) so 5° is trivially satisfied. If W(u) f= 0, u E E so V(u) is defined as is V(1/(t,u)) and
°
~~...!..!.. (A.9)
=
1'(1/(t,u))~: -1'(1/(t, u))f(1/(t, u))g(1/(t, u)) x h(IIV(1/(t, u)!!)ll(1/(t, u)) ~
°
via (ii) of (A.l) and 5° follows for this case. It remains only to verify 6° or a fortiori, 1/( 1, .1 0 +£ \ Ns) C .1 0 -£, If u E .1 c -" then 1(1/(t, u)) ~ C-f by 5°. Thus we need only proveu E Y == .1C+E \(.1c_,UNs) implies that 1/(l,u) E .1 c - E ' Let u E Y. The reasoning that led to 5° showed
(A.lO)
°
d1(1/(t, u)) < dt -'
APPE:\DIX A
84
Since 9 = implies
°
Oil
A c - c. the orbit 1)(1. u) cannot enter A c -€.
1(1)(0. u))
(A.11)
I(I)(t, u)) 'S
E T E:
Therefore
10)
< 2c'
for all t. 2 0. Suppose that u E }' and 1)(£.u) E Z == A c +€ \
for
U
s E [0, tj. This will certainly be the case for small I. Then for such s, 1)(3. u) E (\'ia (A.5)) anel f(7)(S, u)) = 1 = 9(1)(8, u)). By (A.11). 2E 2 =
i;
fO -I'(rl(.,.1I))h(IIF(1)(s.l1))IIW(1)(s,l1))d,~
r h(!W(I)(s, u))II)['(1)(s. 11))II(1)(s, u)) ds
./0
2 {h(IW(I)('S, u))ill'M'(1)(s, u))11 2 ds . () IA.12)
2 b
11
> -b - 2
> -b
u))rI)!II'(1)(,', 11))11 dx
j'l
- 2 , b 2
h(
ds
0
1t
8.
0
b
W(1)(s,u))ds
u) - ul!,
2
where we ,;u(wssively nseo (A.l)(ii). (A.5). and (A.l)(i). Hence by (A. (A.6).
111) l t. u ) -
"
l1i!
ano
S
4f
'S b < '8
Thus the orbit 1)(t,11) cannot leave Z by entering N O/2' Consequently the only way 1)( t. u) can leave Z is to enter A c _.[· We claim this occurs for some t E (0,1) thereby proving 6°. If not. 1)(t, u) E Z for all t E (0,1) and as in (A.12),
(.'1.13)
-.-'-'--'-_.:.-'.. 'S
-hi IW (1) (1, ullli) Ill' (1)( t, u)) 112
If for sOlIle t E (0.1). IW(1)(t. u)) II 'S 1. hlW (1)(t, 11)) II) = 1 and (A.5) and (A.13) imply
(A.14)
On the other hand, if for some t E (0,1), W(r/(t. u)h' 1 so (A.l)(i) and (A.13) yield
(t,l1m > 1, hlIW(1)(t. u))II)
(A.15) COf],;cql1(,nth' for all t E (0. I). we have
( A.Hi)
dI(1) ( t. u))
- mill (\b
2
1)
'=I .
=
APP~~"D1X
8.)
A
Integrating (A.16l and combining the result with (A.1ll gives
which is contrary to (A.6). The proof of Theorem A.4 is complete. REMARKS A.17. (i) :--Jote that the full strength of (PS) was not required in the above proof. We only needed (PS) to conclude that Kc was compact and that (A.5) holds for some b, E > O. But we can also achieve this by merely assuming (PS)loc: there is a 6 > 0 such that IJ(1171) - ci < Sand J'(lLn) -40 as n 00 implies that (1171) is precompaet. Still weaker forms of (PS) have been found (see e.g. ICe, Chi]). (ii) It is sometimes useful to have a bit more flexibility ill the definition of a p.g. vector. If (A. 1) is replaced by
(A.l')
(il
Ilvll:S o:llI'(l1)II,
(ii)
J'(l1)V ~
1!2,
where 0 < {3 < a, it is easy to check that Lemma A.2 holds with 11' chosell so that J'(u)w > 2{3(a+{3l-11lw!! and z = l\iI'(u)w!!. Moreover the proof of Theorem A.4 also is essentially unchanged aside from replacing (A.6) by
(A.6')
0< f < min
(3bS 62
16a'
2'
(iii) There arc problems where instead of rl(t, u) satisfying 5c 7c well as the other conclusions of Theorem A.4), one wants a function ~(L u) to satisfy J(~(l,u)) ~ flu), and if
As := {u E EIJ(u)
~ s},
then ~(l,Ac_" \ 0) c AC+E (with 0 0 if Kc 0). Such an ~ can be obtained by slightly modifying the proof just given, in particular using a positive rather than a negative gradient flow. (iv) There are versions of Theorem A.4 for a functional J defined on a Banach manifold, M, rather than a Banach space [Brl, Pl-2, 81-21. Using the same sort of ideas as in the proof of Theorem A.4, and in particular constructing a p.g. vector field for M which is tangential to M, we can get an analogue of Theorem A.4 for such a setting. E.g. in Chapter 7 we consider the case of M a sphere, say oBJ[O), in a Hilbert space. For this special case, a tangential p.g. vector field can be constructed via V(x) - (v(x) . where V(x) is it p.g. vector field for Jon E. The next result concerns the effect on the mapping 7)(1, u) of making stronger assumptions on the form of I. Such information is useful in particular for highly indefinite functionals such as arise in treating Hamiltonian systems as ill Chapter 6.
APPS:\DIX A
PHOPO:'ITIO:\ A.18.
w1th 1(II)
~ ~:
L Ii. Il)
-~
Suppose E 1" (L rcal Hilbert space and IE ('I(E.R) ), L 18 selfad)oml. and :;:1 IS compar-t. Thrn 1](1.11) =
lL'here 0 <; 0ll.ll) <; 1 orld 1\: [0,1] x E -" E
18
compact.
PROOF. B~' the proof of Theorem A.4, 'I is determined as the solution of the initial \'alnc problem
1](0, u)
(A.19)
°<
u,
<; 1 and Ii is a p.g. \'ector field for l' on E. Actually since f(09(0 = 0 if 1(0 rf:. [c-i,c-i-i] or ifu E SIi/8, j' need onl~' be a p.g. vector field on S =: {~ E E11(0 E Ie i,c + i] ane! E r1 Sli/d. We claim such a F can be chosen so that V(u) Lll W(u) where
wherr
.0'
.0(0 = J(~)!i(Oh(Y(~);I) and
IF is cOlllpact. Assuming this for the momcnt, (A.19) bccoIlles
(A.20) Considering the 1] in the argument of wand W as being known, 1] satisfies an inhomogeneous linear equation ane! therefore it can be represented as
(:\.21 )
u))
11) = exp ( ( [
dS) L)
11
+ K(I, u),
where
K(t, 1/) = - [
,u)) ciT.
[ex p ( [
To see that 1\: iO, 1i x E ~ E is compact, suppose BeE is bounded. \Vithout loss of generality, B = BR for some R > 0. By 4° of Proposition A.4, IICO 1] x B FI ) C BH~l Therefore W(/JUO, 1: x Bit)) C W(BR-i·d W(BR-,-l! which is compact. Let
:3 E [0,11, Z E W(Bltf-d}·
Y=: Since the map
],Z) ......,
is a continuous function on the compact set [0, x W(B It~ 1), its range Y is compact. Therefore the closed convex hull Y' of Y is also compact. Now recalling that <; 0 <; L for each T E ro, 1] and u E B R ,
°
=: exp [ ( [ W(l)(S,U))dS)
Hence for t E [0,1]'
It follows that K is compact.
l
L] W(l)(T,U))W(7)(T,U)) E Y.
ZdTE
Y
87
APPENDIX A
To complete the proof of Propositioll A.IX. we must shOl\ it is possible to choose a p.g. vector field V for l' on S such that F(u) = Lv vF(u). where W is compact. As a first step w(' have: PROPOS1TIO)I A. 23. Let E be (! illibert space lind 1': E··· E be compo.ct.. Then given any ~i > O. there exists aT: E ~ E such that T is compact. locally Lipschitz continuous. and
IIT(u)
(A.24)
T(l1)II::;~:
foralll1EE. Assuming Proposition A.23 for the moment, let, = b/2, where b was defined in (A.5), andlct W be the corresponding mapping given by Proposition A.23 with 1'(11) = p'(l1). We claim F(l1) == Ll1 + W(l1) is a p.g. vector field on S. It is certil.inly IOCillly Lipschitz continuous. We will verify (A. 1') for F with n = 2, (3 = and 11 E S. By Proposition A.2:). (A.5), and om choicc of ,.
1.
W(l1)11
= IILl1 + W(l1)11
::; IILl1 + p'(ll)11 + IIp'(ll)·· W(11)11
:; 111'(11)11 +,:; 111'(11)11
+ b::; 211 1'(11)11.
Therefore (A.l')(i) holds. Similarly,
1'(l1)V(l1) = 1'(l1)(L11 + p'(l1)
+ W(l1)
- p'(11))
:: /11'(11)112 -II1'(l1)lh:: /11'(11)11 2
-
~II1'(11)112 = ~li1'(11)112
Thus we have (A.I')(ii). Finally we can give the PROOF OF PROPOSlTIO;.J A.23. (This result is essentially due to Bend [BeZ].) First we construct a special partition of unity for E and then use it to construct T. For each 11 E E. set Su == E l (11) n {v E Ej 111'(11) - T(v)11 < I}' Then {Su 111 E E} is an open covering of E and therefore it possesses a locally finite refinement {Yd. Let p,(11) denote the distance from u to the complement of Yi and
3,(11) == Pi (11) /
L PJ(11). J
Then p,(l1) is a (locally Lipschitz continuous) partition of unity as in the proof of Lemma A.2. By construction for each Y" there is a w, E E such tbat Yi C Sw,. Therefore - T(w,}, v) < for all 11 E Y, and vEE. Since p,(l1) :: 0 and equals 0 ifu
Pi(l1)(T(l1) - T(Wi), 11) :; iP,(l1) for all 11. vEE. Summing over i yields
r:f Y,.
88
APPEi\DIX A
for all u, vEE. Setting T(u) == L,8,(11)T(w,), it is clear that T is locally Lipschitz continuous and satisfies (A.24). Moreover T is compact. Indeed let BeE be bounded. Without loss of generality, E == ER(O). Let u E BR(O). Then == L, j3,(u)T(w,). If u rf Y,. 8,(u) == 0 while if u E Y u E " and therefore Ilu < 1. Thus the nonzero contribution to T(a) comes from those terms in the sum where w, E E R + 1 · Consequently T(u) is a convex combination of terms in T(BR+1)' Since the convex hull of a compact set is compact, T is a compact mapping.
Appendix B
This final appendix treats the question of verifying in applications some of the hypotheses that are required in the abstract theorems presented earlier. In particular sufficient conditions will be given which insure that certain classes of functionals belong to C 1 (E, R). Furthermore the verification of the Palais-Smale condition will be simplified for these classes. We begin with a useful technical result. PROPOSITION B.!. Let 0 c Rn be a bounded domain and let 9 satisfy (gd 9 E C(O x R, R), and (g2) there are constants r, S :2: 1 and 01, a2 :2: such that
°
for all x E 0, ~ E R. Then the map
U(O),
In Ig(x, u(x))IS dx :s: In (a1 (B.2)
:s: a3
a2IuI T/ 5)8 dx
r (J + lun dx
JrJ
which shows that g: U(O) -; prO). To prove the continuity of this map, observe that it is continuous at
O. We claim there is a 8 > such that ilullu(o) :s: 8 implies Ilg(',u)IIL'(rJ) :s: E. By (gtl and g(x,O) = 0, given any f > 0, there is a 6> such that Ig(x, 01 :s: f if xC 0 and I~I :s: 6. Let u E U(O) with IluIIL'(O) :s: 8, 8 being free for now, and set
°
0 1 == {x
E
Ollu(x)1 :s: 6}
Therefore (B.3) 89
°
APPE\,DIX 13
90
where ill1 i dmotes the mca$ure of Ill, etc. Choo'iC t so that (El" This determines r Let O2 = n \ \/1. Then as in (13.2),
(134)
:S (0:/2)'.
I'dx
f,'
(D.5)
>
r
.In,
or 1\1 21 :S ('\3 1), Combining (134)(11.5) gives
r 19(x, l1(x)JI' deS :::: a .In,
(E.5) Choose {; so that a3(1
, (1 +
+ o-rw :S ([/2)5,
Thus (13.3) and (13.5) imply
119 (- , 11 ) I L' (n) :S [
if
iiull u (n) :S Ii
and the proof is complete. The next result is a standard version of the Sobolev Embedding Theorem. The space W~"(\l) was defined in Chapter 1 and its norm will be denoted by !:
n
PROPOSITIO'\ 13.7. Let
srnooth man1jolrL 1/71 E ~'V({,2 i8 a constant c > 0 SlLe h that
be a bounded domain m Rn whose boundary is a then 11 E L2n/n n 3) and there
(13.8)
u(n)
:S
ellull
for all t E [L2n(n - 2)-1] and for all u E w~·2(n). Moreover the embedding map w~·2(n) ~ £I(n), U --. 1L is compact for t E [1, 2n(n - 2)-1). PROOF. See e.g. [N! or F]. REylARK B.9. If n = 1 or 2, stronger statements can be made. In particular
(13.8) holds for all t E [1, Xl] for n = 1 while for n = 2 it is valid for all t E [1,(0) with c depending on t. "'ow a criterion can be given showing that an important class of functionals which arise in studying second order semilinear elliptic boundary value problems belong to C1(W~,2(n),R), (A related result is given in [BR].) PROPOSITION B.10. Let n be a bounded domain in Rn whose boundary is a smooth manifold. Let p satisfy (pd p(x.O E x R, R), and (P2) there are constants a 1, a2 > 0 such that
Ip(x, ~)i :S where 0 :S s < (n
+ 2) (n -
t
2
1
al
+ a2i~ls,
and n 2: 3 If
P(x,O
foE. p(x, t) dt
APPE:'-iDIX B
91
and
(B.ll)
(B.12)
I(v) ==
Inr (~
- P(x, v)) dx.
I'(v)
In
for all 'P E E == Wd,2(0). Moreover
J(U) ==
(13.13)
is weakly continuous and J' (u)
IS
in
P(x, u(x)) lix
compact.
PROOF. By Proposition B.7, (pd and (P2), I is defined on E and the same is true for I' (u) with the aid of the Holder inequality. It is clear that the first term in I is C 1 (even COO) and its Fn?chet derivative is the first term in I'. Hence we must show
J(u) ==
in
P(x,u(x))dx
belongs to Cl (E, R). This will be accomplished in two steps: First showing that J is Fnkhet differentiable on E (and therefore is Lipschitz continuous) and then proving that J'(u) is continuous. To begin, let u, 'P E E. We claim: given allY E > 0, there exists it [; ~ u) such that (B.14) provided that II'PII ::; 8. Therefore J is Frechet differentiable at u and J' is given by the appropriate term in (B.12). In fact the argument given below shows more: 8 = 8(s, Ilull) and hence J is uniformly differentiable on bounded sets in the sense of Krasnoselski [K I. Setting
w== IP(x, v(x) + 'P(x)) -
P(x, u(x)) - p(x, v(x))'P(x)l,
we have
J(u) - { p(x,u)'P
(B.15)
In
: ; in
W dx.
Define
0 1 == {x E 111lv(xlj 2: p},
O2 == {x
E
1111'P(x)1 2: 'I},
0 3 == {x E 111Iv(x)1 ::; p and 1'P(x)1 ::; 'I} with p and 'I free for the moment. Therefore (B.16)
in
W dx ::;
~ in. Wdx.
92
APPEND IX B
By the
~1ean
(B.17) where
\'alue Theorem. P(x. (+ 71) - P(x.
eE
(0.1). Therefore by (B.I7), (P2). and the Holder in quality.
f
Jo, (JUS)
0 = p(x. (+ e7lh
IP(x.u(x )
-I-
'P(x)) - P(x.l1(x))1 dx
f .In,
-I-
odll1(T)1
< ..
T
!'P(x)WI'P(J;)11 d;r
::; 0110 1
'(0)
+ 031 0 11 1/"[11111It,·q(0) + 11'Pllt'+'(o)III'PliLon;n-'(O). where 1
-+
(LU9)
.)
n- 2
+--=1.
2n O/),ern' that s < (71 -L 2)(- - 2)-1 implies that + 1)-1 + (n - 2)(2n)-1 < 1 ';0 there exists a (J > 1 satisfying (B.19) ,md justifying (E.IS). Combining (E.S) and (B. IS) shows (J
5
dx
iB.20)
+ Similarly
B)'
(B.8) and the Holder inequality,
(B.22)
11111: 2' 06Ijl1llu(fl) 2' 06111111u(Od
:::>
06 310 1 1/ 2 1
Therefore
where "'viI, "'vi2 - 0 as /3 -.,
00.
Combining (B.1S)-(B.23) yields
/' IJt dx ::; 07[kh -1- }vf 1(lluII S + il'PIIS)III'PII· .In, \\'r can assume 6 ::; 1. Further choose 3 so large that °+'11 + M 2 1(llull (13.21)
o,-':l. Hellcr (B.25)
f In,
IJtdJ;<~
-
3
S
+ 1)1 ::;
APPEND IX B
93
Similar estimate s to those above show (B.26)
r iii dx 5 a3 in, l [1 + (lu(x)1 + 1'P(x)WI'P(x)ll dx in2 5
a4
(l
2
+ (lu(x)1 + l
[1
J
, ( L,'p(x )
<: ILs(1 + Ilull' + Hpl/') where m = 2n(n - 2)-1 > s (B.27)
r iii dx 5
+L
~
lP(x,
wheneve r x E 0, I~i 5 and (B.2S) imply:
il,
&>
+ h) - P(x, 0 -
)1/8+1
d:r
S
,
1)
)
0)
0, there exists a ::.; =
Ohl
5 Eihi
and ihl 5 :y In particul ar if ;] == (3 and ~i 5
(B.29)
(B.S)
dx 5
Choose E so that 3a7E 5 E. This determin es (B.16), (B.25), (B.27), and (B,29) yields (B,30)
)"'-(S+I )
+ Ilul!S + II'PII
t\'ext since P E ['1 (0 x R, R), given any E, such that (B.2S)
C'C~X)l
Therefor e
a6,(s-t-l -m)!(s+I )(1
in,
"
1
11'PIIL'c) (il,)
10 iii dx 5 2; II'PII +
~i.
(1
Choose , =
+ HullS
Combini ng
I)
+llpi!S)
Finally choose 5 so small that (B.31)
a6,I-m/ (s+I)(2
+ Ilull s )5(m/(8+1))-1 5
~
thereby obtainin g (B,14), To prove that J' (u) is continuo us, let Urn -+ U in E. Then via (B,S). By the Holder inequali ty and (B.S) again, (B.32)
IIJ'(u m )
-
J'(ulll
sup Ii
II
r(p(x,urn(x))
in
11m. ~ 11
in
LS+ 1 (0)
p(x,'P(x )))p(x)d xi
I
ip(x, 01 5 al + a21~IC<8/Q for any or :2: 1 and all x E 0, ~ E R. Hence Proposit ion B,l implies p E C(L"8(0 ),L"(0)) , Choosin g or = (s + l)S-I, we see the right-ha nd side of (B,32) tends to 0 as m -+ 00 and J' is continuo us, Finally to prove that J is weakly continuo us, let Urn converge weakly to 11 in E. Then by Proposit ion B.7, Urn converges to U in 1'+1(0) since s+ 1 < 2n(n-2) -1
APPE:\DIX B
9-1
\'ia (P2). ('onsequenth' Proposition D.l implies J(l1m) ~ J(u). Since J is \I'eakly continml1ls and .J' is ulliformh' differentiable on bounded subsets of E. an abSlract theorelll X implies .11 i,.; compilrt. (Alternativply, let be boundr:d in E. Then illon:-; a subsequence. 11m COll\'ergcs wrakly to some 11 E E ilnd lim ~ 11 ill U' 1 (ll) via Proposition I3. 7. The proof then concludes via rB_32) illlClthe ,.;('Ilt<:llces which follow it.) RDIARK D.33. The ideas that were used in the proof of Proposition 8.10 call be emplo~'ed to prove related regularity results. E.?!;.: PHOPO"ITIOX D.34_ Let II
n
be a bounded domaIn In R n whose boundary
18
smooth manifold. Let p satisfy ())I) p E elm x R.R), and (P2) there aTC constants al.1l2 > 0 such that
wherr;()
.,«lIt2)(rr-2)-lllndrr2'3. If
P(x,O ==
!'~
t) dt
.0
and I(u) ==
l (~IVll,2
- P(r
11
dr
.0
then 1
('2(\l'(;2(1l).R). l'
r(u)(;:. v) all;. t'
1S
--c
(Jiven liy (I3.12), and
j' ('I;. '1t'-
io
,) dr
W~2(1l).
The next result concerns the verification of (PS). Recall that (P3) requires any sequence ) such that (i) l(11m) is bounded and (ii) 1'(11m) --+ 0 as m --+ 00 to be precompact. According to the following result in the setting of functionals like (B,l1), to get (PS), it suffices to show that (i) and (ii) imply (11m) is a bounded sequence. PROPOSITIOi\ B.35. Let p satisfy (pd-(P2) and let I be defined by (B.ll). If ) lS a bounded sequence in E == WJ2 (ll) such that I' ) ~ 0 as m ~ 00, then (u m ) has a convergent subsequence.
PROOF. Let D: E ~ E' denote the duality map between E and its duaL Then for /1, ;: E E.
(Du):p Thus (D.36)
Inj' VU' vrpdx,
APPE\;DlX B
9
Due to the form of this map, tlw conclusioll of the proposition follows if we ca show J'(urn) has a convergent subsequcllce. Indeed the continuity of D I an, (B.36) then imply
hm=D-IJ'
) + D- I JI(l1",)
-4
lim f)
I
the limit being t.aken along the convergent subsequencI' of )1(lIm)' But sinc (urn) is bounded in E and by Proposition B1O, JI is compact, J'(u m ) has; convergent subsequence and the proof is complerc. In Chapter 6 which studies the existence of periodic solnt ions of Hamilto nian systems, an analogue of Proposition B.lO is needed. The space E =: WI/2,2(Sl,R2n) is defined in Chapter 6, For this space we have PROPOSITION B. 3 7. Let H E Cl (R2n, R) and satisfy (B,38)
for some s E [1,00) and for all (E R2n. Then (B,39)
J(z)
fo21r H(z(t)) dt E CI(E, R)
with JI(Z)W = fo21r Hz(z(t))w(t) dt for wEE, Moreover JI(Z) is compact, PROOF, Given Proposition 6,6 and (B.38) the proof is almost identical te that of Proposition B.IO, We therefore omit the details (see also [BR]).
References
lAg] S. Agrnon. The LP approach to the Dirichlet problem, Ann. Scuola t\orrn . Pisa 13 (19.59). 405--448
.'-;IlP
[ALP] S. A.hmad. A. C. Lazcr and J. 1. Paul. Elementary critical point theory and pertmbatwns of elliptic boundary value problems at resonance. Indiana Univ. \!ath. J. 25 (EJ76). 933 944.
[Ami H..\manll. Saddle points and multiple solutions of differential equations. \!ath. Z. 196 (1979). 127 W6. ,AZ; H .. \llliUlll and E. Zehnder. ;Vuntrivwl solutions for a class of nonresoto non/mear equations. Alln. Scuola 7 (1980). :)39603.
nance
'A: .\ ..\mbrosetti. On the existence of multiple solutions for a class of nonlin(;!lr boundary wive problem,. Rend. Sem. \lat. Univ. Padova 49 (1973), ID5 204.
,AMI, A. Alllbro"c;tti and G. \laneini. On a theorem by Ekeland and Lasry Hamiltonian J. Differential Equathe number of tions 43 (1981). 1 6.
'AM2i _______ . SolutIOns of mmlmal period for a class of convex Hamiltonian ,'l/otems. \lath. Ann. 255 (1981). 405421. ;ARj A. :\mbrosetti and P. H. Rabinowitz, Dual vanatwnal methods m critlUll pomt theory and applications. J. Funet. Anal. 14 (1973). 349-381.
iBa] A. Bahri. TopologIcal results on a certain class of functionals and applimtwns. J. Funcl. Anal. 41 (1981). 397-427. BB] .-\. Bahri and H. Bcrcstych A perturbation method in crzh:cal point thwry and !1ppl1r:ations. Trans. Amcr. \lath. Soc. 267 (1981), 132. 'BLi A. Bahri and r. L. Liolls. Remarks on the variational theory of critical C. R. .\cad. Sci. Paris SCI'. I \!ath. 301 (1985). and 145 1<18. iBe1] Y. Bellci. Some entleal point theorems and applications. Comm. Pure \ppl. \lath 33 (1980). 147 172. :Be21 . On cntzmi p01nt theory jrlr fImciwnals of symmctnes. Trans. Arner. \lath. Soc. 274 (1982). 533-572 96
In
the presence
HEFEIlE :'\CES
97
[Be3] _._. A geometrzcal index Jor the group S 1 and some to the research oj periodic 80lutions oj o.D.E. 's, Comm. Pure Appl. Math. 34
(1981),3 93432. [BF] V. Benci and D. Fortuna to. The dMI method in critical theory: Multiplicity results Jor indefinite Junctionals, Anll. ~lat Pura Appl. (4) 32 (1982). 215 -242. ~BR] V. Benci and P. H. Rabinow itz. Cn/7cul point theorems Jar Junetionals, Invent. !vIath. 52 (1979), 241-27:1. [BLMR J H. Bercstycki. J. M. Lasry, G. Mancini and B. Hul', EXlstcnce oj
multiple periodic orbits on star-shaped Hamiltonian surJaces. Comm. Pure Appl. Math. 38 (1985), 253290. [Bg] M. S. Berger. Nonlinearity and Junctional analysls, Academic Press, I\ew York, 1978. [BW] M. S. Berger and M. S. Berger, Perspectives in nonlmearity. Benjami n, New York, 1968. [Bi] G. D. Birkhoff, Dynamical systems with two degrees oJ Jreedom. Trans. Amer. Ylath. Soc. 18 (1917).1 99 300. [Bo] H. Bohme, Die Losung der Verzwergungsgleichungen nichtlineare Eigenwertprobleme. Math. Z. 127 (1972). 105--126. iBN] H. Brezis and L. Nirenberg, Positive solutions oj nonlinear elliptic eqMt10ns involving critical Sobolev exponents, Comm. Pure Appi. Math. 36 (1983), 437-477. [Brl] F. E. Browder, Infinite dimensional manifolds and nonlinear eigenvalue problems, Ann. of Math. (2) 82 (1965), 459-477. [Br2J __ , Nonlinear eigenvalues and group in variance, Function al Analysis and Related Fields (F. E. Browder, ed.), Springer-Verlag, Berlin and New York, 1970, pp. I-58. [CLJ A. Castro and A. C. Lazer, Applications oj a maximin principle , Rev. Colombi ana Mat. 10 (1976), 141-149. ICe] G. Cerami, Un criterio di esistenza per i punti critici 8U varieta illimitate, Rend. Acad. Sci. Let. 1st. Lombardo 112 (1978), 332-336. [ChI] K. C. Chang, Variational methods for nondifferentiable functiona is their applications to partial differential equatioTl8, J. Math. Anal. Appt
(1981),1 02-129. [Ch2] __ , Morse theory on Banach space and its applicatwT18 to differential equations, preprint. . [e] D. C. Clark, A variant of the Ljusternik-Schnirelmann theory, Univ. Math. ]. 22 (1972),6 5-74. [Col] C. V. Coffman, A minimum-maximum principle JOT a class of eqllation3. J Analyse Math. 22 (1969),391-419. [Co21 --, On a class oj nonlinear elliptic boundary value problems , Mech. 19 (1970), 351-356.
•
9x
HEfEl1E;,\('ES
[CC] C. C. Conley. Isolated l7lVUTwnt sets and the Morse l:nrlex. CB:vlS Regional Conf. Set'. 111 :Vlath .. no. :38. Amer. :Vlath. Soc" Providence, R.. 1.. Ill78. rCF] P. E. Conner and E. E. Floyd. Fixed point free invollLtions and eqlLivuriant maps, Bull. Amer. \lath. Soc. 66 (1966), 416441. [CH: R. Courant anc! D. Hilbert. Methods of mathemcdlwl physics. Vols. I ilnd II. Intf'rscic:rlce. :\ew York. 1953 and 1962. fCR. :V!. C. Crandall and P. H. Rabinowitz. Continuation and variatlonal
methods for the existence of positwe 3011Ltions of nonlmecLT ellIptic c1genval1U: problems. Arch. Rat. :Vlech. Anal. 58 (1975), 201218. [De] S. Deng, A17n·imal JieT1:odic soilltions for a class of H(Lmiltonian eqllations. China University of Science and Technology, preprint. fD1] J. Dieudorme. Foundations of modern analysis, Academic Press. New York, 1960. [DL] G. C. Dong and S. Li, On the existence of infinitely many sollLtions of the Dirichlet problem for some nonl1:near elliptic equations, Sci. Sinica Ser. A 25 (1982).468475 [EH! I. Ekchnd and H. Hofer, I'enodic solut'ions with prescribed rm:nimal pcrwd for convex outonom0118 Hamiltonian systems, Invent. :vlach. 81 (1985). 155 188. [EL] 1. Ekeland and J.-\1. Lasry, On the number of peTlodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. (2) 112 (1980), 283~319.
[FH) E. R. Fadel! and S. Husseini, Relative cohomological index theories, Advances in :Vlath. (to appear). [FHR] E. R. FadelL S. Husseini and P. H. Rabinowitz, Borsuk- Ulam theorems for arbitrary 51 actions and applications, Trans. Amer. Math. Soc. 274 (1982), :345-360. [FR1] E. R. Fadel! and P. H. Rabinowitz, Bifurcation for odd potential operators and an alternatlve topological index, J. Funct. Anal. 26 (1977), 48-67. [FR2] __ , Generalized cohomological index theories for Lie group actions with applicat7:ons to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139-174. [Fr] A. Friedman, Partial differential equations, Holt, Rinehart, and Winston, Inc., New York, 1969. [G] J. V. A. Goncalves, A multiplicity result for a semilinear Dirichlet problem, Houston J. Math. (to appear). [Hel] J. A. Hempel, Superlinear variational boundary value problems and nonuniqueness, thesis, University of New England, Australia, 1970.
[He2] __ , Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1971),983-996. [HoI] H. Hofer, On strongly indefinite functionals with applications, Trans. Amer. Math. Soc. 275 (1983), 185-214.
REFERENCES
[Ho2] __ , A geometric description of t)u: neighborhood of a critu:a/ poin given by the Mountain Pass Theorem, j. London Math. Soc. (2) 31 (1985) 566-570. [K] M. A. Krasnoselski, Topological methods in the theory of nonlmenr tntegra equations, Macmillan, New York, 1964. [LJS] J. Leray and J. Schauder, Topologie ct equations fonctionellcs, Ann Sci. Ecole Norm. Sup. (3) 51 (1934), 45-78.
[LLS] 1. Ljllsternik and 1. Schnirelmann, Methodes topoiogique dans ies prob itimes variationnels, Hermann and Cie, Paris. 1934. [Ma] A. Marino, La biforcazzone ncl caso variazr.onale. Confer. Sem. ;VI at Univ. Bari 132 (1977). [Mi] J. Milnor, Morse theory, Princeton Univ. Press, Princeton, \'. ,1., 1963 [Ni] W. M. Ni, Some minimax principles and their applications in nonlinea elliptic equations, J. Analyse Math. 37 (1980), 248~275. [Nl] L. Nirenberg, On elliptic partial differential equations, Ann. Scuo!. Norm. Sup. Pisa (3) 13 (1959), 148. [N2] __ , Variational ILnd topolo(Jieal methods in nonlinear problems, Bull Amer. Math. Soc. (N.S.) 4 (1981), 267 :)02,
iP 1] R. S. Palais, Lusternik-8chrnrelma.nn theory lin Banach manifolds, Topol ogy 5 (1966), 115·132. [P2] __ , Critr:cai point theory and the minimax principle, Proc. Sympoo Pure Math., vol 15, Amer. Math, Soc., Providence, R, I., 1970, pp. 185·212. [Po] S, L Pohozaev, Eigenfunctions of the equation ::"u Math. 5 (1965), 1408~'141l,
+ Af( 11)
= 0, SoviE
[PSI p, Pucci and ,J, Serrin, Extensions of the mountain pass theorem, Unil of Minnesota Math. Rep., 83-150 (to appear).
[Rl] p, H. Rabinowitz, A note on nonlinear eigenvalue problems for a cla; of differential equations, J. Differential Equations 9 (1971), 536-548, [R2] __ , Variational methods for nonlinear eigenvalue problems, Eigenva ues of Nonlinear Problems (G, Prodi, ed.), C.I.M.E" Edizioni Cremonese, Rom< 1975, pp, 141·195.
IR3] __ , A bifurcation theorem for potential operators, J. Funct. Anal. 2 J1977),412-424,
[R4] __ , Some minimax theorem8 and applications to nonlinear partial d7 ferential equation8, Nonlinear Analysis: A collection of papers in honor of Eric Riithe, Academic Press, New York, 1978, pp. 161·177.
[R5] __ , Some critical point theorems and applications to semilinear ellipt partial differential equatzons, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978 215-223, [R6] __ , Periodic solutions of Hamiltonian systems, Comm. Pme Apr Math, 31 (1978), 157--184,
REFERE:-JCES
lR7j ~_. A variational method for findmg period~c solutions of differential nations, Nonlinear Evolution Equations (M. G. Crandall, Academic Press. 'w York, 1978. pp. 225-251.
iR8] _._. __ . Multiple critical points of perturbed symmetric functwnals, Trans. ncr. ~Iath. Soc. 272 (1982),753-770. [R9] ~_, Periodic solutions of large norm of Hamiltonian systems, J. Difrntial Equations 50 (1983), 33-48. :Sl] J. T. Schwartz, Generalizing the Lusterm:k-Schnirelmann theory of critical ints, Comm. Pure App!. Math. 17 (1964), 307-315. [S2] , Nonlinear functional analysis, Gordon & Breach, ;'\cw York, 1969. iSm] J. Smoller, Shock waves and reaction-diffusion equations, Springerrlag, New York, 1982. 'St!; .\1. Struwe. Infinitely many critical points for janctionais which are t even and applications to superlinear boundary value problems, '\lanuscripta fttL. 32 (1980). 33,5 364 St2j ~_. A note on a result of Ambrosetti and Mancm~, Ann. !\lat. Pura JpL 131 (1982). 107115. :Vai M. M. Vainberg, Variational methods for the study of nonlinear operators. lIden-Day, San Francisco, 1964. VGI E. 'vii. C. van Groesen. Existence of multiple normal mode trajectories of en classl:cal Ham7ltonian systems, J. Differential Equations 57 (1985), 70-89. '\V! A. \Veinstein. Periodic orbits for wnvex l1amiltonian systems, Ann. of ith. (2) 108 (1978). 507 518. [Wh] G. T. V,,"hyburn. Topological analysis, Princeton '\Ialh. Ser., No. 23, inCPiOn lini\'. Press. Princeton. N. ,J., 1958. ,Y; C. T. Yang. On the theorem" of Borsuk-Ulam, Kakutam- Yamabe- YuJobo, rI Dys11l. I and II. Ann. of '\1ath. (2) 60 (1954). 262·282 and 62 (1955). 1180.
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