Infinite dimensional Morse theory and multiple solution problems

i.e., 8, and aD link (homotopically). Since J(x) 2: c +. V x E 8" we conclude that c = inf sup J 0
zeD

:r:eSl

This is a contradiction. The above minimax principle (1.4) under condition (1.6) was obtained by Benci-Rabinowitz IBeRtl, and Ni INil]. The condition (1.6) was weakened to condition (Ll) by Chang IChaS], and to condition (1.7) for k = t by G. J. Qi [Qil]. ActuaUy, under a more general linking definition (aD may

89

I. Topological Linlc

not be compact), the above conclusion remains true, cf. Y. H. Du [Dull and N. Ghoussoub IGhol). Theorem 1.5. Assume that '" E H.(I., f.) is nontrivial, and c" = inf sup f(x).

(1.8)

ZEozElzl

SupJX16e tbat f satisfies the (PS)e" condition and tbat Ke" is isolated. Then there exists Xo EKe' such that C.(I,xo) "10. Proof. It suffices to prove that H.(le'.!" \Ke.) "lOusing Theorem 4.2 in Chapter 1. V E > 03 Z E '" with support IZI c fe'+<' Let U = 8Z be the boundary

of Z; then lui c f. c fe" We shall prove that [U) E H._I(I,.\Ke.) is nontrivial. Indeed, if [u] is trivial, then there exists a singular k-chain T such that Irl C fe' \Ke• and [Or] = [u). Due to the second deformation theorem, 3 'I E C(M, M), satisfying 'I - id and 'I(le' \Kc ') C fe'-' for some E > O. Let rl = 'I(r), we have [TI] = [T) and ITII C fe'-" which is in contradiction with the definition of c· . We observe the exact sequence

-

H. (I"+o.!e' \Ke·)

!!.. H'_I (I,. \Ke·) ..!!.. H._I (Ie' H) -

...

where i : fe' \Ke• - fe"+. is the inclusion. Since i.([Y]) = 0, i.e., [y] E ked. = 1m 8., we conclude

H. (Ie" fe' \Ke·) ~ H. (le'+" fe' \Ke·)

"I O.

Corollary 1.1. Under the assumptions of Theorem 1.5, if, in addition, all critical points in K c. are assumed to be Dondegenerate, then there exists Xo EKe' such that ind(l,xol = k.

Theorem 1.5 can also be extended to G·invariant functions. Let G be a compact Lie group, and let M be a C'·Hilbert Riemannian G-manifold. Suppose that f E Cl(M,II.I) is G invariant, and that a, b are regular values of f. Assume that [Z] E Ht;(I•• f.) is nontrivial. One defines c = inf sup f(x). ZElzl.Elzl

By the same proof, we have Corollary 1.2. Suppose that f satisfies (PS}e and that Ke consists of isolated critical orbits. Then there exists a critical orbit 0 C Ke such that

ct;(g,O)

"I O.

90

Critical Point Theo'71

As 8 consequence, we get an estimate of the Morse index for certain critical orbits in Kc.

Corollary 1.3. Under the 8SSumptions of Corollary 1.2, we 8SSume further that! E C', d' !(x) is a ftedholm operator \I x EM, and that Kc n Fixa = 0. If H"(Ba.,e-) = 0 for * # 0, \I x ¢ Fixe, where 8is the orientation bundle of {-(Gx), then there is a critical orbit a c Kc satisfying m:<S;k:<S;m+n

where m = ind(J,O) and n = dim~O(O).

Proof. According to Theorem 7.8 in Chapter I, we consider the approximation function g, which has only nondegenerate critical orbits. Let Kc = {ai, 0., ... .Ot}, and let m, = ind (J,O,), 11; = dim~(O,). Then 9 bas critical orbits 0'8 with ind(g, Oi.) = qi. E Imi' mi + nil, i = 1,2, ... ,l, s = 1,2, ... ,ii' One concludes that there exists at least a pair (i, 8) such that C~' (g, a,.) '" O. Indeed t

ii

EBEBCe(g,O,.) = He (gc+£,gc-.) = He (Jc+<.!c-.), ;=1 8=1

so there exists a pair (i,8) such that CMg, a,.) F o. According to Theorem 7.5 in Chapter I, C~(g,O,.) = H~-"·(O, .. O-(O,.» = Hk-q"(Ba •• ,O-) where Xo E A,•. By the assumption that H"(Ba.,O-) = o for * # 0 \I x ¢ Fixa; it follows k = Therefore

q,..

m;

:<S;

k

:<S;

m, + ",.

This is our conclusion.

Let us return to Example I, in this case c

= c" = inf sup !(x), ler :tElIO,I}

where r = {f E C(IO, l),X) Il(i) = z" i = 0, l}. According to Theorem 1.5, there exists a point %0 E Kc such that (1.9)

CI (J, xo) '" O.

Due to the geometric meaning, we call a point xo satisfying (1.9) a mountain pass point. The following theorem provides more precise information on mountain pass points.

91

I. Topological Link

Theorem 1.0. Suppose that I E C'(M. R') and that Xo is a mountain pass point. Assume that rP I(xo) is a Fredholm operator and that (1.10)

dimker(rP/(xo)) = 1

il OEI7(d'/(xo ))

then rank C. (I. xo) = 6.,. Proof. Let k = ind (I. xo). 1. If Xo is nondegeoerate. then C.(I.xo) = 6•• G. From (1.9). it follows k = 1. and then C.(/.xo) = 6.,G. 2. If Xo is degenerate, then we have

C. (I, xo) = C._. (i,0)

V q.

where j = liN. and N is the characteristic manifold of I. according to the shifting theorem. From (1.9). it follows that k S 1.

ClJ$e a. If k = 1, then Co(j,8) isolated local minimum of j; then

#

0, which is equivalent to 0 being an

Thus

c. (1.:&0) = 6.,G. ClJ$e b. If k = O. then

C. (I.xo) =

c. (i. 0) •

V q.

By assumption (1.10). the function j is defined on a 1-manifold. And the only poosibility for C, (j.O) # 0 is the case in which Xo is a local maximum of j. Thus

C, (I.xo) = C.

(i. 0) = 6.,G.

The proof is completed.

Remark 1.3. The condition (1.10) is essential. For example. the Monkey saddle is a mountain pass point, with critical groups Cq = Oql G E9 G. Vq EN. Remark1.4. The idea of the homological link was introduced by J. Q. Liu (Liu1]. from which he proved Theorem 1.5. C. Viterbo independently gave an estimate of the Morse indices for critical points defined by equivariant homology classes. i.e.• Corollary 1.3. cr. ]Vitl]. Theorem 1.2 is more or less related to V. Benci (Ben2] and X. F. Yang [Yan1].

92

Criliccl Point Theo'1l

2. Morse Indices or Minimax Critical Points Minimax critical points, by definition, are independent of Morse theory. However, the Morse indices of these critical points play a role 85 a connec-

tion. We shall pay more attention to the estimate of the Morse indices of critical points as determined by some standard minimax methods. All notation appearing in Section 1 will be nsed without change. 2.1. Link Theorem 2.1. Suppose that' E C·(m, lltl) satisJies the (PS)c condition, where c > a is defined in (1.4). Assume that Kc = {XI,X., ... ,x,} with Morse indices {k\, k., . .. ,k,lo and that 0 is not an accumulate point of u(d" I(x.)) whenever 0 E u(d" I(x.)), i = 1,2, ... ,t. Then (2.1)

Min{k;

Ii = 1,2, ...

,f):;; k

Proor. We prove (2.1) by contradiction. Suppose that k < Min {k; 11 :;; i :;; f). According to the splitting theorem, V Xi, there exist 8 neighborhood Nil a homeomorphism c}i : B;' x B6 -+ N i , where B;- and B'6 are r-ba.ll and o-ball in Hi- and H. where H. = ker(d" I(x.)) respectively, and r,6 > 0, such that in each Nil the function J is written in the form

I(x)

= c + IIv+ 112 - lIy_1I2 + j.(Yo),

where x = ~.(y), y = y+ + y_ + Yo, y.l = (y+,y_) E . 1 l ,t. One may assume that f;- 180 :;; .r .

, = 1.2, ...

B;

and Yo E B.,

•

I.), such that IZI c le+ •. Using the deformation theorem, we have ZI E 0 satisfying IZII C IC-'I u U:=I N. \I < > 0, 3 Z E 0, where" E ".(lb,

for some £1 > O. One may assume that £} < ~r2. In order to simplify the notation, we omit the subscripts i l and the transformations I)i , and regard Ni as B; x B'6, if there is no confusion. Let g(y.l) = lIy+11 2 - lIy_1I2; then \I Yo E B~, the sections Ie 1'0='0 9-i(yo)' and

le-'I II/O=YO

=

= 9-. I -i(yo)"

Let B:; C B; be the r-baJJ {y.l = (y+, y_) E B; I y+ = O}. We shall prove that there exists Z2 E 0

satisfying

IZ21 c

('e-£1 \~ N.) u ~~. (Q;)

where Q. = B~' x {O+} )( (B. n Hd, and B~; is the k.-dimensionaJ r.ball, i = 1,2, ... ,to Indeed, let us denote a =

-
j (Yo),

b=

~ ( -I Max{

-0,

O}

+

t2;

a) ,

93

2. MONe Indi= of Mini""", Crilirol Point.<

'P(8, b) = where ctg Q = J2(r' tions as follows:

{

o

ifO<.
(s _ b)ctg Q

if b <.:5

+ a)/(

Jr'••

,;r;-. - JM",,{ -a, O}). We define deformaif y E 1<-., \G

,/,(t,y) = { y

[( )] y_ + Yo + 1 +t Tt:!'P(ly-l, b) - 1

where G = {y E

y+

if y E G,

B; x B.lly+l;:: ",(Iy-I,b)}, and y

Lf.=, N, t)ly_l) f.:J

if y E 1<-., \ y+

+ Yo + (tJr' -ly+I' + (1 if y E

7)2(t, y) = y+

+.~ ~u

+'

(B; x BE) \G

r'-I.+I,+(,-·) .. O.+I.·) 'I/t~+1

if where ,p(', b) = tg" . 8 + b, t E [0,1). The composition of 7)2 and '/, yields Z.

y-

II E G, with

11/+ I :5

Jr'r

= (7)20,/,)(1, Z,), which satisfies

This means that there exists a continuous map q : Die - E such that Z. = u(Dk). One may modify u to be C' on u-'.,(Q,) by the partition of unity because Qi is convex, i = 1,2, ... ,t. Since we have assumed that k < mink" the projection ,,;)Z.I of IZ.I k·

"1

Ok·

onto Br' cannot cover Br . There must be at least one point Pi E Br' missing. Again, we project ",)Z.I onto lIB:' from Pt, 1 :5 i :5 t. Then IZ21 is deformed into 1<-. .,(8B:< x {8+} x (B.nH,», which gives Za E a such that sup I(x) < c.

uu:=,

z€!Zal

This is a contradiction. By the same proof, we have

Theorem 2.1'. Let M be a Hilbert-Riemannian manifold. Suppose that 1 E C'(M,III.') satisfies the (PS) condition in r'[a,b), and that

94

Crilirol Point Th
Knrl[a,b) = {XI,X" ... ,Xt}, with I<; = ind(f,x,) anda< f(x,)
'Irk

(I., f.)

=0

f(lf" k < Min {kd i

= 1,2, ... ,i}.

There is a dual way of defining critical values for Example 2. Instead of the set family :F = {",(D) I", E C(D,X) I", IOD = idOD}, we introduce a new family

!i = Since F

I

compact, VuE C(F, X tl with" IFr10D = idFnllD, we have 0 E o(F)}.

{F C X

c !i (using Brouwer degree theory), !i is nonempty. Let us define

c=

(2.3)

inf sup f(x) FeJ:zeF

then c ~ c. We conclude c ~ d if (1.6) holds. In fact, by taking u equal to th~ projection onto X .. FnX2 '" 0 V F E!i, which implies sUPzePf(x) ~ infi.x, f(x) ~ d. If, further, we assume (1.2) and that f E CI(X,JRI) satisfies the (PS). condition, then

c is again 8. critical value.

Claim. By choosing eo E (O,d - a), it is not difficult to verify that the family!i is invariant under the class of maps 4>'0 (cf. Theorem 1.3). The conclusion follows from Theorem 1.3. In order to obtain a counterpart of Theorem 2.1 for (:, we need

Lemma 2.1. Let A with k > n, such that 0 ii E C(Rn,JRk\{o}).

c JRn be a compact sel, and let" E C(A,JR k ) tf. utA). Then a can be extended continuously to

Proof. Since A i. compact and 0 tf. ,,(A), there exists & > 0 such that a(A)nB,(O) = 0. According to Tietze's theorem, we have a continuous extension U 0(0, fi: lRn _ IRk. Moreover, we may assume 0 E G 1 (Ul"\A,)Ik). By the assumption k > n, it follows from Sard's theorem that b(JRn\A) is measure zero in IIt.k. Therefore Bc(8) is not included in n(Rn), i.e., there exists Xo E B.(8)\b(Rn). We project b(Rn) n B,(8) onto 8B,(8) from Xo. This gives the extension U.

Theorem 2.2. Let X be a Hilbert space with the direct sum decomposition X = XIEllX2, dirnX I < +00. Assume that f E C'(X,JRI) satisfies the (PS). condition, where is defined in (2.3), and that (1.2) and (1.4) hold. Suppose that K. = {XI,X" ... ,xt} with Morse indices {k .. k" ... ,k,} and

c

95

2. MONe Indicu 0/ Minimu Critical Point.

nu1lities {n" n" ... ,nl} respectively, and that,p I(:z;;), i = 1,2, ... Fredholm operators. Then Max {k;

+ n; 11 ::; i

::; I}

,t, are

> dim X •.

Proof. 'if E > 0, 3 F e F such that F C IH<' Applying the first deformation theorem, the splitting lemlIl8, and the argument in Theorem 2.1, tbere is a defonnation "aatisfying ,,180 = idao and

F.

On the one hand, we bave '1( F) e

On the otber hand, let

A= (,,(F)\ UN;) U U+, (8B~

.

1"",1

X

{II+} x (B. n

H;»).

.=1

~

We bave A C I" therefore A ¢:F. Consequently, 3 tr: A - X.\{II} such that tr 180nA = idaonA. Suppose that dimX. > Max {k;+n, 11 ::;i::;t}; then the map tr lul~. +;(8B:: x {I/+} x (B.nH;)) bas .. continuous extension I

ii :

U +; (Q,) -

X. \{II}

£=1

according to Lemma 2.1. Let us define

vxeA V:z; e '1(F)\A.

"( )={tr(:z;) tr :z; ii(:z;)

Then ii': '1(F) - X.\{II} is continuous. If we cboose r > 0 so small such that II.(F)\A

> a,

1::; i ::; t,

then

;, 1.(F)nBO =;,

IAnBO

=

= id,,(F)n8D,

trlAnBO

= idAnBO

96

Crilical Poinl TheorJ/

which implies that ~(F) ¢

F.

This is a contradiction.

2.2. Genus and Cogenus Let X be a Banach space. Let E be the family of compact symmetric subsets of X. Two integer valued functions -y+. -y- : E -+ N U {+oo} are defined as follows: -y+(A) =sup{n E N 13'1':

sn-l

-+

A

odd and continuous}

and -y-(A) = inf {n E N 13 'I' : A

-+

sn- 1

odd and continuous} •

in which, if no such 'I' exists. then we define -y±(A) = +00. -y+ (A) is called the genus of A and -y- (A) is called the cogenus of A. We have the following properties: (1) 'I 'I' : X -+ X odd and continuous,

(2.-4) (2) -y+(A) $ -y-(A). Claim. If not, 3 A E E such that k = -y+(A) > -y-(A) = m, then there exist continuous odd maps ~I and 1.p2 satisfying

According to the Borsuk-Ulam theorem, it is impossible.

Let us define two classes of families of subsets in E: 'I kEN.

Set (2.5)

c~ =

inf sup I(x)

'I kEN.

AEF;%EA

I • = + or -. k = 0,1.2, ... } is finite. and if IE C'(X.IR') is odd and satisfies (PS)e, then c is a critical value of I. according to Theorem

If c E {Ck

1.3. The following relations are obvious: k = 0,1,2, ... ,

and

2. Morse /ndice6

0/ Minima.r

97

Critical Point8

• = ±, k = 0,1,2, ...

.

All a counterpart of the theorems in the previous subsection, we have

Theorem 2.3. Let X be a Hilbert space. and let c be one of the = ±. k = 0.1.2 •...• de6ned in (2.5). Assume that f E C"(X.IR') satiBn.. the (PS). condition. Suppose that K, = {±x" ±x••... • ±xt} with Morse indices {k"k••...• k,} and nullities {n"n ••...• n,} respec· tively. and that 0 ¢ K,. If.p f(x,). ; = 1.2•... • l are Fredholm operators. then

c.. .

Max {k,

+ '" 11

:<; ; :<; l} ~ k - 1

as

* =-,

and

Min{k; 11 :<; i:<; l}:<; k-l

as. =

+.

Proof. We prove them by contradiction. Assume that k> Max{kd n, 11 :<; i:<; t)

+1

as

* =-,

as

* = +.

and k < Min {k, 11 :<; i :<; I} V E > O. 3 Ao E :Fk such that Ao C we have a deformation fJ satisfying

+1

f ...,. As in the proof of Theorem 2.2.

."

where Q, = B.' x {O .. } X (B6 n H,) and r > 0 is small enough such that N, n (-N;) = 0, 1 :<; ; :<; t. Thus '1(Ao) c :Fk and let

,

u

"

0

U (H, (&B~: x {O+} x (B,nH,»)) c f,· i=l

For

* A

= ~

-. "Y-(A) ~ k - I, i.e., there exists an odd continuous map

S·-2. By the Dugundji Theorem, we may extend

98

CritiaJI Point Th.o'1l

'~llI.k+n' Bk-' t' I to 'PI: Ui=l I I -+ con muollS y, where Bk-' is the unit ball in k - I-dimensional Euclidean space. Let

I{J

IU~""I±.;;(8B:tX{9+»)(B.nH;) .p(x) =

~ ('I',(x) -

'I',(-x».

Then.p is an odd extension of 'I' lut•• (±~, (oB:: x {9+} x (B.nH,»)) with r.p : ULIlR.kj+ni -+ Bit-I. By the use of the same argument as in o

Lemma 2.1, we have a pair of points ±p, e Bk-'\.p(llI.k,+n,). LetT,± be the projection ontooBk-' from±p,; thenT,±: Bk-'\{±p,} ~ 8 k- 2 is continuous, and I;-(X) = -Tt( -x), i = 1,2, ... ,I.. Define _ { 'I'(x) ± _( ) 'I'(x) = 'i

xeA x

e (±N,) n 1/ (Ao).

Then.p : 1/(Ao) ~ Sk-2 is odd and continuous. Therefore 1'-(1/(Ao» ::; k - 1. This is impossible. For * = +. We have 1'+(A) ::; k - 1 and 1'+(1/(Ao» ~ k. From the latter, there exists '1'0 : Sk-' .... 1/( Ao) odd and continuous. Denoted by Pi~' * = +. -,0, the orthogonal projections onto the positive, negative and (}.subspaces of rP I(x,) respectively, we conclude that 1',-'1'0 (Sk-' ) cannot cover B:i, because k -1 < Min {ki 11 :5 i :5 i}, l = 1,2, ... ,t. Thus,

3p, e B~' such that {±p,} x (P'++P!')Xn'l'o(Sk-')=0,i=1,2, ... ,f. Let T,+ : B:'\{p,} .... oB:' be the projection onto the sphere from p" let T,-(Z) = -Tt(Z), and let

T,(X) = {T,±(P,-x)+(1',++P!')x ifxe±(B:'\{p,}) x ((P'++P!')X) x

ifx,! ±B:' x

((p,+ + P!') X) ,

= 1,2, ... ,t. The functions T,: X\U~~,({±p,} x (P,+ + P!')X) .... X, i = 1,2, ... ,f are odd and continuous. Therefore 'i 0··· 0'1 0 'Po : 8 k - 1 --I' A is odd and continuous which implies that 1'+(A) ~ k. Again it is imp05Sihle. for i

Remark 2.1. The same conclusion holds if Ie C'(S,llI.'), where S is the unit sphere of a Hilbert space, and if;::t are modified to be families of

subsets 00 S. Remark 2.2. Theorems 2.1 and 2.2 were obtained by Lazer-Solimini [LaS 1), and Theorem 2.1' improves a result due to J. T. Schwartz [ScJI). Lemma 2.1 can be found in L. Nirenberg [Nir1). Theorem 2.3, under the assumption that! satisfies the Palais-Smale condition (not only (PS)c). was

given by A. Baltri [Bah1) for genus, and A. Bahri, P. L. Lions IBaL1) for cogenus. For some extensions, see for instance, Coffman ICofl,2}, S. Solimini

[8012) and N. Ghoussoub [Gho1).

3. Connection.! with Other Theories

99

3. Connections with Other Theories 3.1. Degree Theory Let M and N be oriented n-dimensional manifolds without boundary and let be a smooth map. If M is compact and N is connected, then the Brouwer degree deg(T) is well-defined [Nir 1). First consider an open set U C

]RA I

and a smooth vector field

with isolated zero at the point p E U. The function

Vex) Vex) = lIV(x)1I maps a small sphere centered at p into the unit sphere 8.- 1 • The degree of V is called the index of V at the zero p, denoted by the index (V, p). For a smooth vector field V on an arbitrary manifold M (finite dimensional) with isolated zero at the point p E M, the index of V at the zero p is defined as equal to the index of the corresponding vector field dg- 0 Vog ' at zero 9- 1 (p), where 9 : U - M is a parametrization of a neighborhood ofpinM. The celebrated Poincare-Hopf Theorem studies the relationship between the indices of zeros of a smooth vector field on M and the Euler characteristic of the manifold M. Namely, Theorem 3.1. (Poincare-Hopf). If Mm is a compact manifold with boundary 8M, and if V is a smooth vector field on M which points outward at all boundary points and has only isolated zeros, then the sum of the indices at the zeros Pi, i = 1, ... ,n, of such a vector field is equal to the Euler characteristic of M, i.e., •

L i=l

m

index (V,p;) = X(M) ~ L(-l)· rank H.(M). q=1

We shall extend this theorem to bounded domains on an infinite dimensional Hilbert space for a special case of vector fields, the gradient vector

fields. Let H be a real Hilbert space, and let D be a bounded open set in H. For a vector field V : D ~ H, with 9 ~ V(8D) and", = id - V compact, the Leray-Schauder degree

deg(V,D,8)

100

Crilical Poinl Theory

is well-defined, cf. (Llo1]. Suppose that P is an isolated zero of V; then there exists a ball B, CPo) with radius £ > 0 such that V has no zeros in B, (P) other than p. Hence it is possible to define index (V,p) = deg (V, B,(P), 9) which is independent of £ for £ > 0 sufficiently small in view of the excision property of the Leray-Scbauder degree. The number index(V, p) is called the index of V at the point p. In case dim H < +00, the definition of index(V, p) coincides with tbe previous one. We start with a local result in which tbe connection between the index of the gradient vector field of a function / at its isolated critical point and the critical groups of that point is studied; namely: Theorem 3.2. Let H be a real Hilbert space, and let / E C2(H,ItI.') be a function satisfying the (PS) condition. Assume that

d/(x) = x - T(x), where T is a compact mapping, and that Po is an isolated. critical point of

j. Then we have 00

(3.1)

ind (d/,Po) = L(-l)' rankC,(I,Po)'

,=0

Proof. 1. First, we assume that Po is nondegenerate. Since T is com-

pact, we see that the Hessian

dO /(Po) = id - dTCPo) has only finite index j. By definition, and by Leray's formula index (d/,Po)

= (-1);.

In view of Theorem 4.1 in Chapter I, we have

c. (I,Po) = 6"p. Thus, 00

index (d/,Po) = L(-l)' rank C. (I,Po) q=O

is proved in this special case. 2. For an isolated degenerate Po we may assume for simplicity that Po = 9 and /(1'0) = O. Let (W, W_) be a Gromoll-Meyer pair constructed

101

3. Connections with Other Theoriu

in Chapter I, (5.11) and (5.12), and let 6 > 0 be sufficiently sma.U such that o

B.

C

W n 1- 1 [-i, i), where 'Y is the real number appearing in Chapter I

(5.11). We shall define a function j satisfying the (PS) condition such that

(1) I/(x) - j(x») < i V x € H. (2) d/(x) dj(x) for x in a neighborhood of 8W. (3) In W, j has only nondegenerate critical pOints {p;}i", finite in number, contained in B6 . Once tbe function j is constructed, we obtain immediately

=

W_

= 1-, nW c j_hnw c I-i nw c Ii n W c

jf, n weI, n W

=

W.

However, there are strong deformation retracts:

l,nw-/inW

I\lld

I_inw-I_,nw

provided by the Gromoll-Meyer property. We have

H.(W, W_) = H.

(3.2)

(ih n W, j-h n w)

due to the exactness of the homological group sequence. Thus, index (dl,lI) = deg(cq, W,II) = deg(dj, W,II) m

=

L

index

;=1 m

=

(by (2))

(dj,p;)

(by (3))

00

LL(-I)q rank Cq (i,p;)

(by 1).

;=1 q=O

Applying Theorem 4.3 and Remark 4.1 of Chapter I to the function j on W, we have m

00

LL(-I)q rank Cq (i,p;)

00

=

;=1.,=0

L(-l)q rank Hq (ih nw,j_h n w) .,=0 00

=L(-l)q rank Hq(W, W_) q=. 00

=

L(-I)q rankCq(/,II). '1=0

102

Criticol Poiqt Theory

This is due to the fact that the negative gradient flow of j directs inward on 8W\W_, and hence also on 8W\(W nj-'(-h)). 3. Finally, we shall construct a function j, satisfying the (PS) condition as well as conditions (1)-(3). We define

j(x) = f(x) where p E

+ p(lIxll) (xo, x),

C' (lR~, lR') is a function satisfying pet) = {

~

with 0 :5 p :5 I and Ip'(t)! :5 ~, Xo E H is determined later. Let

then

i3 > O.

We choose Xo E H such that

Then we have If(x) - j(x)1 <

~

'if x E H

Ildj(x)11 ~ ~

'if x E

dj(x) = df(x)

'if x

B,\B'/2


For smaller IIxoll, dj is a k-set contraction mapping vector field with k < I (cf. Lloyd [Llol]). Therefore deg(dj, W,O) is well-defined. The (PS) condition for the function j is verified directly. On account of the SardSmale theorem, a. suitable ::to can be chosen such that j is nondegenerate. The proof is complete. Corollary 3.1. Under the assumptions of Theorcrn 1.6, if Po is a mountain pass point, and if the smallest eigenvalue >., of d' f(PO) is simple whenever Al = 0, then Al :5 0, and

index (df,PO)

= -1.

Proof. If.A1 > 0, then PO must be a local minimum, which implies C,(J,po) = O. This contradicts the assumption that PO is a mountain pass point. Therefore >., :5 O.

103

3. Con_lions wiIh 00.,. Theories

In the case where A, = 0, we have assumed dim lrer d" I(xo) = 1. It follows from Theorem 1.6 that C.(f,1'o) = 6. , G. In the case where A, < 0, tbe Morse index ind(f,1'o) = k > O. From the mountain pass point assumption, it must be k :5 1. Consequently, k = 1. Again, we apply Theorem 1.6, C.(f,1'o) = 6. , G. In botb cases our conclusion follows from Theorem 3.2. We generalize Theorem 3.2 as follows: Theorem 3.3. Under the assumption in Theorem 3.2, suppose that W is a bounded domain in H on which I is hounded. Assume that

r'ta)

a,

(1) W_ ~ {x e aw I '1(t, x) rt W V t > O} = W n for some where '1(t,x) is the negative gradient /low of I emanating from x;

(2) -d/lew\w_ directs inward. Then we have deg(df, W,II) = x(W, W_).

(3.3)

Proor. Due to assumptions (1) and (2), II rt
deg(dl, W,II) =

L

index (dl,p;)

j=1 m

00

= LL(-l)'rankC.(f,p;) j=lq=o 00

= L(-l)· rank H.(W, W_). q=O

The last equality follows from assumption (2), Theorem 4.3 of Chapter I, and the fact that l(p;) > a, whicb is a consequence of assumption (1). If I is degenerate, we perturb it as in Theorem 3.2. Since the critical

•

set K is compact in W, we construct a (fl. function pIx), satisfying:

pIx) = { where 6 and

~

> 0 such that dist(K, aW)

n. =

~

26, K is the critical set of I in W,

{x e W I dist (x,aW) < 6}.

104

Critical Point TheorJl

We may assume that l1'(x)1 :S 1 and

Ip'(x)l:S M2

< +00.

Let M, = sup{lIxlll x E W},

b = inf {f(x) 1x E W\n.}

and (1 = min { inf 1I
Then, by our assumptions, b > a and (1

j(x)

> O.

One defines

= !(x) + p(x) (xo, x)

for suitable Xo E H, with

. {b-a (1

0< IIxoll < mm

M,'

(1

}

3' 3M,M2 .

By the Sard-Smale Theorem, Xo can be chosen such that ate.~ Thus,

Ildj(x) II

~

is nondegener-

IId!(x)1I -lixoll -llp'(x)lIlIxll ·lIxoll 1

~ (1 - 3(1 -

>

j

1

3(1

>0

M,M211 xoil V x En,..

Now the function j satisfies the (PS) condition and assumptions (1) and (2). Indeed, the (PS) condition is easily verified by the above estimate, assumption (2) is trivial, and assumption (1) is verified by the following inequality:

-

!(x) ~ !(x) -lixollllxll ~ !(x) -

b-a

M, . M, > a

Since

deg(d!, W,O) = deg

'V x E W\n•.

(d], W,O)

(the RHS is the generalized Leray-Schauder degree for a k-set contraction mapping vector field with k < I), the proof is completed. Remark 3.1. Theorem 3.2 was given by E. Rothe IRotIl under stronger assumptions, and Corollary 3.! was obtained by Hofer IHof41 and Tian ITia!1 by a combination of the Poincare-Hopf Theorem with the splitting theorem.

3. Connection.! with Other Theories

105

3.2. Ljusternik-Scbnlrelman Theory Let X be a topological space. The category of a closed subset A of X is defined by Cat x (A) = inf{m E N U {+oo} I 3 contractible closed subsets Ft, ... ,Fm in X. such that A C Uf,:,l F,}. In the following let M be a CI-Finsler manifold. We write simply Cat(M) = CatM(M). Ljusternlk-Schnirelman Theorem. Suppose that f E C1(M.IRI) is a (unction bounded from below, satisfying the (PS) condition. Then f has a least Cat( M) critical points. The topological invariant Cat(M) can be estimated by other topological invariants, for instance, the cup length of M. Namely, we have Cat(M)

~

CL(M) + 1.

For a proof and a more general result, see Corollary 3.5 below. Now we extend this relation to estimate the number of critical points. It is known that for a nontrivial class ITI E H,(I• .I.), a < b, one can determine a critical value

c = inf sup f(x)

(3.4)

TElr] zEITl

if c E (a. b) and if f satisfies the (PS), condition. It is natural to ask, if we have two homology classes ITII. [T.I E H. (1• .1.), both nontrivial, and if Cl,C2 are defined in the same way as in (3.4), are there two distinct critical points? Generally speaking, no, but we have Lemma 3.1. Suppose that f E C1(M,IRI), has regular ""lues a < b. Let hI. IT.I E H,(I•• f.) be nontrivial classes. Suppose that 3 w E H'(I.), with dimw > 0, such that

Set (3.5)

c;

=

inf

sup f(x),

i

= 1,2.

riEIT.) zEIT,1

f satisfies (PS),,, i = 1.2, and that 3 a neighborhood N DE 8IJd a singular cochain wE w such that suppwnN = 0. Then Cl < C2 are two distinct critjcal values. Assume that

KC2

Proof. By definition

Cl

S

C2·

It remains to prove

Cl

<

C2.

106

CritiCO/ Point

:n.eo..y

'IE > 0, 3 a singular relative closed chain,.. E [T.] such that IT.I c I .. +<. We choose a neighborhood N' of K .. , such that N' C N' c N, and subdivide T. into r4 + Tf, such that IToI eN and IT~'I c I".+.\N'. According to the assumption, supp wn N = 0, and we have T1 = T2nW =~/nw,

which implies IT,I C I". ... \N'. Since a and b are assumed to he regular, then a < c, :S eo < b. According to the first deformation lemma, there exist constants g < Min{b-eo,c, - o} and 'I E e([O, I] x M,M) satisfying 'I (1,1..... \N')

°<

E

<

c 1.. _..

and

,,(1,·)

~

id

in

(f., I.).

Therefore 'I (1, lTd)

c 1.. _•.

However, '1(1, T,) E [T,], and it follows

We emphasize that the positive lower bound of the difference eo-c, depends on the behavior of df near K ... Corollary 3.2. Suppose that there are positive constants d, {, and 6 satisfying

and

(3.7) where (Kc )' = {x E M

«

Min

{26d',6d}.

I dist (x, Kc) :S 6}. c,

<

c. -

Then

(/3.

Proof. The proof is the same as above. The only thing changed is that the first deformation theorem is replaced by Theorem 3.4 from Chapter 1. Let us recall the notion [T,] < [1'2], cf. Definition 1.1, Chapter 1. Corollary 3.3. Suppose that IE C'(M,JR') and that 0 < b are regular values. Assume that [T.] < [1'2] are nontrivial c111SBeS in H.(f., I.), and

3. Connecti .... with Other Th
107

that (PS)c; holds, where c; is defined in (3.5), i = 1,2. If #K.. < 00, then

c, < CO are two distinct critical values. Proof. We follow the proof of Lemma 3.1. Since #K.. < 00, there exist two contractible neighborhoods of K .. , N' C N' c N. We write 1"2 = To + T!f where IToI c N and 1T!f1 C / ..+.\N'. Let w E W(f.) be the cohomology class satis£ying dim w > 0

and

(TIJ

= (1"2J n w.

The cap product does not cliange if we shrink N to a point becaUBe there exists W E (wJ which applies to any chain having support in N gives O. Therefore (T.J n w = O. Tbe remainder of the proof is the same as Lemma 3.1. Theorem 3.4. Suppose that / E C'(M,IIt') and that a < b are regular values. Assume that / satisfies the (PS) condition, and that / has only isolated critical points in r'(a,bJ. If there are m-nontrivial homology c l _ (TI) < (1"2) < ... < ITmJ in H.(f., fa), then / has at least m-distinct critical values. Proof. It follows directly from Corollary 3.3. Corollary 3.4. Suppose that / E CI(M,IIt') is bounded from below and that / satisfies the (PS) condition. Then / has at least CL(M) + 1 critical points. Proof. According to Theorem 1.1 from Chapter I, L(X, Y) = CL(X, Y)+ 1 for any topological pair (X, Y). We have m CL(M) + 1 nontrivial homology classes (TI) < (1"2), •.. ,IT.. ), by takingY = 0, and X = M. (fa = 0 if a < inf fl. The conclusion follows from Theorem 3.4.

=

Theorem 3.5. Suppose that / E C' (M, lit') and that a < b are regular ..alues. Assume that (TI) < (T,) < ... < (Tm) are nontrivial homology classes in H.(f.,Ja). Let (3.8)

c,= inf

If c = c, = CO = ...

sup/ex),

',-h) .el'
= em, and if /

i=1,2, ... ,m.

satisfies the (PS)e condition, then we

have

CalM (Ke)

~

m.

Proof. Since Ke is compact, we may choose neighborhoods N' C N' C N of Ke satisfying the following conditions: (1) CatM(N) = CatM(Ke ); (2) there exist constants 0 < e < l < Min {b - c,c - a} and 'I E C( M, M) such that

108

Critical Point Th
7l1/c -t" = id le-t. f} '"

id,

and

" (Ie+< \N') C

le-"

according to the fitst deformation theorem. We prove our theorem by contradiction. H CatM(Ke ) :<: m - 1, then there are (m-l) contractible closed sets {Bj}j"-I cover N, i.e., Uj:,1 B j :J N. Since [.,.<1 < [.,-,[ < ... < ["'m[, there exist. "",w:., ... ,Wm e H'(Ib), dim Wj > 0 such that

["'j-IJ = [.,.;J n Wj,

j = 2,3, ... ,m.

Since dimwj > 0 and Bj is contractible, we always may choose Wj E with /Wj/ nBj = 0, j = 2,3, ... ,m.

We choose .,. T

=

0'1

e ["'mJ

which has support /T/ C

Wj.

Ie+<' Subdividing.,. into

+ 0'2 + . , . + Om such that /0'1/ C

'. one has

/e+<\N', and /0';/

C

Bj ,

j = 2,3, ... ,m,

.,.' =.,.n (w, U ... UWm ) = 0'1

n (w,

Hence /.,.'/ C /e+, \N' and '1(/.,.'/) C 1)("") e ["'IJ, which implies Cl

U ... U

wm ).

le-,'

However.,.'

:5 C -

e [.,.d; therefore

E.

This is a contradiction.

Corollary 3.5. Suppose that / e CI(M, JlI.I) and that a < b are regular values. Assume that hI < ["'2J < ... < ["'mJ are nontrivial homology classes in H.(I.'/.). Let c; be defined as in (3.8), i = 1,2, ... ,m. Suppose that there are two open neighborhoods N' C N of Ui'! 1 K Ci satisfying dist (N',&N) ~ ~6, 6> 0 and that there are constants b,;: > e > 0 and c such that (1) IId/(x)II ~ b V x e /e+<\(le-' UN'), (2) 0<;:< Min06b',k6b}, (3) c - e < CI :<: C, :<: ... :<: em < C + e.

Then CatM(N)

~

m.

Proof. This is a combination of the proofs of Theorem 3.5 and Corollary

3.2.

3. Connections with Other Theoriu

109

3.3. Relative Category Definition 3.1. Let (X. Y) be a topological pair. and let Y be closed. A closed subset A of X is said to be of m category relative to Y. and we write Catx.y(A) = m. if

m = inf{n E N U {+oo}

13 closed subsets Fo. F" ... • Fn. and 3 '" E C((O.IJ x F,; X).

such that

(1) "'(O.x) = x, V x E F" 0 ~ i ~ m, (2) ho(l. x) E Y, V x E Fo. and ho(t. x) = x Vet. x) E [O.IJ x (YnF;), (3) h,(l.x) x,. V x E F•• for some x. E X. 1 ~ i ~ m}. We defined the cuplength of a topological pair (X. Y) in Section I Chapter I and related it to a notion concerning the length of the chain of subordinate nontrivial homology classes L(X. Y), L(X, Y) 2: CL(X. Y) + 1. What is the relationship between Catx.y(X) and CL(X. Y) ?

=

Theorem 3.6. Catx.y(X) 2: L(X. V).

Proof. Let m = Catx.y(X). Next. assume that there are [TiJ E H,(X. Y), 0 ~ i ~ m. with [ToJ < [T.I < ... < [TmJ. We h8ve to prove [ToJ = o. Assume, by oontradiction. that [ToJ f< O. By definition we have h-d = hJnwj,i = 1.2•...• m, withwj E H'(X). dimwj > O. Moreover there is a class Wo E H'(X. Y) such th8t Of< ([ToJ.Wo) = ([T,] nw"Wo) = ... =< [Tml,wm UWm _. U ... UWQ >. Consequently. (3.9)

Wm

Uw m _. U •.. UWo f< O.

Consider the exact sequence

-. H' (X. F.)

~ H'(X) .!i.. H' (F;) .......

for i = 1,2, ... 1m, and

,

-. H' (X,Fo UY) ~ H'(X. Y)

10

-->

H' (Fo UY. Y) .... ....

We claim that 11; is surjective for * ~ I, i ~ 1 and that '10. is surjective for. 2: O. Indeed. provided by assumption (3). the injection j;: F, -. X is homotopic to the constant map Fi ..... Xi, and we conclude that it :

110

Critical Point Theory

H'(X) _ H'(F,) is trivial for and i ~ 1. Next we define g(I,x) = {

*~

'I.

1, and then

is surjective for

*~

1

if (t,x) E [O,lJ x Fo

~(t,X)

if(t, x) E [O,lJ x Y;

because of assumption (2), g is well-defined and continuous. Then the injectionjo·(FoUY, Y) - (X, Y) is homotopic to the map g(1, .), which maps FoUY into Y. Again, by exactness, 'hi: H'(X,FouY) - H'(X,Y) is surjective for * ~ O. This proves the claim. Combining with the commutative diagram W(X,FoUYl® H'(X,F,j® ... ® W(X,Fml

10: W(X, Yl

1.: ® W(X)

1rr:. ® ... ®

We find that Wm U Wm_l U " . . finishes the proof.

Corollary 3.6. Cat(X)

~

,u

H'(Xl

Wo

CL(X)

~H' (X'YUQF;) ,,{e} ~

1 H'(X, Yl

= 0, which contradicts (3.9).

This

+ 1.

Claim. By definitions, Catx,e(X) = Cat (X), and CL(X, 0) = CL(X). It is obvious that the following relative properties hold.

(1) If Y = 0, then Fo = 0, and Catx,e(A) = Catx(A). (2) Let A, B be closed subsets of X, then

Catx,y(A U B)::; Catx,y(A)

+

Catx(B).

(3) Catx,y(Aj = 0.,. 3 h : [O,lJ x A _ X such that (i) h(O, . ) = idA, (ii) h(l, A) C Y, (iii) hIt, . )\AnY = idAny. (4) If 3 h E C([O,lJ x A,X) such that hIt, . )lAnY = Catx,y(A):5 Catx,y(B), where B = h(l, A).

idAny, then

Remark 3.2. The relative category was initially studied by E. Fadell (Fad1J. The notion was consequently used by G. Fournier and M. Willem (FoW1J. Theorem 3.5 was independently proved by Chang, Long, Zehnder [CLZ1] and Foumier-Willem [FoW2J. A different definition was given by A. Szulkin [Szu1J. Similarly, but with a different idea, see J. Q. Liu [Liu2J.

111

4. invariant FUnctions

4. Invariant Functions For a. G ma.nifold M, general1y speaking, invariant functions possess more critical points than functions without symmetry. That is, i£ the action is free. As a matter of fact, an invariant function is regarded as a function defined on the quotient manifold MIG, and, in many cases, the quotient manifold MIG has a richer topology than the manifold M, e.g, M = 8", G = Zo, which contains the antipodal map as an action, MIG = P", the real projective space. If the action is not free, then things are not so simple, but the fact that there are more critical points remains true. In tbis respect there are many interesting theories, among tbem, LjustemikSchnirelman category theory, genus and cogenus theory, cohomology index theory and pseudo index theory. The purpose of this section is to present a different approacb which is consistent with our Morse Theory. That is, in replacing index theory, we use tbe subordinate relative homology classes to obtain multiple critical points. Furthermore, a Galerkin approximation method is employed to avoid the pseudo index theory. For the sake of simplicity, we are only concerned with the two groups G, = Z. and G. = 8'. It is well-known that E = 8"" is the universal total G-space for G = G" i = 1,2, and that the classifying spaces B are J>"> and Cpoo respectively. The cohomological rings H'(POO,Z2) and H'(CJ>">,Q) read as follows:

H' (POO,Z2) = Zo(w),

dimw = 1,

H' (CPOO, Q) = Q(w),

dimw = 2,

and where R(w) is a polynomial ring generated by w with the coefficient ring R, and Q is the rational field. Let X be a Hilbert space, and let G = G, and G. act on X orthogonally (in the case where G" X is real) or unitarily (in the case where G., X is complex). For G = ~, one can find orthogonal 5ubspaces Xl and X2 = sucb that X, = FixG and that the G action has the representation x -x 'I x E X •. Thus G acts freely on the invariant subspace X •. For G = 8', according to the Stone Theorem, 'I 9 = e E [0,211'), the unitary representation has the following spectral decomposition:

xt

e'·,

T(g) =

i:

e'''dE"

where E, is the spectral family with spectrum (1 C Z. Let Xl = (E+o E_oJX, which is the fixed point set Fixe, and let X. = Xr. It is worth noting that 'I x E X 2 \{9}, the isotropy group G. is a finite group. Indeed, let V C X2 be a finite dimensional invariant subspace.

112

Ghtic4l Poinl Th<M!l

Then V is isomorphic to the unitary space Ck, and the group action is represented by the diagonal matrices

T(g)

= diag {eil19,eiA2', ... ,e iAk '}

where At, ... ,At are nonzero integers. Let m be the greatest common factor of A" ... ,Ak; then G. = \I x e V\{9}.

z..,

Theorem 4.1. Assume that (fl ) ! e CI(X, 1It1) is G-invari811t, G = G I or G2 • (f2) 3 two regular values a < b, such that (PS)c hold for all c e [a, b]. (fa) There exist G.-invariant sub6paces X+ and X_ with . I cod'1m X +<m=dI d ·1m X _<+00, J==d

"

Then

!

d

= lor 2,

where i and m are integers, such that (1) Fixe C X+o FixG n X_ = {9}, (2) !(x) > a \I x e X+, (3) !(x) < b \I x e X_ n Sp for some p > 0, (4) Fixenrl[a,b] =0. has at least m - j distinct critical orbits.

The proof is separated into two cases: G

= Z. and G = SI.

Cose I. G = Z2. From (f3) (2), (3) and (1), it follows that !. \XI C X\X+ and X_ nSp c !.\XI. We have the injections i,j and k 88 follows: (X-nSp ,0)/G

_

i

~

(X\XI,X\X+) IG.

From (f3) (1) and (4), the three pairs (X_ n Sp, 0), (J. \X I ,!.\XI ) and (X\XI,X\X+) are G-free. We shall figure out the homology groups of these pairs. We have

H. «X_ n Sp) IG, 0) = H. (p..-I) and

H.

«(J.\X I ) IG,(J.\XI)/G) =

H. (J.IG, !.IG»

provided by the excision property. V x EX, we have the orthogonal decomposition x y E

Xi, e x+. z;

Let '1(t, x) = y

+ tz

\I t

e [0, I].

=

y

+ Z,

where

4. Invariant .F\mction..

113

Tben 1/ : (X\X+) -+ X;\{6}. which is G-equivaciant. We obtain H,«X\XIl/G. (X\X+)/G) = H,«X\X,)/G. (Xt\{6})/G). Similarly.

H, (X\X,) /G. (X.;\{6}) /G) = H, (X2\{6}) /G. (Xt\{9}) /G). In summary. the following commutative diagram holds:

••

H, (t./G././G)

1;. H,

(P"-'.pH)

where n = dim X. ?: m. As to the cohomology rings. we have

It is easily seen that

"'w' = ("'W)2 •

where w is the generator of H'(pn-I) = Ej':-~~. and that dimk'w f. O. Thus. we have nontrivial classes (z,] E H,(J~...-I) for t 0.1 •...• m-I. satisfying (Z'_I) (z,] n l 1.2•...• m - 1. We shall prove that I.:,(z,( f. 0 for t = i.i + I •...• m - 1. Indeed. it suffices to prove that Iz,1 n X+ f. 0. V z, E (z,]. l = i.i + I •...• m - 1. If this is not true. i.e.• there exists to E [i. m - I) such that Iz,.1 n X+ = 0. then Iz,.1 is deformed in xt n SI. It is impossible. Since k$ = i· . P t we have

=

"'w.

=

=

;,(Z'_I) = i, (lz,] n k'w) = i,(z,( nj'w. and dimj'w

f. O. Then we obtain m -

j subordinate classes

Theorem 3.4 is applied to provide m - i distinct critical orbits of I with values in (a. b). because I-I[a. b)/G is a manifold. The proof is complete.

Case II. G = 8' . If we follow the proof given above, then the argument is stuck by the

fact that the pairs (X_ nSp .0). (t.\X"/.\XIl and (X\X,.X\X+) are no longer G-free.

114

CritiaJl Point Theo'll

Although f-'(a,bJlG is not a manifold, the minimax principle is still valid, due to the G-deformation theorem, cf. Chapter I, Theorem 7.1. More algebraic topology is needed to derive tbe chain of subordinate classes. In tbe following, if' denotes the Cech-Alexander-Spanier coh<>mology functor. It is known (cf. Spanier (Spal) pp. 340 and 420) tbat if' (Y) = H' (Y), tbe singular cohomology if Y is a CW complex. We shall fignre out H'«X\X,)/G) and H'«X_ n S.)/G) via their counterparts in

fro Let us recall tbe following «Spal) p. 344).

Theorem 4.2. (Vietoris Begle). Let f : X' .... X be a clo6ed continuous surjective map between paracompact Hausdorff spaces. Assume that there is n :<: 0 such that ifll'(r'(:r:» = 0 V x E X, and for q < n. Then

f' : if9(X) .... if9(X') is an isomorphism for q < n and a monomorphism for q

= n.

From this we relate if'(Y/G, Q) with the G-cohomology ifa(y, Q) where Q is the rational field and Y = X\X, or X_ n S•. Lemma 4.1. Suppose that Y is a paracompact Hausdorff G space, and that V x E Y, the isotropy group G. i.finite. Then the map E XG Y .... Y/G induced by the projective j : Ex Y .... Y induces isomorphisms

J:

~.

- q

. - q

J : H (Y/G,Q) .... HdY,Q)

in all dimensions q, where E is the unjversal total G space.

Proor. We make use of the filtration E' C E2 C ... C E'" C E"'+' C ... of the universal total space E, and consider the diagram [or each m:

E'"

XG

Y

I:

Y

1:.

Y/G

1

Note that]m is closed because E'" is compact and that Gm)-'x = E'" /G. V x E Y, where G. is the isotropy group. Since if 9 (E"'/G.,Q) = 0 for q < m, applying the Vietoris Begle Theorem,

4. invariant hnction.t

115

,0

are isomorphisms. Then is just the composition or Om)o and the is<>morphism iIO(E'" Xc Y, Q) ~ iIO(E Xc Y, Q), q < m. Let p : { - B he an orientable r-dimensional vector bundle, and let D{, S{ he the associate disk and sphere bundles. According to Thorn'. isomorphism theorem, iIr(D{, S{) ~ Z possesses a generator which induces the isomorphism

tee),

where U is the cup product. Since the disk is contractible, iIO(D{) ~ under this isomorphism. The exactness or the cohomology sequence reads as rollows:

iIo (B) , let a({) E iIO(B) he the image or

tee)

... - iIO(D{,S{) - iIO(D{) - bO(se) _ .... It turns out to he the rollowing Gysin sequence:

where p is the projection Se - B.

Lemma 4.2. If Ifln-l

c

X 2 , then q = 2;, 0"$;"$ n - 1 otherwise

and

dimX2-1

L

iISl (X2\{/J},Q) =

w1.

;=0

Proor. Consider the rollowing diagram:

-.l... -

•

E = SO" B

!

= CP"",

where p is the projection induced by the projection p. We look at the vector bundle .. : { = E X en - B = ep~. The associate 2n - 1 sphere bundle Se turns out to he p: E XSI Ifln-l _ B. We have the Gysin sequence

... - flO (De, Se) -

II

iIO-

2n

flq(B) /«() (B)

-

flq(so - ...

116

Critical Point Theory

Thus. for q < 2n - 1. we have the isomorphisms fI«B) ~ fI«8{); and for q

= 2n -

1, Crom

fI' n (D{.8{) _ fI·n(B)

0- fI' n -' (B)

/I

- I[

o

8"(B)

/.«)

it follows fI' n - ' (8{) ~ o. For q > 2n - 1. from Lemma 4.1.

fI«8{.Q)

= flo (E x s '

8'n -'.Q)

(8'n-'. Q) flo (8'n-'/8'.Q) = O.

= fI;, =

Therefore,

q = 2j.

0$ j $ n- 1

otherwise;

and then q = 2j,

j

= 0,1,2, ...

otherwise. The conclusion fonows. After these preparations, the previous arguments can be modified to suit

the case G = 8'. Indeed. since (X 2 \{9})/8' and (X_ n 8 p )/8' are CW complex. the singular cohomology and the Cech-Alexander-Spanicr coh()mology are isomorphic. We have HO(X_ n8p )/8'.Q) ~ HO(Cpm-'.Q). and H"(X.\{9})/8'.Q) ~ H"(cpn-'.Q). Again, from the commutative diagrams

and

..------..

H o (X_ n8p )/8'.0)

...!!..

H o (1./8'././8')

1;·

Ho (X\XIl /8'. (X\X+) /8')

we find nontrivial classes [.,j E H,«X_ n 8 p /8'). l = 0.1 •...• m - 1. satisfying [.,_,1 = [',1 n kOw. f = 1.2 •...• m - 1 and kO[.,j i 0 for - ;~\1-'1-"':!fl.I

~;:".

. "':" ~" :!.-" "

"-..,., .

4. Invariant Functi0n6

117

+ I, ... ,m - 1. Thus i.(z;) < i.(z;+I) < ... < i.(z",_,) are the required m - j subordinate classes. The proof is complete.

1 = j,j

Corollary 4.1. Under the assumptions of Theorem 4.1, there are eriti· ca!orbits{O, I/=j,j+l, ... ,m-I} such that

do (/, 0,) to,

1 = j,j + I, ... ,m - 1.

A dual form of Theorem 4.1 cau be formulated as follows. In replacing (f2) and (f,), we state (f~) There is a sequence of Cd invariant finite dimensional vector sulr spaces:

X,

eX' c ... eX· c ... ,

dimX·

= nd,

with U~=l xn = X satisfying (1) «PS)· condition) 3 no E (li( sum that 'I n ?: no, the restric· tion f" = II X" satisfies the (PS)c condition 'I c E (a, b). (2) «PS)' condition) For any sequence x. E X·, n 1,2, ... , 1

=

a ::;

I (x.) ::; b and df" (x.) - 0

imply a convergent subsequence. (fa) There exist G d invariant subspaces X+ and X_ with j =

~ codim X+ < m

=

~dimX_ < +00,

where j and m are integers such that (I) FixG C X_, FixG n X+ = {OJ, (2) I(x) > a 'I x E X+ n Sp for some p> (3) I(x) < b 'I x E X_, (4) FixGnr'(a,b) =0.

d= 1,2,

°

Theorem 4.3. Under the assumptions (t.), (/2), and (/3) where a, b are regular values, the function f has at Jeast m - j distinct critical orbits.

Proof. First, we change I to - I, -a to band - b to a. Assumptions (2) and (3) of (fa) are changed to be the following: (2') I(x) < b 'I x E X+ n Sp for some p (3') I(x) > a 'I x E X_.

> 0, and

118

Cri&ol Poinl

neorv

According to (f.). we choose n large enongh. such tbst both Xi and X_ are included in Xfl, and then restrict ourselves on the finite dimensional Gd-invariant subspace xn. Thus

r(z)
r(z) > a 'liz E X_.

codim X~ = (n- m)d and

dimX~ =

(n- j)d.

Also. xn n Fixe C X_ = X~. (xn n Fixe) n X~ = Fixe n X+ = {8}. Now we apply Theorem 4.1 to and conclude that there are (m - j) subordinate homology classes

r

which correspond to critical values a ~

cj ::5 cj+l :5 ... :S c:!a_t :5 b

Let CI =

lim

n_oo

c'l, t =

for n large.

j,j + 1, ... ,m - 1.

Then a :S

Cj

:S Cj+l ::S ••• ::5 Cm-l :S b.

According to assumption (f.). (PS)". values. It remains to prove that if C

= Ck = Ck+l = ... = CI,

Ct.

j ~ f ~ m - 1. are all critical

j::5 k ::S

t

::S m - I,

then #Ke ~ f. - k + 1. We prove it by contradiction. If Ke = {OI.O••... • O.} • • < f. - k + 1. then we choose neighborhoods VD. of OQ, 1 :S Q ::S 8, such that (1) UQ n U~ = 0 for 0 i' /3. (2) U:=I UQ C rl[a. bJ. (3) Each UQ is in a G-tubular neighborhood of OQ in /-1 [a. bJ. We omit the subscripts o. Let" : U - 0 be the projection. and let S = ,,-I(P) lip E O. It is well known (cf. Chapter I. Lemma 7.15) that S is a Gp-manifold, where G p is the isotropy group at p. The orthogonal projection onto (xn).J. is denoted by P;-. U IG. - X be the injection; then

Let i :

4. invariant Fbndions

119

is a finite dimensional submanifold, provided n is large enough. Again, one may assume that the geodesic segment connecting p and the nearest point pn exnnS is in S. We choose suitable coordinates: Y Yi Ell Y2 , dim y, = nd, '{J : Y n B, - S, such that '(J(Y, n BIl c S n X n , '(J(9) = pn, p = '(J(ii2) and '(J(tii2) e S \I t e (0,1), where B, is the unit ball and ii2 e Yo. Then 3 D > 0 such that cp(V) C S, where V {(y" 112) e Yi Ell Yo I IIy,II < 6, 11112 - wll < 6, 3t e (0,1], w = tii2}. Thus N = '(J(V) is an open neighborboud of p with the following prop. erty: 3 a deformation retract

=

=

'I:

NIGp _xnnN =Nn.

Indeed, 'I = '{J 0 " 0 '{J-', where" is the projection onto Y,. Noticing that Gp is a finite rotation group with fixed point p, we may choose 6 > 0 80 small that Nn = Nn IGpLet us introduce

•

•

N= UG.Na and J./ft= UG.N:. 0'=1

6

We choose > 0 small enough such that there exists G-invariant open neighborhood N' of K. with dist (8N,N') ~ ~6. We conclude 3 b > 0, ! > 0 such that

IIdr(x)II ~ b \I X

E

(r)m \ (r)._. u (J./ft)') and

O
=

xn n N'. J(x n )

-

c

U6ij2,~6b},

Iudeed, if not, 3 Xn

IIdfn (Xn) II

and

-

E

xn \(Nn)'

0,

then hy (PS)', 3 x'
c- e

< ci S C~+l

and

~

.••

:S c'l < c + e

t

U K.~ c (J./ft)' . We exploit Corollary 3.5 and oonclude that

•

L 0'=1

Catli!: (N':) ~ 1- k+ 1,

Critical Point 'I'II<m1J

120

because N:; = N:;/GPa = G· N:;/G. However, CatR/:(Jir.:) = 1. Indeed Va is contractible, and so is No. Henee N:; is contractible in Na/GPa' i.e., 3 {a : [0,1] X N:; - No/GPo satisfying (0(0,.) = idRft, {a(I,') =_ Po E No/Gpo. Let us define To = a "101 0 Then Ta is a contraction in N:;.

eo.

Hence, we obtain a contradiction:

•

• =

L

CatR/:

(N;:) ~ l - k + 1.

0=1

Corol1ary 4.2. Suppose that! E C ' (X,R') is even with 1(6) = 0, and that (1) 3 p, Q > 0, and a finite dimensional linear subspace E such that liE.insp <': £t, (2) 3 .. sequence of linear subspace Em, dim Em = m, and 3 R... > such that !(x):-:; 0 'Ix E Em\BRm, m = 1,2, ....

If, further,

1 satisfies (PS)"

with respect to

{Em I m

= I,2, ... }, then

°

f

'possesses infinitely many distinct critical points corresponding to positive

critical values. Proof. If not, f has at most f critical points. Let dim E = jj we choose m - j > f. Now, (f, ) is obviously true, with Fixe = {6}. Let xn = En; the (PS)~ condition holds, using (2), so (f;) is satisfied. Let X+ = E.i, X_ = Em, a = cr, and b = Max' EEm I(x) + 1 then (fJ) holds. We apply Theorem 4.3. There are at least m - j pairs of critical points. This is 8. contradiction. Moreover, all critical values, obtained by the minimax principle, are greater than a, so they are positive.

Remark 4.1. Corollary 4.2 was given by Ambrosetti-Rabinowitz [AmRI j, where the function 1 is assumed satisfying the (PS) condition rather than the (PS)' condition. Remark 4.2. Theorems 4.1 and 4.3 were proved by many authors via index theory and pseudo index theory, and the index for G = :Eo is genus. The associated theorems were given hy Clark [Clal[, Ambrosetti-Rabinowitz [AmRI[, and V. Benci [Ben4]. S'-index theory was first introduced by Fadell-Rabinowitz [FaR2]; see also V. Benci [Ben3J, L. Nirenberg [Nir3j, Costa-Willem [CoWI[, Fadell, Husseini, Rabinowitz [FHRIJ. Pseudo index theory was introduced by V. Benci [Ben4[; see also K. C. Chang JChaI3J. Other approaches can be found in Fadell [FadIJ (relative category) and Liu [Liu4J (pseudo category). The new result is Corollary 4.1.

5. Some Ab.olmct Oriticol Point 1'11.........

121

5. Some Abstract Critical Point Theorems In tbis section, we shall give several abstract critical point theorems using Morse theory. Their applications will be studied in subsequent chapters.

C" Hilbert-Riemannian manifold, and let

Theorem 5.1. Let M be a

f

i 0 for some kEN, wbere b > a are regular values, and that {x"x., ... ,xt} c K n f-l[a, b] with Fredholm operators cP f(x,), i = 1,2, ... ,t. If either E C'(M,JR I ) satisfy the (PS) condition. Suppose tbat Hk(J• .Ja)

(5.1)

ind(J,x,) > k

ar ind (f,x,)

+ dimkercPf(x,) < k

for i = 1,2, ... ,l, then J has at Jeast one more critical point Xo with Ck(J, xo) i o.

Proof. If not, K = {XI,X., ... ,xt}. From (5.1), it follows Ck(J, x,) = 0, i = 1,2, ... ,t, provided hy the shifting theorem. We apply the k'" Morse inequality: t

0= M.(a, b, J) =

L

rank C. (J, x,) 2:

= rank H. (J., fa)

13.(a, b, f)

> O.

Tbis is impossihle. Corollary 5.1. Suppose that tbe boundary IJD of a k ball D and S homologically link in M, and that {Xl, x.,.:. ,x,} are critical points of f satisfying (5.1). Then the conclusion of Theorem 5.1 bolds. This is a comhination of Theorems 1.2 and 5. L

Remark 5.1. Corollary 5.1 includes a theorem due to Lazer Solimini [LaSl] as a special case, in which S and fJD are as in Section I, Example

2. Let H be a real Hilbert space, and let A be a bounded self-adjoint operator defined 00 H. According to its spectral decomposition, H = H+$Ho$H_, wbere H""Ho are invariant suhspaces corresponding to the positive/negative, and zero spectrum of A respectively. Let P"" Po be the projections (orthogonal) of these suhspaces. The following assumptions are given: (HI) A±:= A IH± has a bounded inverse on H±. (H,) 7:= dim(H_ $ Hol < 00. (H.) 9 E CI(H,JRI) has a bounded and compact differential dg(x). In addition, if dim Ho i 0, we assume

122

Critical Point 'I'hem'y

9 (POx) -

-00

as IlPoxll -

00.

We shall study the number of critical points of the function lex) =

1

"2 (Ax, x) + g(x),

or equivalently, the number of solutions of the operator equation

Ax + dg(x)

= fl.

Lemma 5.1. Under the assumptions (H,), (H.) and (H,), we have that (1) / satisfies (PS) condition, and (2) H.(H, /.) = e..,G for -a large enough, as /a n K = 0. Proof. 1. First, we verify that / satisfies (PS). For {xnH" C H, df(x.) fI, and /(x.) bounded, we shall find a convergent subsequence. In fact, from d/(x n ) - fI, it follows V e > 0, 3 N = N(e) such that for n > N

where x; = P±xn . Hence

Ilx; II, and then (Axnxn)

are bounded. Since

Ig (PoX n ) I :0; Ig (x.) - 9 (Poxn)1

+ Ig (xn)1 :0; m (\lx~1I + Ix;;1) + Ig(xn)1

where m = sup{lIdg(x)1I I x E H}. If f(xn) is bounded, then Ig(xn)l, and therefore Ig(Pox.)I, is bounded. Thus IIPoxnll is bounded. Since dg is compact there is a subsequence x ni such that dg(x ni ) is convergent. By

X;i

and by the boundedness of A±l we conclude that is convergent. Since dim Ho is finite, there is a convergent subsequence Pox ni . The (PS) condition i. verified.

2. Denote

e*

R + = m±l. let £+

= inf {(Ax±, x±)

I

IIx± II

=

I} which is positive, and

I

From (5.2)

(d/(x),x±) = (Ax±,x±) - (dg(x),x±)

"= e± IIx±II' -mllx±1I

5. Some Ab./nJcI OritiCGl Poinl Theore"..

123

we know that I has no critical point outside M, and that -df(x) points inward to M on 8M. Noticing that

1

- :iIlAllllx_1I2 - m(lix-il + 14) + 9 (Pox) 1

$/(x) $ :iIlAIIR~

1

- :iE_lIx_1I2 +m (lix-li + 14) + g(Pox) ,

we obtain I(x) -

-00

0$=}

IIx_ + Poxll- 00

uniformly in

x+.

a, < R2 > 0 such that (H+ nBR+) x (Ho (DH_)\BR,) c I •• nM C (H+ nB~)

Thus, 't T > 0, 3

x (Ho(DH_)\BR,) C

1. 2 nM

Also we find T > 0 such that K n I-T = 0. The negative gradient flow of I defines a strong deformation retract

Another strong defonnation retract in

is defined by

T2

J02 n M

= {(I,,), where

IIxo +x_1I ~ R. if IIxo + x_II $ R •. if

We compose these two strong deformation retracts, obtain 8 strong deformation retract

T

=

T2 0 Tl,

and then

and, again, the following deformation: 'I (I, x+

+ x_ + xo) x+ +xo+x_ = {

if IIx+1I

., R+

B~!II (t14 + (1 - 1)lIx+lI) + Xo + x_ if 11"'+11 > 14

124

Critical Point 7'heoty

is .. strong deformation retract of the topologic&l p&ir from

(H,f",,)

to

(M,Mnf•• )

provided by (5.2). 3. Fin&lly, we bave

H. (H,f•• ) g; H. (M,M

nf•• )

H. ((H+nBR+) x (HoffJH_) ,

g;

(H+

n BR+) x (Ho ffJ H_)\BR.)

g;

H. (Ho ffJ H+, (Ho ffJ H_) \BR.)

g;

H. (Ho ffJ H_) n BRI'O (Ho ffJ H_) n B R+))

g; 6..,G.

Theorem 5.2. Under the assumptions (H.), (H2) Md (Ha ), if f bIOS critical points {p;}f=. with ,

n

"f

¢

U jm_ (P;), m_ (P;) + mo (P;)J i=1

where m_(p) = index(j,p) Md mo(p) = dimkerd'f(p), then f hlOS a critical point Po different from Pl, ... ,Pn, with C,(j, Po) # O. Proof. Directly follows from Lemm.. 5.1 and Theorem 5.1. Jlem4rk 5.2. In Lemma 5.1 and Theorem 5.1, if dim Ho = 0, the boundedness of dg(x) can be repl&ced by the following condition: (5.3)

IIdg(x)1I

= oOlxl1)

lOS

IIxll -

00.

Proof. Condition (5.3) implies that / bIOS no critic&i point outside a big ball BR(O), R> O. Now we define a new function fIx) where

pIt)

1 = 2(Ax,x) + p(lIxll)g(x),

={ ~

0::;

t::; R.

t>R2

0< I

<1

-P- ,

and satisfies the estimate

Ip'(t)1 ::; 3 R 1 R ' where R2 > R. > R are suitably chosen. 2 2 •

5. Some Abtmcl Critical Point Theorems The new function

j

125

pc: : 8: :: the same critical points as the function

J,

and satisfies the (PS) condition as well. In fact, f(x) = i(x) for IIxll :5 Rio and df(x) = Ax for Ilxll > R 2 , we only want to verify that IIdi(x)1I "16 for x E B R .(6)\BR1 (8). Let 0 = IIA-11I-l by assumption (5.3); 3 Ro > 0 such that

!

IIdg(xlil < ollxll

'I x ¢ BRo'

The compactness of dg(x) implies that 3 M,

Ildg(x)1I < ollxll + M,

> 0 such that 'I x E H.

Thus Ig(x)1

< ollxll' + MEllxl1 + Ig(8)1·

Let Rl

> max { R, Ro, ~ (4ME + 3) }

R,

= max {2,1 + Ig(6)1} R1 ;

,

we ha.ve

Ildi(x)11 = IIAx + p'(lIxll)g(x) 11:11 + pCllxIDdg(x) I ~

IIA-T' IIxll 3

(ollxll + M,)

1

- '2 R, _ R, (ollxll' + M,lIxll + Ig(8)1) ~ 1

'IxeBR,\BRl .

As to the (PS) condition, suppose that di(x.) - 6; then {x.} C BR, except for 6nitely many points, according to condition (5.3) and the invertibility of A. Since dg is compact, there exists a convergent subsequence dg(x.). Comparing this with the assumption df(x.) = di(x.) - 6, and the boundedness of A-', we obtain a convergent subsequence. Remark 5.3. In the case H = }tN, n = I, P' = 6, and dimHo = O. This theorem is due to Amann Zehnder [AmZl[. The above lemma, and the general statement with the condition 'Y < m_ CPi), i = 1, ... ,n, is due to Chang [ChaI). The above version is due to Z. Q. Wang [WaZ2[.

Corollary 5.2. Under the 8SSumptions (H,), (H.) and (H,), if f has .. nondegenerate critical point Po with Morse index m-CPo) "I 1, then f has a critical point PI "I Po.

126

Crilical Point Theory

Moreover, if (5.4)

then f has one more critical point 1'2 #< Po, Pl. Proof. The first conclusion foUows directly from Theorem 5.2 and then we have C, (f,Pl) #< O. However, by the shifting theorem .., E [m_

(p,), m_ (P,) + mo (P,)],

Condition (5.4) implies one of three possibilities: (I) m_(Po) rt [m_(p,), m_(p,) + mo(P,)] and .., E (m-(p,), m_(p,) + mo(P,) , (2) .., = ffl_(Pl), (3) 'Y = ffl_(P,) + mo(P,). Using the splitting theorem and the critical group characterization of the local minimum and the local maximum, we see in both cases (2) and (3),

The Morse inequalities in combination with the Betti numbers for the top<>logical pair (H, f.) (see Lemma 5.1), gives the existence ofthe critical point 1'2. In case (I), again by the splitting theorem and the critical group characterization of the local minimum and the local maximum, we obtain

and

fl. (H, f.) = "" for -a large enough, by Lemma 5.1. However, case (I) implies either m_(Po) < m_(p,) orm_(Po) > m_(Pl)+ mo(P,). If there were no other critical points, then in the 6rst case, the m_ (Po) + I 'h Morse inequality would read as -I

~

O.

This is a contradiction. And in the second case, both the m_(p,)+mo(P,)I'h and the m_(p,)+mo(P,),h Morse inequalities would imply the equality m_ (PI )+mo(Pl)

L

q=m_(PI)

(-1)0 (rank Co (f,Pl) - "") = O.

5. Some Ab&tract Oriticol Point Theorem. Again, the m_(po)

+ l'b

127

Morse inequality would read as

-1 2: 0, and this is also a contradiction. To sum up, we have proved tbe existence of the third critical point.

Now we tum to a variant or Lemma 5.1 wbich provides more information on the numher of critical points if the function I is defined on H attached by a compact manifold Y.

Lemma 5.2. Suppose that (H.) and (H2 ) bold. Lei yR be a C' compact manifold wilbout boundary. Assume tbat 9 E C'(H x YR,IlI.') is a fonction baving a bounded (ifdimHo = 0, IIdg(x,v)1I = o(lIxll) V v E Y) and compact dg, satisfying 9 (Pox, v) - -00 as Lei

IIPoxll - +00,

1 I(x,v) = 2(Ax,x)

if dim Ho

'I 0.

+ g(x,v).

Then (1) I aatisties (PS) condition, (2) H,(H x YR,I.) ~ H,_,(VR) for -a large enougb, wilb K n I. = 0. The proof is similar to tbe proof of the previous one. Now define

M

= (H~ nBR+) x (HoffJH_) x yR.

By the aame method, we eventually obtain

H. (M,I. nM) ~ H.

(B"S'-') ®H. (VR),

by the Kiinneth formula. Thus

H,(H x YR,I.)

Q! ~

H.(M,I. nM) (yR).

H,_,

Theorem 5.3. Under the assumptions of the above lemma, the function has at least CL(yR) + 1 critical points. If furlher, 9 E C', and I is nondegenerate, then I has a least L~-o II. (YR) critical points, where 1I.(yn) is the ,ih Betti number ofyn, i = 0, 1, ... ,n.

I

Proof. Since

128

Critirol Point

'I'heot7i

and

H' (H x V")

~

H' (V"),

we obtain 1+1 nontrivial singular homology relative classes IZt+11

... < IZI], with ZI, ... ,ZHI E H.(H x V",f.), where I

< [Zt] <

= OLIVO).

The first conclusion follows from Theorem 3.4. Moreover, if f is nondegenerate, by the same argument, we may assume K n f. = 0 for - 4 large enough. The Morse inequalities now read as

m, ~ rank H.(H x V".!.) = rank H._~(V") = P,-~(V"), q =0,1, .... Therefore there are at least E;~oP;(V") critical points. The same idea can be applied to study functions bounded from below. We have Theorem 5.4. Suppose that M is a C'-Finsler manifold. Assume that IE OI(M,IItI), satisfying the (PS) condition, is bounded below. Suppose that there exists a crjtjcaJ point po, which is not the global minimum of f,. with linite E;'o(-I)' rank C,(J,Po) f. X(M) -1. Then I bas at least

three critical points. Proof. According to the (PS) condition and lower semi-boundedness,

I

has a global minimum PI. Let c; = I(p;), i = 0,1. If I had no critical points other than Po and PI then for arbitrary b > Co there would be no

critical point in

M\f.,

and the following identity would hold:

where 0 < e < CO - Ct- Since there exists a strong deformation retract deforming Minto fb1 and JCO-f: into PI, we would have

x (J.) = x(M), and But

X (I., 1",,-.) =

=

E(-I)' rank O. (J,Po) q=O

because Po is the unique critical point in diction.

I-Ilea -

e, b). This is a contra-

Corollary 5.3. Let H be a Hilbert space, and I E CI(H,RI) be bounded below with the (PS) condition. Suppose that d/(x) = x - T(x)

5. Some Abstroct Critictll Point Theorem.

129

is a compact vector field, and Po is an isolated critical point but not the global minimum with index (d/,Po) = ±1. Then / bas at least three critical points. Remark 5.4. In the case H = JRn , the coroUary was proved by Kr.... nosclskii via degree theory, but was rediscovered by Castro Lazer in ICaLI) by a homology method. The ahove version was given by Chang ICbal). A tittle later, Amann presented a degree theoretic proof IAmal) for Corollary 5.3. Theorems 5.3 and 5.4 are due to Chang IChal,5).

Now we tum to tbe study of bifurcation problems. Let H be a Hilbert space and n be a neighborhood of 0 in H. Suppose tbat L is a bounded self-adjoint operator on H, and that G E C(n, H) with G(u) = o(lIu)1) at u = O. We assume that G is a potential operator, Le., 3 9 E e'(n,JR'), such tbat dg = G. Find solutions of the following equation with a parameter ,\ E JR':

Lu + G(u) = '\u.

(5.5)

Obviously u = B, for all ,\ E R', is a solution of (5.5). We are concerned with tbe nontrivial solutions of (5.5) witb small lIull. Because (5.5) is the Euler equation of a function with parameter ,I, tbe bifurcation phenomenon has its specific featnre. We shall prove tbe following tbeorem due to Krasnoselskii IKra11 and Rabinowitz lRab21 via Morse theory, cr. ICha6). Theorem 5.5. Suppose that / E C 2 (n,IR') with df(u) = Lu+G(u), L being linear and G(u) = 0(11,,11) at" = O. If I' is &l! isolated eigenvalue of L of finite multiplicity, then (1',8) is a bifurcation point for (5.5). Moreover, at least one of the following alternative occurs: (I) (1',0) is not an isolated solution of (5.5) in {/l} x n. (2) There is a one-sided neighborhood A of I' such that for all ,I E A{/l}. (5.5) posse8SfJS at least two distinct nontrivial solutions. (3) There is a neighborhood I of I' such thai for all ,I E I\{I'}. (5.5) possesses at least one nontrivial solution. The proof depends upon the Lyapunov-Schmidt reduction. Let X = ker(L -1'/), with dimX = n; and let p,pl. be the orthogonal projections onto X and Xl., respectively. Then (5.5) is equivalent to a pair of equations (5.6)

(5.7)

1'''' + PG (x + xl.) =,\x

Lxl.

+ pl.G (x + xl.) = ,\xl.

wbere " = x + XL, x E X, ",1. E XL. Equation (5.7) is uniquely solvable in a small bounded neighborhood 0 of (1',8) E JR' x X, say xl. = cp(A,x)

130

cntioJl Point Theory

for (A,x) E 0, where 'P E C'{O,X.l). Substitute x.l = 'P{A, x) into (5.6), and (5.8)

/AX

+ PG{x + 'P{A, x»

= AX,

which is again a variational problem on the finite dimensional space X. Let J,{x) = I{x

(5.9) =

+ 'P{A, x»

-

1

A

2 (lIxll2 + MA,X)II') 1

A

2{/' - A)lIxll' + 2{L'P,'P) - 211'PII' + g{x + 'P)

where dg = G, with g(8) = O. It is easy to verify that (5.8) is the Euler equation of J" and that 'P{A, x) = o{lIxl) at x = 8. The problem is reduced to finding the critical points of J, near x = 9 for fixed A near /" where J, E C'{n"IIt'), n, isa neighborhoodof8 in X. Proof of Theorem 5.5. Clearly x = 8 is a critical point of J, V A such that (A, 0) E 0. UO is not an isolated critical point of J~, which corresponds to case (I) in the theorem, then there are only two possibilities: (i) x = 0 is either a local maximum or a local minimum of J p ; (ii) x = 8 is neither a local maximum nor a local minimum of JpIn case (i), suppose that 8 is a local minimum of Jp- For some 0 > 0, W = (Jp ). = {x E n, 1J~{x) :S o} is a neighborhood of 8, containing 8 as the unique critical point. The negative gradient flow of J p preserves W, and therefore the negative gradient flow of J, preserves W for IA -/'I small. Since W is contractible, X{M) = I, x = (J is a local maximum of J).., for A > /' and J, is bounded from helow on W; we obtain two nontrivial critical points, according to Theorem 5.4, in particular, Corollary 5.3. Therefore, for each A in a small right-baud side neighborhood of /" there exist at least two distinct nontrivial solutions of (5.5). Similarly, we prove that there exist at least two distinct nontrivial solutions of (5.5) for each A in a small left-baud side neighborhood of /" if 8 is a local maximum of J p">

In case (ii), 9 is neither We see that

8.

local maximum nor a local minimum of Jw

(5.1O) according to Example 4 in Section 4 of Chapter I. Since (5.11) (5.12)

Co {J" 8) Cn (J" 8)

= 1, =1

for A < /" for >. > /"

and

we conclude that there is a neighborhood I of /' such that for>. E I\{/,}, J;.. possesses 8. nontrivial critical point. If not, 3 Am - p, say .xm > III

131

6. Perlurbatima Theory

such tbat J~m bas tbe unique critical point 9, tben Cn(J~.. ,8) = I, m = 1,2, ... , implies Cn(J~,8) I by Tboorem S.6 of Cbapter I. Tbis contradicts (S.lO). Similarly for >'.. < p. Tbis completes the proof.

=

Remark 5.S. A weaker result tbat (1',8) is a bifurcation point was proved by a simpler argument, cf. Berger (BerIJ.

More information on the number of distinct solutions can be obtained if we assume, in addition, tbat the function f is G-invariant on some G manifold. For G Z. tbe reader is referred to E. Fadel! and P. H. Rabinowitz (FaRI]; for G = 5" see E. Fadell and P. H. Rabinowitz (FaR2J, and A. Floer, E. Zehnder (FIZlJ. As to tbe general compact Lie group G, see T. Bartsch and M. Clapp (BaCIJ.

=

6. Perturbation Tbeory We study two problems in tbis section: (I) Given a C2-function f, let E he a nondegenerate critical manifold of f. What becomes of E if we perturb f to f + 9 where 9 is small? In the first part of this section, we shall study this problem under the various metrics of 9 : CO, C' and C2. (2) For a given function f, which does not bave the (PS) condition, we perturb it to f. = f + 09 such that for each 0 > 0, f. possesses tbe (PS) condition. Under what conditions can one extend the critical point theory for the perturbed functions to the original one? This will he studied in tbe second part of this section. 6.1. Perturbation on Critical ManifoldS

We start with the CO-perturbation, i.e., 9 is assumed to be small in the CO-norm. Because of the very Hexibility of g, one cannot expect any strong

conclusion. Lemma 6.1. Let A eYe B c A' C X C B' be topological spaces. Suppoee that H.(B,A) '" H.(B',A') '" O. Then h. : H.(A',A) H.(X, Y) is an injection. Proof. Observing the following diagrams:

H,(A',A)

.. ""-

..s

H,(B',A)

/~.

H,(X,A)

_

H,(B',A')

132

Critical Point Th""'ll

and

H.(B,A)

_

(il).

H.(X,A)

_

(a.). ' "

H.(X,B) (P.).

/

H.(X,Y) where i : (A',A) --+ (U,A), il, at P, Qb PI, are incursion maps. From the exactness of these sequences, and the assumptions H. (B, A) '" H.(B', A') = 0, i. and (il). are isomorphisms. However, i. = fJ. 0 a. and (i,). = (P,). 0 (0,).. Therefore (0,). and a. are injections, so is h. = (0'1). 0 Q •• Theorem 6.1. Suppose that IE C'(M,IR') satisfies tbe (PS) condition, witb an isolated critical value c. Assume tbat (a, b) is an interval containing c. Tben there exists an _ > 0 such that [or (6.1)

Sup {19(z) - l(z)11 z

E r'[a,bl} < _/3.

'Ye bave an injection h. : H.(fc+<.!c-<) -

H.(gc+;,gc_;)'

Proof. Choose _ > 0 such that c is the only critical value of f in [c - _, c + _[ c (a,b). (6.1) implies

Ic-< C gc-, C fc-t C Ic+~

c

gc+~ C fc+<·

Applying Lemma 6.1, we obtain

Theorem 6.2. Suppose that I E C'(M,llI.'), satisfying the (PS) condition, has only Dnitely many critical points in I-'[a,b[, wbere a,b are regular values o( I. Then there exists an _ > 0 such that

M.(f)

~

M.(g)

q = 0,1,2, ...

(or all 9 E C' (M,llI.') satislYing (6.1) and the (PS) condition, where Mq( . ), are the Morse type numbers with respect to (a, b).

"q,

Proof. Straightforward. Theorem 6.2 implies that the Morse type numbers are lower semi-continuous under CO-perturbation. As a direct consequence, we have a result due to

Ambrosetti-Coti Zelati-Ekeland [ACEI[.

6. Pert_lion on Gnlico/ Mtmi/oldJJ

133

Corollary 6.1. A..sume that f, 9 E C'(M,JR') satisfy the (PS) condi· tion, with c = inf I > -00, d = inf 9 > -co, and that there exists q > 0 such that H. (KcU» #< 0 where KcU) is the critical set of I with critical value c.

Suppose that there exists E > 0 such that KU) n r'(c, c + E]

=0

and

Sup {]g(x) -/(x)llx E Ie+,} <

E

2'

Then 9 has at least two critical points.

Proof. Obviously K.(g) #< 0. We prove by contradiction that if K(g) Kd(g) = single point, then M.(g) = O. But

=

M.(g) ~ M.U) = rank H. Ue+,) = rank H. (KcU» > O.

This is impossible. Next, we tum to C'-perturbation. It is equivalent to the C'-perlurbation of the variational equation df(x) = 9. The inverse function theorem is applied.

Theorem 6.3. Let M be a C'-Hilbert-Riemannian manifold, and let bea C'-function. Let E be a compact nondegenerate critical manifold for f O = 1(0, . ). Assume that d" is a Fredholm operator V x E E. Then there exist an e> 0 and a neighborhood U ofE such that V 0 < lei < e. The function I' = feE, • ) has at least Cat(E) critical points in U.

I: [-I, I] x M _Il'

rex)

Proof. We regard I' as a family of functions defined on the fibers of a has a nondegenerate critical normal disk bundle over E. The function point on each fiber. We shall prove by the inverse function theorem that I' has the same number of critical points.

r

1. We choose a tubular neighborhood W of E, which is diffeomorphic to a normal disk bundle NE(r), r > 0, in the following sense: V x E W, 3 a unique decomposition, % = Px E E and v = Qx E N.(E)(r) such that x = exp.l 'V. Suppose that M is modelled on a Hilbert space H, If % E E. We consider the orthogonal projection >r. :

H _ 1m d" 1°(%).

Therefore If x E W

df'(x)

= 9 <= {

>r,d!,(x) = 9, (1 - "') df'(x) = 9.

(6.2) (6.3)

134

Critical Point Theory

2. Let

¢(E, z, v) We have ¢(O, z, 0)

= ",d/O(z) = 0,

=",dr(x). V z E E, and

which is invertible from 1m ,p I"(z) into 1m ,p I"(z'). By the implicit function theorem, one has € > 0, a neighborhood Uo C NE(rh and a aI_map u: (-E,E) x E - Uo such that

¢(E,Z,U(E,Z)) = 0 and u(o,z) = 0, which solves equation (6.2) uniquely in Uo. We shall prove that the section z - U(E,Z) of NE(r) provides the critical points of 1'. 3. Indeed, let

E, = {x = exp, q(E, z) I z E E}. Then E, is a oompact oonnected manifold, with Cat(E,) = Cat (E) since exp(u(_, . )) is a diffeomorphism between E and E,. ··.Note that and

H ~ ker d2 /(z) Ell 1m df 2 (z), where Ee denotes orthogonal direct sum, we have

kerd2 /(z) = T,(E) and 1m d'/(z) = N,(E). Let U

cW

be the pull back of Uo C NE(r), then

un K (J') =

{x E E, I in which (6.3) holds} = {x E E, I df'(x) E T,(E)} = {x E E, I df'(x) E T. (E,)} = K (I'll::,),

where K(J) denotes the critical set of I. This is due to the fact that as I-I> 0 small, T,(E) is closed to T.(E,). 4. V 0 < lEi < E, the function 1'1>:, has at least Cat(E,) = Cat (El critical points, according to the Ljusternik-Schnirelman Theorem. The conclusion follows. Finally, we study the aI-perturbation.

6. Perluriction on Critia&l ManiJoldt

135

Theorem 6.4. LeI M bea Hilbert-Riemannian manifold, and Jell, 9 e CI(M.RI). Suppase thst both I and f + 9 BBlisfy the (PS) condition, and thaI E is a connected nondegenerate critical manifold of I with finite index. If 9 is sulliciently small in the CI-norm in a neighborhood U of E in M, then #(K(f + g) n U) ;?: CL(E) + 1. FUrthermore, if I. 9

e C',

and I

+ 9 is non degenerate in U,

then

""

#K(f + g) ;?: L:/3;(E) . • =0

Proof. According to the (PS) condition, E is compact. With no loss of generality, we assume K, = E. There exist 6 > 0 and two tubular neighborhoods II; W of E satisfying (1) V eVe We W c U, (2) V"" NE(rl). W "" NE(r2), 0 < rl < r2 < rl + 1, where NE(r) is the normal disk bundle of E, (3) We int(f'+I\!'_I)and/-l(c-6,c+6JnK=E. According to Theorem 7.4 of Chapter I,

where (C(r),C(r» is the negative sphere bundle over E. Let us define cp e C'(M, RI) satisfying cp(x) = 1 for x e v. cp(x) x '1 W, 0:5 cp(x) :5 1 V x e M, and 11d 1. According to the (PS) condition, there exiSts eo > 0 such that

IIdl(x)1I ;?: eo Fixing 0 < e < Min

= 0 for

V x e W\V:

H, ;i2; }, we obtain

IId(f + "'9)(x)1I ;?: eo -lIclg(x)II-IId 0 V x e W\V, if IIglb(U) < e. Thus the function F(x) = (f + cpg)(x) satisfies the (PS) condition and pOSSES ES the same critical points as J + 9 in W. Moreover, we have (Fe+" Fe_o) = (fo+o'/,-o). Using Thorn's isomorphism theorem,

where k = ind (f, E) .

136

Critical Point The<W!l

Since

H. (Fc+6.Fc_') '" H. (fc+ •• lc-')

'" H. (IC+6\/c-•. r'(c- 6») '" H.

(c(r).k-(r»).

and

H'

(Ic+.\/c-.) '" H' (e-(r» '" H'(E),

preserving the ring structure. there exist w"Wo •...• Wl E H·(E). t = GL(E). with dimw, > O. i = 1.2•... • t. such that w. UWoU",UWl # O. from which we obtain (Zo].(Z.] •... • (Zd E H.(Fc+o• Fc _') such that (Zo] < (Z.] < ... (Zd· Therefore there are at least l + 1 distinct critical points of F. and then. of 1 + g. in U. provided by Theorem 3.4. Thrthermore. if 1 + 9 is nondegenerate. tben F = 1 + 'Pg is also. Since H.(Fc Fc-o) '" H._k(E). we have at least 1:;':0 A(E) critical points. p~ovided by the Morse inequalities.

+6.

Remark 6.1. Theorems 6.1 and 6.2 were given by Marino Prodi (MaPI]. Theorem 6.3 was obtained by Reeken (Reel]. A special case of Theorem 6.4. in which M is assumed to be finite dimensional. is a version of a theorem under the name of Conley Zehnder by A. Weinstein (Wei2]. cf.(ChaI5].

6.2. Uhlenbeck'. Perturbation Method Lemma6.2. LetM be a G'-Fins/er manifold. and Jet I. g E G 1 (M.1ll. 1 ). Suppo6e that f is hounded below and that 9 ~ O. For e E (0.11. let = f + eg. Assume that (1) IIdgll is hounded on sets on which g i. hounded. (2) satisfies the (PS)c condition. for some c. Then the function h = g(c - J)-1 satisfies the (PS).-. condition and K,_l (h) = Kc(J'). where K.(f) = the critical set of 1 at JeveJ b.

r

r

Proof. By computation dh

= (c- J)-ldg + (c = (c - J)-'g(df

Thus

=.-'

h(xo) xo E K,-t{h) # { dh(xo) = 0

J)-'gdf

+ h- dg). 1

6. Perturbation on Crilic4I Mani/old6

137

I'(xo) = I(xo) + h(xo)-'g(xo) = c .,. { dJ'(xo) = dJ(xo) + h(xo)-'dg(xo) = (c -/(xo))2dh(xo) = 9 .,. Xo E Ke (I') . Suppose that {xi} is a sequence along which

h(xi)

--+

e-'

and dh(xi)

--+

9.

By the same algebraic computation, we have

I'(xi)

IW' (xi)11

--+

c and

= II(c-/(xi))'g(Xi)-' dh(xi) $

Ig (xi) h-' (xi)1

Since 0 $ 9(xi) $

IIdh (xi)11

+ (e - h(Xi)-') dg(xi)II

+ 0 (lIdg (xi)II)·

(~+

l)(c-/(xi» is bounded, provided by the fact that we obtain dJ'(xi) --+ 9. Again, using assumption (2), it follows the (PS),_. condition for the function h.

I is bounded from below. Combining this with assumption (I),

Corollary 6.2. Under the conditions of the above theorem, if, further, Ke(l') = 0 VEE (0, EoJ, then (I'O)e is a strong deformation retract of Ie. Proof. Since 9 ;:: 0 and eo > 0, (I'O)e C Ie. From the theorem,

there is therefore no critical value of h between EO' and +00. The (PS),_. condition for the function h II E E (O,EoJ, implies that (I'O)e = h _. is a strong deformation of Ie

= hoc.

'0

Definition 6.1. Let M be a Finsler manifold, and let I E C'(M,IlI.'). We say that I satisfies the e-deformation property, if (1) I(K(I)) is closed; (2) for any interval la, hJ C lit' on which I has no critical value, i.e., K (I) n I-I la, bJ = 0, there exist. a family of functions I' E C'(M,IlI.') with I' ;:: IV E E (O,lJ such that the level set (I'). is .. strong deformation retract of I. II e E (O,lJ, i.e., 3 'I' : 10,lJ x I. --+ I., satisfying (i) 'I' (0, . ) = id, (ii) '1'(1,/.) c (f')., and (iii) '1'(t, . ) 1(1'). = id(l').'

138

Critical Point

1'hwrlI

Theorem 6.5. (E-Minimax Principle) Let:F be a family of suboets of M, and let f E C'(M,JR') satisfy tbeE-
If (1) c is finite, (2) :F is invariant with respect to 1/' liE E (O,lJ where 11' is tbe strong deformation retract satisfying (i)-(iii) with any interval [a, bJ containing c as an interior point. Then c is a critical value of J. Proof. Suppose that c is not a critical value provided by the closeness of f(K(f», there exists 6 > such that K(f) n r'lc - 6,c + 6J = 0. Choosing Fo E :F such that Fo C fc+', we have if(l, Fo) C g_" but

°

f(x):5 r(x):5 c- 6 II x E 1/'(l,Fo) E:F. This is a contradiction.

Theorem 6.6. Let M be a C'-FinsJer manifold modeled on a separable Banach space with differentiable norms. Let f, 9 E C'(M, JR'), satisfy the following assumptions: '.,(1) f ~ -m > -00, 9 > and = f + eg satisfies tbe (PS) condition, lie > 0. (2) IIdg(x)1I is bounded on sets on which 9 is bounded.

° r

(3) UO<'';'o K(f') n (f<) 'la, bJ is compact for some 60 > 0. (4) a,b ¢ f(K(f». And II Xo E K(f) c r'(a,b), 3 a neighborhood U such that K(f') n U consists of a curve X(E), with x(O) = xo, E E 10,6J, and X(E) are nondegenerl1te (in the sense of Section 4, Chapter I) and of the fixed index. Then f. with handles, adjoined in a corresponding fashion to the critical points of f with values in (a, b), is 11 deformation retract of fb' The dimensions of the handles correspond to the dimensions of the indices at these critical points.

°

Proof. First, assume b < +00. Then 3 6 > such that a, b ¢ f'(K(f')) II e E 10,6]. We claim that (f<)-'la,bJnK(f') consists of all these families of curves

x(e). In fact, if not, then 3 Ej -+ 0, Xj E K(f'j) n (f'j)-'la,bJ; but x; do not lie on any families x(e) defined in (4). However, by assumption (3), x; -+ x' E K(f) n r'la, b). This contradicts assumption (4). Then Corollary 6.2 is applied: (I'). ~ f., (f')b ~ fb. Since the Morse handle body theorem for f' is known (Chapter I, Section 4)

•

(I'). u U h;(Dmj) ~ (I')b' j=1

6. Perturbation on CriIic4l Mani/old6

where h;(Dm;) denotes the attached handle, i = 1,2, ...

139 ,8.

We obtain

•

I. u U h; (Dm;) ~ I.· j""'. Second, if b = +00, then 3 bJ - +00 such that b; ¢ I(K'), and E; ! 0 with the following properties. (1) VEE lo,EJ)' bJ- .. bJ are not critical values of /" (2) V "'0 E KU') n U,)-lla,b;), it lies on the unique one parameter family of nondegenerate critical points of /' near the critical points of I. Then (1';).; can be retracted to (I'J).;_I with bandies corresponding to critical points of 1'; with values in (bJ_I,bj ). In addition, U';-I).;_I is a deformation retract of U'j ).j. The sequence of retractions gives the desired result for 1+00.

Remark 6.2. The material of this subsection is taken from K. Uhlenbeck IUhI2).

CHAPTER

III

Applications to Semilinear Elliptic Boundary Value Problems

Semilinear elliptic boundary value problems have attracted great interest in the applications of critical point theory because they are good models to deal with multiple solutions problems with respect to both results and methods. 1. Preliminaries

Let us turn to some notation and basic facts in the theory of partial differential equations. ) .•et !l C llI.n be a bounded open domain with smooth boundary 00. For a nonnegative integer vector Q' = (all' .. I an) we write

to denote the differential operator, with lal = a, + ... + an. Let V(!l) be the function space consisting of Coo functions with compact support in !l, and let V'(!l) be the dual of V(!l), i.e., the Schwartz distribution space. For each integer m ~ 0, we denote

cm(fi) = {u: fi ..... llI.' I {J"u is oontinuous on fi, lal with norm

lIuli m =

L

~ m},

su!'.l{J"u(x)l·

1D:I~m zEO

For p

~

I, and an integer m 2:: 0, we denote

W;'(!l)

= {u E LP(!l) I (J"u E LP(!l),

lal ~ m} ,

where V is the p-th power integrable Lebesgue space, and ential operator in the distribution sense, with norm

{J'"

is the differ-

141

1. Preliminaries

w;,(n) is called the Sobol•• space. In particular, if p = 2, Hm(n) stands for W2'(n). The closure of V(n) in the space W;,(n) (Hm(n) and C"'(i'l) o

_

is denoted by w;:,(n) (HO'(rl), CQ"(n) respectively). o

The dual space of W;:'(rl) (and HO'(n» is denoted by w,;m(rl) (and H-m(rl) resp.), where + ~ = 1. The following inequalities are applied very frequently.

*

Poincare inequality

,

,

(!olul" dx)' ~ c(n) (!olvul"dx)'

o

Vu E w!(n)

where Vu denotes the gradient of u, and C(rl) is a constant independent ofu. Sobolev inequality. Suppose that for I ~ p, r < 00 and integers t <: m <: 0, we have (I) If ~ ~+ then the embedding W:(rl) <-+ W;'(rl) is continuous. If the inequality ~ is replaced by a strict inequality <, then the embedding is compact. (2) If ~ then the embedding W:(rl) <-+ C"'(i'l) is continuous. If the inequality ~ is replaced by a strict inequality <, then the embedding is compact.

*

*

':m,

':m,

For a function f E C(i'l • lit' ,lit' ), suppose that there exist constants a, C > 0 such that If(x, t)1 ~ C (I + IW). Then the following nonlinear operator: u ~ f(x,u(x»

maps boundedly and continuously from £P(rl) to L'(n), with p = aq, for example, cf. [Berl]. The operator is called the Nem"tcJci operulor. Applying this result, in combination with the Sobolev inequality. we see tha.t the functional J(u) =

!o f(x, u(x)) dx

is well-defined and continuous on the Sobolev space HJ(n) if

2n

a<--

- n-2'

If we further assume that growth condition:

If:(x, t)1

f

E

c' (i'l x lit'. R')

~ C (1 + 111°)

with"

satisfies the following

~ : ~ ~,

142

Semilinear Elliptic Boundary Value Probknu

then the funct.ional J is C' on the space HJ(Il). with differential

(dJ(u).v) = Furthermore if IE

L

f.(.,.u(.,»v(.,)dx.

"v E HJ(Il).

C'(n x JR'.JR'). satisfies

1/:;(.,.t)1 ~ C(1 +IW)

with 0 S

n~2'

then J is C' on HJ(Il). with

rP J(u)(v. w) =

In I:a.,.

u(x»v(.,)w(.,) dx "v. wE HJ(Il).

As for the Laplacian -A defined on L'(Il). with domain D( -A) =

H2(1l)

n HJ{Il).

it is a self-adjoint operator. The operator -A can he

w:

o

extended from n W!(Il) to 0'(11). 1 < p < 00. continuously. It is known that ker(-A) = {9}. and that K = (_11.)-' maps 0'(11) int!> itself continuously. and is 8 compact operator. Also the operator K maps L'(Il) into HJ(Il). such that

The following boundary value prohlem will he considered: -Au = g(x ...(x)) in n { ..100 = O.

(1.1)

where the following growth condition on 9 E C(!! x JR'. 11.1 ) is assumed: (1.2)

Ig(x.t)1 S C(1

(and if n :s; 2,

Q

+ ItIO)

0

~ :~~.

ifn

has no restriction) for constants C,

Q

~3 > O. We say that

... E HJ(Il) is a weak solution of (1.1) if

L

[V... · Vv - g(x ....(.,))v(x)] dx = 0 "v E HJ(!1).

If we define the functional (1.3)

143

1. Preliminariu

where

G(x,t)

(1.4)

then J is

=

J.'

g(x,{)d{,

e 1 on HJ (0) with

(dJ(u),v) = klvu.vv-g(x,,,(x)).v(x)]dx VveHJ(O). This means that the differential equation (1.1) is just the Euler equation of the functional J and the weak solutions of (1.1) are critieal points of J, and conversely. Since it is well-known that weak solutions of (1.1) are classleal solutions of (1.1), if the function 9 is smooth enough (cf. IGiTIIl, it is enough to look for weak solutions of (1.1), i.e., the critieal points of J. The notion of sub- (or super- ) solutions of the equation (1.1) is also important. We say y e CO(O) n ern) (or a) is a sub- (or super- resp.) solution if

{

-Ay:5 g(x,y(x» in 0

(or - Aa

Yloo :5 0

aloo ~ 0).

~

g(x, a(x)) in 0

If furthermore y < a, then we say they are a pair of suI>- or supersolutions. They are ealled strict if they are not solutions. Fbr second order elliptic operators, (especially for -A) the maximum principle plays an important role in both qualitative and existence studies. Theorem (Maximum Principle). Suppose that" equation

(1.5)

e CO(O) satisJies the

-A" :5 0 in O.

If u attains a maximum M at a point of 0, then" '" M in O. Theorem. (Strong Maximum Principle) Suppose that" e COCO) satisfies (1.5). Suppose that u :5 M in 0, and" = M at a boundary point P. Assume that P Ii... on the boundary of a ball K 1 in D. If" is continuous in 0 U P and an outward directional derill8tive &u/Bn exists at P, then &u/Bn > 0 at P unless u '" M.

Corollary 1.1. The operator K = (_.0.)-1 is positive, in the sense that it map" nonnegative functions to nonnegative functions. Particularly, K : LP(O) -+ eJ(n) for p > i, maps nonnegative functions to the interior of the positive cone in eJ (n). For positive operators, we have the Krein-Rutman Theorem which asserts that the first eigenvalue (-A) is simple. More generally, we have

144

Semilinear Elliptic Boundary Value Problerru

Theorem. (Kat<>-Hess [KaRl]) Suppose that m E C(i"i), /Uld that there is a point Xo E f! such that m(xo) > O. Then the equation

-6u(x) = Am(x)u(x) { ul8ll = 0

xEf!, AER'

admits a principle eigenvalueA,(m) > 0, characterized by being the unique positive eigenvalue having a positive eigenfunction. Moreover, A, (m) has the following properties: (1) if>' E C is an eigenvalue with Re >. > 0, then Re >. 2: A,(m). (2) J.!,(m) := l/A, (m) is an eigenvalue of the operator K·(m·) : L2(f!) ~ L2(f!) with algebraic multiplicity 1. In the applications, sometimes we would consider the restriction J of J on a smaller Banach space CJ (n) , where J is defined in (1.3). The functional I may lose the (PS) condition (on CHn), even if J has on HHf!)). However, hy a bootstrap iteration, the following is proved in

[Cha3). Theorem 1.1. Under assumption (1.2) with a < ~, ifn > 2, suppose ~at 9 E C' , and that J satisfies the (PS) condition; then the functional I possesses the following properties: (1) I(K) is a closed subset.

(2) For each pair a < b, K n I-'(a, b) = 0 implies that 1. is a strong deformation retract of J"\Kb, where K is the critical set of J ( and also

1). Thus for any isolated Po E K, we have Corollary 1.2. c.(I,Po) = C.(J,Po) with integral coefficients.

Claim. For any open neighborhood U of Po, let V = U'ea' ~(t, U), where '1 is the negative gradient flow of J. We have c. (J,Po)

= H. (Jo n V, (Jo\{Po}) n V;Z) = H. = H.

(L, nv,.L,nv;z)

= C.

(3,+, n v';,_,V;z)

(I,Po) ,

using the Palais Theorem at the end of Chapter I, Section 1, where c = !(Po) and < > 0 is suitably small. 2. Superlinear Problems

The classification of the semilinear elliptic BVPs into snperlinear, asymptotical1y linear, and sublinear is very vague. Roughly speaking, it describes

2. Superlinear Pro6lems

145

the growth of the function g(x, u) with respective to u in (1.1). But sometimes g(x, u) is superlinear in one direction, but subUnear in the other, so that it is not easy to classify them very clearly. Nevertheless, we follow the customary notation in the literature. In the following, (1.2) is assumed (0 < ~, subcritical, 0 = ~ is called critical). Our first result in this section is tbe following.

Theorem 2.1. Assume that the functional J defined in (1.3) satisfies the (PS) condition on the space HJ(O), and that J is unbounded below. Moreover, if there exists a pair of strict suI>- and supersolutions of equation (1.1), then (1.1) po....... at least two distinct solutions. Before going into the proof, we recall a weD-known result (cf. Amann IAmal)) that if there is a pair of suI>- and super- solutions y < il of (1.1), then there is a solution Uo e C of (1.1). One asks whether we can characterize the solution by the corresponding functional J? Now we shall prove that J is bounded from helow on C x = C n cJ(Il), where C = {u E HJ(O) I y(x) ::; ,,(x) ::; il(x) a.e.}, and then attains its minimum, which is the variational characterization of tlQ. Applying Example 1 from Chapter I, Section 4, we obtain the critical groups of Uo: k=O

(2.1)

k~O,

if it is isolated. Lemma 2.1. Suppose that y < il is a pair of strict suI>- and supersolutions of (1.1). Then there is a point Uo E. CX which is a local minimum of the functional J = JlcJ(l!)" Moreover, if it is isolated, then (2.2) Proof. One may assume that y(x) Define a new function

_ g(x,~)

< Ii(x), without loss of generality.

{ g(x, Ii(x» II (-~il(x», = g(x, ~), g(x,y(x» V (-~y(x»,

>y(x) ::; ~ ::; Ii(x)


where a V b = max{a,b}, and a II b = min{a,b}. By definition,

C(Il X lII. 1 ) is bounded and satisfies: g(x,{) =

g(x,~)

for

y(x)::;

~::;

fi(x).

g(x,~)

E

146

Semi/in""r Elliptic Bounda'll Value Problems

Let

- {) G(x, Then

= J.{ 0 g(x, t) dt.

8 E el(n x Il'), and the functional

defined on HJ (0) is bounded from below and satisfies the (PS) condition. Hence there is a minimum Uo which satisfies

i.e., Uo satisfies the equation

-t.Uo = g(x, Uo) { UoIIll! = O. According to the lJ' regularity of solutions of elliptic BVP and the strong o _

ma.ximum principle, we see that Uo E ex, the interior of ex in the CJ(O) top";,logy. (See Remark 2.1 below.) However, flex = Jicx = flex; therefore Uo is a local minimum of f. (2.2) follows from Example 1, Chapter I, Section 4. Under condition (1.2) J is well-defined On HJ(O). Since eJ(n) is dense in Hl(O), Uo must be also a local minimum of J. In the case when it is isolated, (2.1) holds. o

Remark 2.1. We verify Uo E ex. Claim. Since!! is 8 strict sub-solution,

-t. (Uo -!!) (x) { Uo - !!11lI! ~ O.

~

0, but not identical to 0, in 0,

It follows from the strong maximum principle, that Uo > !h and (uo - !!)11lI! < 0, where t. is the outward normal derivative. Similarly, we have Uo < il, and t.(il- Uo)11lI! < O. Therefore Uo is an interior point of ex in eJ topology.

I..

Proof of Theorem 2.1. We already have a local minimum so that it suffices to find Mother critical point. Since J is assumed to be unbounded below, 3 U, E HJ(O) such that J(UI) < J(Uo). A weak version of a link (mountain pass) is easy to see. JIBB(....) ~ J(Uo)

max{J (Ul). J (Uo)) :5 J (uo).

for h > 0 small.

2. Superiinear Pmblenu

147

Exploiting Theorem 1.2 (or Remark 1.2) from Cbapter II, there exists a different critical point. We present an example for the application of Tbeorem 2.1. Assume tbat (g,) (1.2) with a < ~; (g.) 3 9 > 2 and M > 0 such that

9G(x, t) $ t g(x, t) "x E fI, for It I ~ M; (ga) "E

L~ (fI) is nonnegative, but not zero.

Theorem 2.2. Under assumptions (g,), (g.), and (ga), the equation (2.3)

{

-~u =

g(x,u) -" in fI

uloo =

0

possesses at least two solutions, i[g(x,t) ~ 0 "(x,t) E fI x 0, g(x, to) > 0 and g(x, 0) = O.

lit', 3 to>

The proof is just a veri6cation of Theorem 2.1. Lemma 2.2. Under lJSSumptions (gil and (go), [or any" E L"""(fI), the functional J(u) =

(2.4)

10 [~ll7ul' - G(x,u(x)) + "u(X)] dx

satisfies the (PS) condition on HJ(fI). Proof. Let {u.} be a Bequence along which IJ(".)/ $ C, and dJ(u.)

~

.

~

First, {".} is bounded. In fact, 3 C2 , C3 , C, > 0 sucb that

-1. ~ ~II"./l2 _ ~ 1.

C, ~ ~IIu.II2

G(x, ".(x»

dx

IUk(z}I2':M

-1"1' /1".11 - C2

".(x)g (x, ".(x)) dx -Ihl'

IUk(z)I2':M

~ G- ~) 11"./1' + ~ 10 (V"kV'U. - Ihl . /lu.II -

~

11".11 -

C.

9 (x, "') u.) dx

C3

G-~ -.) 11"./1'

+ ~ (dJ (u.), u.) - C"

wbere 11·11, 1·1 and (, ) stand for HJ norm, L~ norm, and tbe HJ(fI) inner product respectively. Since dJ(u.) ~ 9, l(dJ(".), u.)1 $ .11".11 if we choose 2e < ~ - ~, then IIu. /I is bounded.

148

Semilinear Elliptic Bounda'1l Val.., Problem8

Let P = or + 1, and consider the following maps:

Yen)

,(s,-)

where ~ + } = 1. j is a compact embedding, as is i'. Both (_~)-I and g(x,·) are continuous. The boundednes8 in HJ(O) of {Uk} implies a convergent subsequence (-~)-I·i' .g(·,Uk')' Since dJ (u",) =

(_~)-I . j ' . g(. Uk')

Uk' -

--+

8 in HJ,

finally, we obtain a convergent subsequence {Uk'}' Proof of Theorem 2.2. It suffices to verify (1) J is unbounded below. (2) 3 a pair of strict suI>- and supersolutions for (2.3).

Claim (1). Since g(x, t) ~ 0 and g(x, to) > 0, so G(x, t) > 0 \I t > Max {to, M} we have g(x, t) 8 ->G(x,t) - t'

~

to.

For t

Hence G(x, t) ~ Ct" for some constant C > O. There exists a constant C 1 > 0 sucb that

J(u) $

In {~lvuI2 - Cu" + h· U}

dx+

1

C

for any nonnegative U E HJ (0). Noticing 8 > 2, say, if we choose u = tIP., where !PI > 0 is the first eigenvector of -~ with O-Dirichlet boundary data, and t > O,and let t --+ +00, then

J(tcptl--+

-00

Claim (2). The equation (2.3) has a strict supersolution 0, and a strict subsolution 11: -~11 = -h in n { ul8ll = O. By the Maximum Principle 11 < O. All conditions in Theorem 2.1 are fulfilled. The proof is complete. Example 1. The equation -~U = u 2 -

(2.6)

{

uI8Il = 0

h

inn

possesses at least two solutions, if (1!3) is satisfied.

2. Superlin""r Problems

149

Theorem 2.3. Suppa;e (g,), (g2) with G(x, t) > 0,

ItI ;:: M,

and

(g.) 9 E C'(i1 X lit') with g(x,O) = g,(x,O) = O.

Then equation (1.1) J1OO5' '" at least three nontrivial solutions. We need Lemma 2.3. Under the assumptions of Theorem 2.3, there exists a constant A > 0, such that J. '" SO" , the unit sphere in HJ(O) for

-0

> A where J is the functional (1.3).

Proof. By the same deduction, but by assuming G(x, t) > 0 If t, M, we conclude G(x, t) ;:: Cltl' If t, It I ;:: M.

ItI ;::

Thus If u E S"", J(tu)

We want to prove: 3 A f,J(tu) < O. In fact, set A= 2

If J(tu)(= " -

--+ -00

as

> 0 such that

2MIOI

t If

--+ +00. d

< -A, if J(tu) :0:

max Ig(x, t)1 (',')Ellxi-M,MI

d,

then

+ 1.

I" G(x, t,,(x»
d d/(t,,) = (dJ(tu), u)

-In ~ {In

= t

u(x) . g(x, t,,(x»
:0:

G(x, tn(x»
~

In

t,,(x) . g(x, t,,(x»
o}

:0:

~t {(-01_~) f tU(X)g(X,tU(X»
:0:

~t {(-01- ~) CO { Itl'lu(x)I'
The impUcit function theorem is employed to obtain a unique T( u) E C( S"" , It') such that J(T(u)u) = a If u E S"".

150

Semilinear Elliptic Boundary Value Problems

,

Next, we claim that IIT(u)]] possesses a positive lower bound < > 0. In fact, by (&4), g(x,O) = gax,O) = 0, J(t,u) = " - ott') VuE 8"". The conclusion follows.

Finally, let us define a deformation retract 7/ : [0,11 x (H\B,(8» H\B,(8), where H = HJ(fI), and B,(8) is the <-ball with center 8, by

~

7/(B, u) = (1 - B)U + BT(u)u. VuE H\B,(8). This proves H\B,(8) '" J., i.e. J. '" 8"".

Proof of Theorem 2.5. 1. Provided by (&4)

(2.7)

1

J(u) = 211ull' + 0 (i]uIl 2 ),

so, 8 is a local minimum, and C. (J, 8) = 6"" G. 2. We find two nontrivial solutions. Let us define g+(x, t)

={

g(x, t) 0

and

where

G+(x,t)

= /.' g+(x,s)ds.

Again, J+ E C 2 (HJ(fI),JR') satisfies (PS), using Lemma 2.2. As in the proof of Theorem 2.2, we also have

where '1" > 0 is the first eigenvector of -6. with O-Dirichlet data. On the other hand 3 6 > 0 such that

provided by (2.7). The mountain pass lemma (Theorem 1.4 from Chapter II) is applied to obtain a critical point u+ E HJ (fI). with critical value c+ > 0, which satisfies

= g+(x,u+) "+]"" = O. -.6.u+

{

2. Super/in..... Problems

By using the Maximum Principle, u+ J. Analogously, we define

g_(x, t) = {

~

151

0, so it is again a critical point of

~(x, f)

t:o;O f

> 0,

and obtain a critical point ,,_ :0; 0, with critical value c_ > O. Chapter II (Theorem 1.7) and tbe Kato-Hess Theorem imply that C.(h, u±) = 6,1 G. According to the Palais Theorem (cf. Section I), we have

C.(h,u±) = C. (J±,u±) = C. (J,u±) , where

J = JlcJ; and again, C.(J,,,±) =

C.(J,u±). Therefore

3. Suppose that there were no more critical points of J. The Morse type numbers over the pair (HJ(fl), J.) would be

Mo = 1, M) = 2, Mq = 0, q ~ 2, but tbe Betti numbers (J.=O Vq=0,1,2, ... ,

since H.(HJ(fl),J.) '" H.(HJ(fl),S~) '" O. This is a. contradiction. Example 2. The function t~O

f
Theorem 2.4. Under assumption. (gl), (g,) with G(x, t) > 0 for M,and (go) g(x, t) = -9(X, -f)

V (x, f) E fl

It I ~

X IlI.l.

Equation (1.1) possesses infinitely many pairs ofso/ution•.

Proof. The functional defined in (1.3), J E Cl (HJ (fl), Ill. 1 ), is even, and satisfies the (PS) condition.

152

Semilin""r Ellip!ic Boundary Value Prob/e"..

According to (g,), we have (2.8)

J(u)

~ ~lIull2 -

C

(1 + lIull~~~,)

where C > 0 is a suitable constant. By using the Gagliardo-Nirenberg inequality, (2.9) where C, is a constant, and 0 < {J < 1 is defined by

1

-={J 0+1

(I2 nI) ---

1

+(l-{J)·-. 2

Substituting (2.9) in (2.8), for u E IJBp(O), we have J(u)

~ ~p2 - C2p(o+')~lIulli2-P)(0+1) -

C•.

Let A, < ),2 $ ),. $ ... , be the eigenvalues of (-Ll.), associated with eigenvectors fPl,Cf':Z,¥'3, .. ·, and let Ej = span {CP1,Ip:Zl'" ,'Pj}. i = 1,2, .... The variational characterization of the eigenvalues provides the estimates

Hence

where 6 = -~(I - (J)(1 + 0) < O. Since ),j - +00 as j _ 00, we choose p, jo such that 1- 2C2p o+' ~~, { p2 > SC•.

o-'A1

Thus

1 J(u) ~ gp2 > 0 VuE IJBp(O) n E/;,.

Since all norms on a finite dimensional space are equivalent, and since it was already known that G(x, t) ~

Cltl"

V t, It I ~ M,

there exists R j > P such that

J(u) $ 0 VuE Ej\BRj(O)

j

= 1,2, ....

3. A"lI"'ptotiaJl/y Lin .... Problem.

153

According to Corollary 4.2 in Chapter II. the proof will be finished if the (PS), condition with respect to the subspaces {Ej/j = 1.2•... } is verified. However. the verification is similar to that of the (PS) condition given in Leroma2.2. Remark 2.2. Theorems 2.1 and 2.2 are taken from K. C. Chang [ChaJ-

4]. and Theorem 2.3 from Z. Q. Wang [WaZ2]. Remark 2.3. There is a beautiful application of the Morse index estimates to the following perturbation result: Tbe equation

-Ilu = lulp-lu - /(x) { ul80 = 0

in fl

n:,.

has infinitely many solutions. if / E LO(fl). and 1 < p < cf. A. Bahri. H. Berestycki [BaB1). A. Bahri. P-L. Lions [BaLl.2). Dong Li [DoLl). M. Strowe [Strl) and P. H. Rabinowitz [Rab4). 3. Asymptotically Linear Problems

3.1. Nonre80nance and Resonance with the Landesman-Lazer Condition First, we assume that the function 9 is of the form

g(x. t) =

(3.1)

,i.t + rp(x. t).

where 'I' E C(n x JRI.JRI). satisfying rp(x. t) =

aUt I)

as

Itl- 00

uniformly in

x E fl.

We study the BVP (1.1) via the abstract theorems of Section 5 in Chapter II. Set H = HJ (n). A = id - X( -Il)-l.

l' = -10

~(x.t) = and F(u)

rp(x •• )d.

oI>(x. u(x)) dx.

Problem (1.1) is equivalent to finding critical points of the functional 1 J(u) = 2(Au.u)

+ F(u).

154

Semuit\ear Eluptic Boundary Value Problerru

Theorem 3.1. If AIt d( -£1), the spectrum of (-£1), then (1.1) sesses a solution. If we further assume that (3.2)

{

'I'(x,O) = 0

'I' E G I


l'I'ax,t)1

(0 X

p0s-

]RI,IlI.I),

~ G (1 + Itl~)

and 3 AE d( -£1) such that

(3.3)

then (1.1) possesses a nontrivial solution. Theorem 3.2. If XE u( -£1), and 'I'(x, t) is bounded, and we assume the Landesman·Lazer condition that (3.4)

1~ (x, t n

ti 'l'i

(X»)

dx

--+

+00

as

)=1

where Span{'I'I,'!'2, .. ' ,'I'no} = ker(-£1 - XI).

sol~tion. FUrthermore, if (3.2) holds, and if 3

); < AO, then (1.1) possesses a

Then (1.1) possesses a such that X<

XE u( -£1)

nontrivial solution.

Proof. Let Al < A. ~ >.a ~ ... be the eigenvalues associated with the eigenvectors '1'1,'1'.,'1'., ... of (-£1) with O-Dirichlet data. Let H+ = span {'I'j' I Aj

Ho = ker (-£1-

> X},

AI),

H _ = span {'I'j I Aj <

and

X} .

Then (HI) and (H.) of Theorem 5.2 of Chapter II hold. Since dF(u) = (-£1)-I'I'(X,U(x)), F E GI(H,IlI), and then IldF(u)1I = o(lIull) if (3.1), and IIdF(u)1I ~ G, a constant, if 'I'(x,t) is bounded. Note that 'P(z,'} _--=--~, L2 So dF is compact. Therefore (H.) also holds. We apply Theorem 5.2 of Chapter II (with Ho = 0) in the case n = 0, to obtain the existence of a solution in Theorem 3.1, and, in the case n = 1, to prove the existence of a nontrivial solution. In fact, '"Y = dim(Ho Ell H_) = dimH_ = Max (j I Aj < AI, and m_(O) = Max (j I Aj < Aol, mo(O) = dim ker( -£1- Aol). Condition (3.3) is equivalent to '"Y It (m_(O),m_(O) + mo(O)].

155

3. A.oymplotica/ly Linear Probl....

Similarly, Theorem 3.2 is proved by the fact that the Landesman-Lazer condition (3.4) implies F (Pou)

--< -00

as

IIPoull -

00,

where Po is the orthogonal projection onto Ho. Remark 3.1. Problems in wbich X ¢ u( -6) are called nonresonance, and in which XE u( -6) are called resonance.

Theorems 3.1 and 3.2 are due to Amann-Zehnder [AmZl] and Cbang [Cba2] respectively. See also Liu [Liu3]. For a similar problem from the two point boundary value problem in ordinary differential systems, we can get a hetter estimate of the number of solutions. Let us assume that G E C'([O,,,] x IItn,IIt') satisfying (1) G(t,9) = 0, G.(t,8) = 9, (2) 3 integer k such that

k' In < A",,(t) < (k + I)' In, where In is the n x n identity matrix and the limit Aoo(t) = limlul_oo
(3) 3

ic E :1:+ such that PIn < Ao(t)

=
< (ic +

1)

2

In

where the notation B < A means that the matrix A - B is positive definite.

Theorem 3.3. [WaZ3] Under assumptions (1)-(3), if k '" equation -u = Gu(t.u) .

{ u(O) =

ie,

then the

u(,,) = 8

possesses at least two nontrivial solutions in C'([O...].lItn ).

Proof. Again, we use the abstract framework. H = HJ([O. ,,].lItn ). A = id - (-1.~rlA",,(t) .• H+ = span {e;sinjt I i = 1.2.... • n. j 2: k+ I} and H_ = span {e;sinjt I i = 1.2•... • n. O:=; j:=; k}, where {e;}~ is the basis of IItn. Condition (2) implies that Gu(t.u) is asymptotically linear. Indeed. for any 1lo E lit" we have IIG.(t.u) - A",,(t) . ull ••

:=; IIG.(t. Uo) - A",,(t)UoII.· + IIG.(t.u) - G.(t.Uo) - A",,(t)(tl - Uo)II.n :=; IIG.(t. Uo) - A",,(t)UoII.·

+ Sups E [0.1) I(tPG(t. Stl + (1 -

s)"o) - A",,(t)II'

lIu -1lon. •.

156

Semilinear Elliptic Boundary Value Problems

Thus

IIGu(I, u) Let F(u) = -

-

A=(I)u!lan =

0

(liull •• )

as

lIull •• -

J; G(I, u(I)) dl + ! J; (A=(I)u(I), u(I) IIdF(u)IIH =

00.

dl; then

o(llullH)

and dF( u) is compact. The following functional is considered: 1 J(u) = 2(Au,u)+ F(u).

By simple computations, we have (cPJ(8)u,u) =

1 r 210

[1 .. (1)12 - (Ao(l)u(I),u(I)j dl.

Condition (3) implies that 8 is nondegenerate with m_(8) = (k + l)n. But 'Y = dim H _ = (k + l)n f. m_ (8). We conclude that there is a nontrivial solution "0(1). Moreover I note that

ker (d 2 J("o))

= {u E HJ (10, "l.ll!.n) I u(O) = tI(,,) = 8 and

-it = d!G(I, u(I))u in (0, ,,)} , which is of dimension:::;: n. The condition

is fulfilled. Corollary 5.2 in Chapter II can be applied to obtain the second nontrivial solution.

3.2 Strong Resonance Now we study a new class of resonance problem, called strong resonance, in

which the Landesman-L820r condition is dropped so that the (PS) condition does not hold. The function 9 is again written in the form (3.1):

g(X, I)

= ).1 + cp(x, I) >. E a( -1'1).

The Landesman-Laoer condition (3.4) placed on cp is replaced by the following:

(H) V ~j E 111.'"0,

I~jl

_ 00,

VUj _ u in HJ(O) , and V v E HJ(O);

we have

(3.5)

Iim (cp(X,Uj(X)+ 3-+00

in

~~;ei(X))V(X)dx=O, i=l

3. AsymptotiC41ly Linear /'robl....

157

and lim

(3.6)

f

J-oo}n

4> (x, Uj(x)

+

I:

{je,(x») vex) d:!: = 0

i=l

where {e,(x)};"O is an orthononnal basis of the eigenspace ker(-~ - XI), and {j = ({j,{1, ... ,{j"). Again we study the critical points of

1 J(u) = 2(Au, u) + F(u) as above. The only difference is that the assumptions F(Pou) _ -00 as IWoull - 00, and the boundedness of IIdF(u)1I are replaced by F(u) - 0, and dF(u) _ IJ as IlPoull - 00, where Po is the orthogonal projection onto ker( -~ - XI).

Lemma 3.1. Under assumptions (H.) and (Ho) of Theorem 5.2 of Chapter II, if we assume (H~) (113) F

E

C'(H,R'). dF is bounded and compact.

F(u.) - 0,

and dF(u n ) _IJ as

IlPounll- 00.

Then J satisfies the (PS)c condition \I C F O. Moreover, if J(Hn) - 0, dJ(u.) - IJ along a sequence Un, then 3 a subsequence (still denoted by un.), with the property that either 'Un converges, or

11(1 -

Po)u.lI- 0

as

IlPounll- 00.

Proof. Suppose that (3.7) (3.8)

1

= 2 (Au n , un) + F(u n ) - c dJ (Un) = AU n + dF( Un) - O. J(u n )

Decompose Un into u~ + u; + u~, where u; is the orthogonal projection of Un onto H± I and u~ == Pou". Then

(3.9)

!(Au;,u;)! =

HAHn,u;)!

= !(dJ(un )-dF(un ),u;)1 $Cliu;li·

Since A± has a bounded inverse on H h (3.9) implies the boundedness of u~. If u~ is bounded, so is u,u and then it has a weakly convergent subsequence. By the compactness of dF, and the finite dimensional conditions on Ho, we get 8 strong convergent subsequence. On the contrary, if

158

SemUinear Elliptic Boundary Value Problems

Ilu~lI-+

(3.8),

00

-+ 0, dF(u n ) -+ O. -+ 0, i.e., c = 0.

(we ignore a subsequence) then F(u,.)

u; -+ O.

Finally, (3.7) implies that J(u n )

From

A new idea is employed to avoid the difficulty arising from the lack of compactness (the (PS). condition). We compactify the space H by adding some 00 points, extend the critical point theory (in particular, tbe deformation tbeorem) to the enlarged space, and distinguisb tbe genuine and the pseudo critical points that come from the 00 points. We proceed as follows: 1. Compactify tbe subspace H.

E = H. U {oo} '"

= ker A.

Let

sm., mo = dim H.,

and

E = Hif Define

X

E

where Hif = H+

_( ) _ { J(v + s) J x , ,(Av,v)

E!)

H_.

(v,s) E Ht X H. (v,s) E Ht x {oo},

where x = (v, s) E E. The assumption J(v + 8) -+ as IIslI-+ 00 implies tbat Jis continuous (but not differentiable in general). Although we cannot comment on the critical set of J, we call K = KU{(O,oo)} the pseudo critical sct, where K is the critical set of J. Points in K are said to he genuine, and the single point (0,00) to be pseudo. Thus

°

-

-

K< := K

__ I

nJ

{

(c) =

K<

K.U

{()} 0,00

ifc';'O, if c = 0.

2. The first deformation theorem can be extended to study the pseudo critical set. Namely, we have

Lemma 3.2. Under the assumptions (H,), (H.) of Theorem 5.2 of Chapter II, and (H;), if K. is bounded in II, then V c E R', V closed neighborhoods N of K< in E, there exist 'I E C«O, I) x E, E) and constants < e < !" such that

°

(1) '1(0,·) = id, (2) 'I(t, -) 10 1-'1<-<,<+<1 = id 10 )-'[<-<,<+<1' (3) 'I(t,·): E -+ E is a homeomorphism V t E [0, I), (4) 'I(I,.J.,+c\N) C .J.,-" and if K< 0 then 'I(I,.J.,+c) C .J.,-c'

=

Proof. The proof is similar to the standard one (cf. Theorem 3.3 of Chapter I). The main point is to construct a flow. Let {(t, v, 8) be the flow defined on Ht x H. by the vector field V(x) (d. Theorem 3.4 of

3. A!ymptotically Linear ProblefM

159

Chapter I). Let the functions p, q have the same meaning as thooe in the referred theorem. Let W he a vector field on H if x {oo} defined by W(v) = -p (~(AV, v») q(IIAvIIlAv.

The associated flow

(= W«(), (O,v) = v, is denoted by (t, v).

Set

if (v, x) E Hif x Ho if (v,s) E Hif x fool.

,(t,v,s) 'I(t, v, a) = { «((t,v),oo)

Obviously this mapping satisfies all the conditions stated above. It remains to verify its continuity at points on Hif x fool. Claim. First, we show that

IIV(v + s) - W(v)1I _ 0

as v remains on a bounded set and lIall- 00.

This is due to the facts IJ(v + a) -

~(Av,v)1 = IF(v + s)l- 0,

IIX(v + a) - Avll =

and

L 'Ip(v + s)dF (xp)

->

0,

~

where {'I~} is the partition of unity in the construction of p.g.v.f. X, in combining with the following inequalities: Ip(J(v + a)) - p G(AV, v»)

I~

C IJ(V + a) -

Iq(IIX(x)lll- q(IIAvllll ~ CIIX(x) - Avll Next, assume that tn - t, Vn --. V, Sn and Sn E Ho. We want to show that

poe (tn' v.

+ sn) -

00

and

-+ 00,

~(Av,

v)l-

0

o.

where

(I - Po){ (t., v.

tn

E (0, I),

+ s.) -

Vn

E

Ht

t

(t, v).

Consider the equation

1I{(t, Vn + so) - {CO, v. + s.)11 ~ 1, it follows 11(1 - Po) . IIv.1I + 1 ~ C, 11P0{(t,vn + s.)11 ;:>: ISnl- 1 - 00. We conclude IIY(e(t,v. + an» - W«l- po)e(t,v. + sn))ll- 0 from above. Since

IIYII

~

W,v n + s.)11

1,

~

160

Semilinear

Ellip~ic

Boundary Value Probleml

We turn to the differential inequation:

d dt 11«1, v) - (I -

pole (t, lin + sn)1I

:5 IIW«(t, v)) - (I - Po)V (e (t, lin + s.»11 = IIW «(t, v)) - W«I - pole (t, v. + &.))11 + 0(1) :5 C 11((1, II) - (I - pole (t,v. + &.)11 + 0(\). According to Gronwall's inequality, we have

1I((1,v) - (1- Po)W,v. + &.)11 :5 CII(O,v) - (I - PO){(O,II. + 8.)11 +0(1) = C IIv -

v.1I + 0(1).

On the other hand

lie (t,v. + ") -e(t.,v. + •• )11:5 follows from

IIV II :5 1.

It - t.1

Thus

1I«t,v.) - (1- Po){(t.,v. +8.)11- o.

Theorem 3.4. Under assumptions (HI)' (H.) of Theorem 5.2 of Chapter II and (H), equation (1.1) pass,"". a solution. If, furtber, we assume that 'I'(x,O) = 0, 'I' E C l (fi X ]RI,]RI) and that either (1) 4I(x,O) = 0 or (2) 41(:1:,0) > 0, and 'l'ax,O) u( -6), A; < or (3) 41(:1:,0) < 0, and 'l'ax, 0) It

h

It (->: + ;\,0], where;\

la, X- >:], where X=

= max{A; IE

min{A; IE u( -6),

X> A;} , then there is a Dontdvial solutjon. Proof. One may assume that Ko is bounded, since otherwise we would he done. Now Theorem 5.3 of Chapter II, is applied to assure at least CL(S .... ) + 1 = 2 pseudo critical points. Therefore there is at least one genuine critical point, which is the desired solution. Suppose 'I'(x,O) = 0; then 9 must be a critical point. However by the minimax principle for subordinate classes, 3 a critical value C F O. In case (I), 4I(x, 0) = 0, 9 is on the level ;-1(0), so a point Uo in K, is a nontrivial solution.

3. Asymptotically Linear Probl.",..

161

If c < 0, we may choose "0 to be the one, which corresponds to a m_ reI&tive homology class. According to Theorem 1.S of Chapter II, Cm _ (J, '(0) f 0, and if c > 0, we may choose it to be the one, which corresponds to a m_ + rna relative homology class. Therefore Cm_+m.(J,"o) f o.

However, C.(J,O)

< ~ + cp;(x, 0) < >'0+, ~ + cpax, 0) = >"+1 and q ¢

= Oqm

if >'.

=0

if

1m, m]

where ),1: < )."+1 is a pair of consecutive eigenvalues of -.6, and k

m= Ldimker(-t.->'jI), j=1

k+l

m= Ldimker(-t.->'jI). j=1

Thus

and

Cm_+m.(J,O)=O if cp;(x, 0)
The proof is finished. Several sufficient conditions can be given to assure (II). Namely,

(I) cp(x,t) - 0 and .(z, I) - 0 as 111- 00, if.i. is simple and the nodal set of the associate eigenfunction has measure zero.

(2) cp(x, t) = cpo(t) + hex), where h E kerr -to - ~I).L, and <poet) together with its primitive .o(t) = J~ <po(s)ds. are bounded and uniformly continuous on Ill'. And dim(fI) = I As a special case of (2), we assume that CPo is a T-periodic function, with Indeed, the verification of (1) is trivial. (2) needs a little real analysis, so we refer to J. Mawhin [Mawl]. See also Solimini [SoI3] and Ward [War I]. Theorem 3.4 is due to Chang and Liu [ChL2].

Ji CPo(t)dt = O.

3.3. A Bifurcation Problem For simplicity, the function g(x, t) in (1.1) is replaced by get). We assume that (I) Iiml'I~= ~ < >' .. the first eigenvalue of -to with O-Dirichlet boundary value. (2) g(O) = 0, and 9 E c' (Ill').

Theorem 3.5. Let>. = 9'(0), then (i) Fbr>. > >' .. the BVP (1.1) has at least two nontrivial solutions.

162

Semilinear Elliptic Boundarv Value Problema

(ii) Fbr A > .1.2, or A = .1.2 with i!p ~ A ill a neighborhood U o( t = 0, (1.1) has at least three nontrivial &alutions. (iii) Fbr A > .1.2, we assume that i( A E u( -t.) either g(l) > A or t -

g(l) < A holds t -

(or t f 0 in a neighborhood U 0(0, then (1.1) has at least (our nontrivial solutions.

Proof. By condition (I), there exists an a E (0, Ad and a constant Co > 0 such that g(t) ~ at + Co if t > 0, and g(t) ~ -at - Co if t < O. Let 'Po be the solution of the following equation -t.'Po = a'Po + Co { 'Pol = o. llO

in 0

Then, by the maximum principle, 'Po > 0, and hence, -f{Jo < 'Po is a pair of sub- and super-solutions of (1.1). . According to the cut-off technique, we may assume that g(l) is bounded, and define the following functional

J(u)

= 10

[('7;)2 - G(U)] dx

on

HJ(O) ,

which is bounded from below so that the P.S. condition is satisfied. (i) Let 'P, be the first eigenfunction, with max.E!!'P, = 1, and 'PI > O. We may choose £ > 0 so small that -'Po

< -£'P' and £'P, < 'Po

are two pairs of strict sub- and super-solutions of (1.1). According to Lemma 2.1, we have two distinct solutions %\, %2 E HJ (0), satisfying C.(J, %;)

={ ~

k=O k f 0'

i = 1,2,

if they are isolated. (ii) We may assume that there are at most finitely many solutions. The weak version of the mountain pass lemma is employed. We obtain a third solution %3_ According to the Kato-Hess Theorem and Theorem 1.6 in Chapter II, we have

k=1 k

f 1.

163 We shall prove that

Z3

i

8. Indeed, if A >

e.(J,8) = e'_j(J, 8),

,1.2,

then

where j = iud,p J(8) ;:: 2.

So Z3 i 8. If A = ,1.2, with eip 2: ,1.2 for It I i 0 small, then ind(J,II) = I, II is degenerate, and is a local maximum of J on the characteristic submanifold Nat 8. Since dimN = dim ker( -t. - AI) = mo, we have

{~

e.(J,O) =

k= l+mo, ki l+mo·

Again, the critical groups single out Z3 from 8. (iii) In the case ,I.E u(-t.), 11 is a degenerate critical point. But, g(t)/t 2: A for t E U \ {O} implies that J = JiN :5 0, where N is a neighborhood in the characteristic submanifold at 0, dimN = dimker(-t. - AI), equal to mo. Let m_ be the Morse index of J at II. We bave m 2: 2, and

e.(J,II) = {

~

because II is a local maximum of J. Similarly, in tbe cases either A ¢ u( -t.) or A E u( -t.) but g(t)/t :5 A for t E U \ {O}, 0 is a local minimum of J; thus

e.(J, 0)

={ ~

If there were no other critical points, then a contradiction would occur due to the Morse inequalities: f30 = I, 13. = 0, k i o. In fact, for k > j, one would have

The LHS is even, but the RHS is odd. Therefore there arc at least four nontrivial solutions. The theorem is proved. A special case of this problem is due to the fact that the function is of the following fonn: (3.10)

g(u)

= AU -

h(u),

where A is a real parameter and h(u) satisfies the following oonditions: hE e'(R'), h(O) = h'(O) = 0 and limlul_~ +00. In this sense, we call it a bifurcation problem.

¥ =

164

Semilinear Elliptic Boundary Value Problems

Example l. The special form (3.10) of Theorem 3.5 has heen studied by many authors. Cf. Ambrosetti (AmbI]. Struwe (Str]. for at least three solutions. Hofer (Bo£]. Tian (TiaI] and Dancer (Dan!] for at least four solutions in cases (i) and (iii). 3.4. Jumping Nonlinearlties

Elliptic equations with jumping nonlinearities were first studied by Ambrosetti and Prodi (AmP!] and followed by many others: cf. Amann and Hess (AmHI]. Berger and Podolak (DePI]. FUcik (FUel]. Kazdan and Warner (KaWI]. Hess (Hesl]. Dancer (Dan!]. H. Berestycki and P. L. Lions (BeLl] and author (CbM]. After an observation due to Laxer and McKenna (LaMI]. more solutions were obtained. In this respect. the reader is referred to Solimini (Soil]. Ambrosetti (Ambl]. Hofer (Hon] and Dancer (Danl]. We consider the following BVP with a real parameter tEll'.

(p,){ -Llou = f(x.,,) +!
ul oo = 0

where 'P' is the first eigenfunction. with 'P'(x) > 0 'Ix E n. Assume that C' (n X lit'). satisfying the following conditions: (1) 1i11l(_+~f(x.{) = 7 uniformly in x En. and 7 E (,\;.,\;+1) for some j ~ 1. where {>.; Ii = 1.2•... } = u( -Llo). (2) Iim( __ ~ ~ ::; >., - 6. uniformly in x E for some 6 > o.

IE

n.

(3) There exists a constant M such that

If(x.{)1 ::; M

(1 + I{I~) .

We note that condition (1) implies that lim f(x.{) = 7. ("_+00

e

Theorem 3.6. Suppme that the conditions (1)-(3) are fulfilled. Then there exists t' E lit' such that (Pt ) has (1) no solulion. iEt > t'; (2) at least one solution. if t = t'; (3) at least two solutions. if t < t·. Iffurther, we assume j ~ 2, i.e., i > "\2, the second eigenvalue of -d, then there exists t·· < t· such that (Pt ) has at least four solutjons ift < t··, The proof depends on the following lemmas.

165

3. A8/1mptotically Linear hob,.",.

Lemma 3.3. Assume conditions (I), (2), and I E C(l'i X JRI). Let

In

J.(u) = where F(x,~) = condition.

[IV;I' - F(x,u) - "PIU] dx U E HJ(rI)

J; I(x,a)d..

Then for all t E JRI, J. S8tis1les the P.S.

Proof. For each function U E L!~(rI) we denote ,,+ = max{u,O}, and u- = U - u+. Assume that {un} C HJ(rI) is a sequence satisfying

In

(3.11)

where

/I . /I

In

(V .... Vv - f(x,un)v - t
v E BJ(rI)

is the HJ (rI) norm. Then we obtain

[Vu!Vv - I(x, un)+v - (t
V v E HJ(rI).

Let Pn = p;t - P;:, where P~ = U~

By condition (I),

-

(-LW I

[/(x. Un )± + (t
/II (x, u n)+ -1'u!IIL' =

o(lIU!/lL'); but

u! = (id - 1'( -a)-I) -I {( -a)-I [(f(x, Un )+ -1'U!) it follows that

+ t+
{/I"!/I} is bounded. From conditions (1) and (2),

we have

6:> 0 and C:> 0 such that

(3.12)

f(x,{) - AI{:>

61~1-

C.

Let us choose p < Al such that Al - P < 6 then we have (-a - pHun - Pn)

= f(x,un) + t
Pn)

+ PPn·

By the weak. Maximum Principle, one deduces

noticing Pn - 0 (HJ(n». Combining (3.13) with the boundedn... of u!, we obtain that lI"nIlL' is bounded. Substituting this fact in (3.11), we get that {/I"nll} is bounded. After a standard procedure, the (PS) condition is verified.

166

Semili....... Elliptic Bounda'1l Value Proble....

Lemma 3.4. Under conditions (I) and (2) there exists a subsolution !!t for the BVP (P,) such that for each solution '" oft?,) we have '" > !!t. Proof. According to (3.13), if we define !!t to be the solution of the following BVP: -L),." - "" = -G + t'l'l in n { = 0 the conclusion follows from the weak Maximum Principle, and the inequality (3.12).

,,180

Lemma 3.5. Under conditions (I) and (2) there exists to E lIt 1 such that if (P,) is solvable, then t < to. Proof. By (3.12), we have 6 > 0, G

!(x,{) - A1{

> 0 such that

~

61{1- G.

Thus, if u, is a solution of (P,), tben mUltiplying by '1'1 on both sides of the equation, and by integration, we obtain

A1

In

",'1'1 dx =

In

!(x, "')'P1 dx + t

In 'P~

dx.

From this one deduces

t

In 'P~

dx

+ 6ln1u,1. 'P1 dx -

that is

t<

G

(In 'P~dx) Gin -1

In

'P1 dx :5 0,

'P1dx.

Lemma 3.6. Under conditions (I), (2) and (3), there exists t1 E lIt 1 such that (P,) possesses a positive solution U, which is a nondegenerate critical point of J, with index d"J,(u,) = hj for t < t1, where hj = LkSj dimker(-L),. - AlI). Proof. Let

g(x,{) = -"({ + !(x,{)

{~O.

We have g(x, {) = o(JW uniformly in x E n, as { ~ 00. We extend the function 9 to be a function 9 such that 9 E G 1 (l"! X lIt1), with Ig(x,{)1 = o(IW uniformly in x E l"i. According to Theorem 3.1, the equation -L),.u = "(" + g(x, u) + 1'1'1 {

"Illfl = 0

possesses a solution Ut. Define

(3.14)

_ Vt ==Ut -

t'P1 A1 -"(

--.

in

n

167

3. A"1I"'l'lolic4l1y Linear Problem&

We obtain

-~v,

=

+ g(x, llt)

')'Vt

{ Vt1 00 = o.

Thus, the V a priori bounds for

Ilv,lIc'

=

VI

in

n,

are employed to deduce

o(ltl) as Itl- 00.

Substituting tbe estimate in (3.14), we obtain

il, > 0 for t < tl wbere -tl is a real number large enough. This proves that llt is a solution of tbe problem (P,). Again, we have

il,(x) = v,(x) +

~'I'I(X) - +00, AI --r

a.e. as t _ -00.

This implies tbat

I(x, llt(,,» Since

rP J(il,)

-

= id -

-r

t - -00.

B.e. as

(_~)-I !(x, Ii,(x»,

by tbe Holder inequality as well as the Sobolev embedding tbeorem, we see that

IIrP J(llt) - (id - -r( wbere I'

= ~.

-~)-I)IIL:(HJ(n»

$

(In 1I(x, il,(x» - -rI

Pdz) lIP,

Applying the Lebesgue domina.nce tbeorem, we arrive at

IlrP J(il,) - (id - -r( -~)-I>lIL:(HJ(n)) = 0(1). Since -r It u( -~) is assumed, id nondegenerate with Morse index hj .

-r( _~)-I

is invertible. Thus

u,

is

Proof of Theorem 3.6. Define to = sup{ t E IR I 1 (P,) is solvable}. Combining Lemma 3.5 witb Lemma 3.6, to E IRI, hence (1) is proved. Now we come to look for a strict snpersolution for t < t·. In factI choose t' E (t, to) and let be a solution of (P,.). Then ut. is a strict supersolution of (P,) wbich satisfies ut. > !!to >!!t. Hence !!!t,u,.) is a pair of strict sub- and supersolutions of (P,). Lemma 2.1 is employed to deduce a solution u, of (P,) wbich is a local minimum of tbe functional = JdcJ(lIl' so tbat

H,.

J.

k

= 0,

k

r! o.

168

Semili.....r Elliptic Bound.", Val... Problenu

Noticing that the functional J, is unbounded from below along the ray u. = 8'1'1, 8 > 0, Theorem 2.1 is applied. We lind a second solution ti, with critical gronps k= 1,

k,H. The conclusion (3) is proved. As for conclnsion (2), we prove, by tbe same method as in Lemma 3.3, that the set {uti t e (t' -I,t'n, where u, is the solution of (P,) obtained by the previous sub- and supersolutions, is bounded in HJ(O). We obtain a sequence t, --+ t' such that u'; weakly converges in HJ (0), say to u·. Then u" is a solution of (P,"). Finally, we assume "I e (>.;, ).j+I), with j 2: 2. According to Lemma 3.6, there is a t" < t" such that there exists a third solution ii, of (P,) such that iit is nondegenerate, with

One more solution wiD then be obtained by a computation of the LerayScbauder degree. In fact, by Lemma 3.3, we conclude that all solutions of the equation (3.15) o

are bounded in an open ball B Rp where Rh the radius, depends on continuously. By the homotopy invariance of the Leray-Schauder degree, one has deg(id - (-Il)-I F"RR .. 0) cons!. \I t e JR I ,

=

where

F,u

= f(x,u(x)) + II'I (x).

But, from conclusion (I), if I

> I", (3.15) has no solution. It follows that

deg(id- (-Il)-IF"R"O) =0,

\It eJR 1

t·.,

If t < suppose that there are only three solutions Ut, Ut and Utt then by Theorem 3.2 of Chapter II, the Leray-Schauder degree would be o

deg(id - (1l)-IF"B"O) = (_I)". This will be a contradiction.

Remark 3.2. Lemmas 3.4 and 3.5 are due to Kazdan and Warner IKaWI!. and Lemma 3.6 is due to Ambrosetti IAmbl] and Lazer and McKenna

3. A6Jl"'Pwtical1y Linear Froblerru

169

[LaMIJ. The idea of the proof is taken from Hofer [HonJ, Dancer [DanlJ and Chang ICbMJ. An extension, in which lilll(_-oo < A;, i > I, has been studied by many others. The reader is referred to the survey paper by Lazer [LazlJ; see also Lazer and McKenna [LaM2] and Dancer [Danl].

¥

3.S. Other examples Suppose that 9 E C'(R') satis6es the following conditions: (1) g(O) = 0, O:S 9'(0) < A,; (2) 9'(t) > 0 and strictly increasing in t for t > 0; (3) 9'(00) = liml'l_oo 9'(t) exists and lies in (At. A.). Theorem 3.7. Under conditions (I), (2), (3) the equation

-8u

(3.16)

{

= g(u)

in

n

ul oo = 0

has at least three distinct solutions. Proof. 1. It is obvious that 8 is a solution, which is also a strict local minimum of the functional J(u) =

1. [~("yu)'

- G(U)] dz on HJ(fI),

where G is the primitive of g, with G(O) = O. 2. Modify 9 to be a new function

get) = {

~(t)

t~O

t < 0,

and consider a new functional

where G(t) = J~ g(t)dt. It is easily seen that 8 is also astrict local minimum of J, which is a C'-functional with a (PS) condition. Since J is unbounded from below, along the ray u. = ' 0, Theorem 2.1 yields a critical point uo f< 8 of J which solves the equation

-8u = g(u) x E fI, { Since g(u) of (3.16).

~

ul oo = o.

0, by the Maximum Principle, Uo 2: 0, hence Uo is a solution

170

Semili...... Elliptic Boundal'!/ Value Problema

3. Now we shall prove that -A - g'(",,(x» has a bounded inverse operator on L2(0), which is equivalent to the fact that Jd-( -A)-Ig'(",,(x» has a bounded inverse on HJ (0), i.e., "" is nondegenerate. Since "" satisfies (3.16), it is also a solution of the equation -A"" - q(x)",,(x) ~ 0, where q(x) =

Let 1'1

l

1Jo180 =

0,

g'(t",,(x»dt.

< 1'2 < ... be eigenvalues of the prohlem -Aw - I'g'(IJo(x»w = 0, { wl80 = O.

We shall prove that 1'1 < 1 < 1'2. This implies the invertihility of the operator -A - g'(",,(x». In fact, according to 8SSumption (2), we have q(x)

< g'(",,(x» V x E 0

so that .

1'1

= mm f

f(Vw}' g'(",,)w2

.

f(Vw)'

< mm f q(x)w' :5 1.

Again, hy assumptions (2) 8Jld (3), we have

y'(u.(x)) < A2 V x E O. According to the Rayleigh quotient characterization of the eigenvalues . 1'2 = sup mf

EI UlEEt

f(Vw)2 fg'( (» 2 Uo x w

1

.

> ,sup mf 1\2 EI weEt

f(Vwj2

fW'

= 1

where EI is 8Jly one-dimensional subspace in HJ(O). 4. The Morse identity yields 8Jl odd numher of critical points. Therefore there are at least three solutions of (3.16). Finally, we tum to the following example. Theorem 3.8. Suppooe that g E GI(1lI. 1 ) satisfies the following condi· tions: (1) g(O) = 0, and A. < g'(0) < A3; (2) g'(oo) = lim,_±~ g'(t) exists, and g'(oo) Ii! a( -A), with g'(oo) > >'3; (3) \g(t)\ < 1 and 0 :5 g'(t) < >'3 in the interval [-c, cJ, where c = max,.!! e(x), and .(x) is the solution of the BVP:

-A, = 1 in 0 {

el 80 = o.

3. A.oymptoticallV Linear J'

171

Then equation (3.16) possess at least 8"" nontrivial solution&.

Proof. Define g(e) g(t) { g(-e)

g(t) = and let

I(u) where

Crt)

=

ift>e if ItI :$ e if t < -e

fa [~(VU)2 - C(u)] dz,

= J~ g(s)d8. The truncated equation

'-t.u = g(u)

(3.17)

in 0

ul 80 =0

{

possesses at least three solutions 9, Ull U2, because there are two pairs of sub- and supersolutioDS [",p.,el and I-e,-E'I'.I, where '1" is the first eigenfunction of -t., with '1',(",) > 0, and E > 0 a sOIBII enongh constant. By the weak version of the Mountain PBSS Lemma, there is a mountain pass point U3. That "3 i 9 follows from the fact that

-

C.(J,U3) =

{G0

k=1 k

i

1.

But from condition (1)

-

C.(J, 9)

= {G 0

k=m. kim.

+mo

+mo,

where m; = dimker(-t. - )..,1), i = 1,2, .... By Lemma 2.1, one has

-

C.(J,u,)

= {G0

k=O klO'

i = 1,2.

Noticing that I is bounded from below, we conclude that there is at least another critical point U4Obviously, all these critical points "i, i = 1,2,3,4, are solutions of equation (3.17). On account of the first condition in (3), in combination with the Maximum Principle, all solutions of (3.17) arc bounded in the interval [-c, cl. Therefore they are solutions of (3.16); moreover, all these solutions u, because of their ranges, are included in [-c, c), and we conclude: 2

ind(J, u) + dimker(d'J(u» :$ m:= dimEB(-t. - ).;1), j=1

172

Semilinear Elliptic Boundary Val... Prob,.,...

provided by the second condition in (3). Because of condition (2), we learned from TIleorem 3.1, TIleorem 5.2 of Chapter II is applicable, with 'Y > m, because 9'(00) > ~. Therefore there exists another critical point u., which yields the fifth nontrivial solution for the equation (3.16). Cf. Chang (Cha12].

4. Bounded Nonlinearities 4.1. F\mctlonala Bounded from Below

The functionals J associated with equation (1.1) in this section are considered to be bounded from below. We shall study several cases which occurred in PDE about numbers of solutions. First we assume (go) 3", < )",/2, and P > 0 such that G(x, t)

= 10' g(x, .)d. :5 Qt' + P

''Yhere ),., is the first eigenvalue of -a with O-Dirichlet data; (grllg:Cx,t)i:5 G(l + Itl)', 'Y < n~.' ifn > 2.

Theorem 4.1. Under llSSumptions (go) and (g7), suppose that

(4.1)

g(x,O) = 0, and 3m ~ 1 such that

),.m

< g;(x,O) < ),.m+!.

where {),.,,),. ••... } = u(-a). Then (1.1) has at least three solutions.

Proof. Again, we consider the functional (1.3) J(u) =

.£. [~lvuI2 - G(X,U)] dx

whicb is well-defined and c" on HJ(O) provided by (g7)' (go) implies that J is bounded from below: (4.2)

J(u)

~ ~ (1 - ~7) '£'IVul'dx - P mcs(O).

And 9 is a nonminimum, nondegenerate critical point with finite Morse index of J provided by llSSumption (4.1). In order to apply TIleorem 5.4 of Chapter II it suffices to verify the (PS) conditions. In fact, the coercive condition (4.2) in conjunction with the boundedness of J(u n) imply that {Un} is bounded, and hence is weakly compact. From (g7), we see that Ig(x. t)1

:5 G, (1 + Itl)",

n+2

1'-<--2' n-

if n > 2.

173

4. Bounded Nonlinearitiu

We use the same argument as in Lemma 2.2, from

We conclude that the sequence

Un

is subconvergent.

4.2. Oscillating Nonlinearity

The following technique is often used to reduce problems such as (1.1) to problems with bounded nonlinear terms.

Lemma 4.1. Suppose that the function 9 in (1.2) satisfies (go) 3 { ~ 0 such that g(x, {) ~ 0 V x E 11, and let

g(x,t)

= {g(X,{) g(x, t)

Assume that u E

HHn)

if t >{ if t ~ {.

is a solution of the following equation:

-t.u = g(x, u(x» x E n { ul = o. oo

(4.3)

Then u(x)

~

{, so it is also" solution of (1.1).

Proof. By the standard regularity theorem u E w;(n) V p < +00 so u E CI(n). Considering the domain D = {x E n I u(z) > {}, we have

-t.u(x) :5: 0 V ZED, {

ulaD :5:

e·

By the Maximum Principle, we have D(X) < {in D, and hence D = 0, i.e., D(X) :5: {. Corollary 4.1. If in the above lemma, (go) is replBced by

e

(ga)' 3 ~ 0 such that g(x,{) < 0 V x E Then D(X) < { V x E

n.

n.

Similarly, we introduce the assumption

(ga)· 3 { $ 0 such that g(x, {) ~ 0 (or> 0) V x E 11, and consider the truncation _ { g(z, t) g(x,{)

9( x,t ) We have

if t if t

~

{,

< {.

174

Semilinear Elliptic Boundary Value Problems

Corollary 4.2. Under (1.2) and (I!s)., if u E HJ(O) is a solution of -~u(x)

{ then u(x)

x E0

= g(x, u(x))

ul8
<: eV x E fI, and is also a solution of (1.1).

By the same trick, if one looks for positive solutions, the function 9 de6ned on x R~, is extended continuously to he g: x !lI.' ..... IR' , with g nonnegative for t < O. Keeping this in mind, we consider some examples. For the sake of simplicity, we assume g(x, t) = g(t), and study the eigenvalue problem

n

(4.4)

n

-~u(",)

{

= }..g(u(x»

xE0

ul8
Theorem 4.2. Suppose 9 E C' (!lI.~) and g(O) <: o. Assume that (go) There exists 0 < a, < a, < ... < am, such that g(ai) = 0, i 1,2, ... ,m. ", (glO) G(t) = J~ g(s)ds satislies m",,{G(t) I 0 ~ t ~ ai-'} < G(ai),

i = 2,3, ... ,m

=

and G(a,) > O.

Then 3 }..O > 0 such that Eor >. > }..O, (4.4) poesesses at least (2m - 1) nontrivial solutions. FUrthermore, if g(O) = 0 and g'(O) ~ 0 then 3 }.., > 0 such that for}.. > }.. .. there are at least 2m nontrivial solutions for (4.4). Proof. By the truncation trick, we consider the (unctions

gi(t) =

{

gi(O)

t
~i(t)

O~t~ai

t

> ai

gi E C(R'), and the functionals with parameter }..

Given the above explanation, we know that tbe critical set K i (}..) of J,(.,}..) must he a subset of the critical set Ki+I(}..), i = 1,2, ... ,m - 1, and by Lemma 4.1, VuE K i (}..), 0 ~ u(x) ~ ai, i = 1,2, ... V x E First, we see that J i (·,}..) possesses a global minimum Ui(}..), provided by the bounded ness from below plus the (PS) condition. Thus Hi(}..) E K i (}..), i = 1,2, ... ,m. Second, "i(}..) ¢ K i _,(}..) for>. large, i = 2,3, ... ,m.

n.

4. Bounded Nonlinearitiu

175

Indeed, we only want to show 3~, > 0 and W E HJ(Il), with 0:<:; w(%) :<:; "', such that J,(w,~) < J,_,(",_,(~),~) \I ~ > ~,. Let a = G(",) - Max{G(t) / 0 :<:; t :<:; o,_,}. By (glO), 0 > o. Let 116 = {% E Il J dist(%,8!l) < 6} for 6 > 0, and let

W6 E C8"(Il),

0 :S W6 :<:; OJ,

= OJ

W6(%)

\I % E Il \~.

Thus

{ G(W6)tk

in

=(

10\0,

G(",)tk +

r G(W6(%»)tk

10 ,

~

G(Oj)JIl/ - 2C,J1l6/

~

in G(v,_,(~»tk +

oJIl/- 2C,J1l6/,

where C, = max{G(t) /0:<:; t :<:; a'}. Hence J'(W6,~)

- Ji-l(V'_'(~)'~)

=

~ fa

:<:;

~

in

[J'I7w6/ 2 -

J'I7u,_,(~)J21 tk - ~ fa [G(W6) - G(v,_,(~») tk

J'I7 w6)'tk -

~(o/IlJ -

2C,/1l6)) < 0

for. = " > 0 small, and ~ > ~, large enough. The function W = W6 is just what we need. One may assume~, :<:; ~2 :<:; ... :S ~m. Third, from G(o,) > 0, we have

using the above argument, so v,(~) i o. One may assume #Km(~) < +00. Tben the Morse equality is applied to the bounded from below function J,. Noticing that HJ(Il) is contractible, we have Po = I, p, = {3, = ... = 0,

and

E(-I);MJ(~)

= I,

j

where Mj(~), j = 0,1, ... are the Morse type numbers for K,(~). But the Morse equality also holds for JH " and we have known that u,+I(~) E K'+I(~)\K,(~) for ~ > ~,+ .. and that V'+1 is the global minimum of J HIo so C.(JHh u,+1) = ' .. G, i.e., the contribution of U'+1(~) in the alternative summation Ej(-I)iMr'(~), is 1. If there were no other critical point in KH'(~) \ K,(~) for ~ > ~'+1, then the equality would lose the balance. Therefore, we concludes

#

(KHl(~) \ K,(~» ~ 2

if ~ > ~Hl'

i = 1,2, ... ,m - 1.

176

Semmnear Elliptic Boundary Value ProbJenu

In the cases 9(0) = 0 and 9'(0) > 0, 0 has no contribution in critical groups. This is proved by the standard perturbation technique in combining witb tbe homotopy invariance property (cf [ChaI6». In summary, we bave #(K.,,().) \ {8}) ~ 2m - 1, if), > )..". The first conclusion is proved. Assume that g(O) = 0, and g'(O) 50; then V >., J,(u,>.) ~ 0 = J,(O,>.) for HuH small. 8 is a local minimum of J,(., ).), but not the global minimum u,(>.) V >. > >.,. We apply tbe Morse equality to J" that tbere must be one more point in K,(>.) for>. > >." i.e., #(K,(>.) \ {8}) ~ 2, >. > >., so is #(K.,,(>.) \ {O}) ~ 2m, if >. > >..". 4.3. Even Functionais

Theorem 4.3. Suppose that g(x, t) is ofthe form a(x)t +p(x,t) where and that (g.), a E C(n), and p E C'(O X \li,1 ,\li,'). Assume that a> 0 in (gu) p(x, t) o(lt» uniformly with respect to x E and (g12) p(x, t) -p(x, -t) V (x, t) E i1 X \li,', hold. Then the equation

n,

= =

-6u(x) {

= >.g(x,,,(x»

-Ll.v(x) = Jlo(x)v(x) {

in 0

"\"" = 0

has at least k distinct pairs of solutions, if>. eigenvalue of the eigenvalue problem (4.5)

n,

> >.., where >..

is the kth

in 0

v\"" = o.

Proof. V)", the functional is written as

where P(x, t) = f~ p(x, s)ds is a.n even function with respect to t, provided by (g12). Thus J, is an even functional. According to (go) and Lemma 4.1,1, is bounded from below. And a > 0 plus (gu) imply that there exists p > 0, such that J'\spnE, < 0 for >. > >.., where Sp is the sphere with radius p centered at 8 in HJ (0), and E. is tbe direct sum of eigenspaces with eigenvalues 5 ).. of tbe problem (4.5). The verification of the (PS) condition is omitted. Now we apply Theorem 4.1 of Chapter II. There are at least k pairs of distinct solutions.

4. Bounded Nonlinearitiu

177

4.4. Variational Inequalities A variety of variational problems with side constraints arlSlng from mechanics and physics are called variational inequalities. They have been extensively studied since the 1960s. See, for instance, Duvaut and J. L. Lions [DuLl]. A typical example is as follows: Given a closed convex set C in HJ (n), a continuous g: n x 111.' _ 111.' and h E L~ (n), find uo E C such that (4.6)

l

['I7UO' '17(" - UO) - (g(x,,,o(x» - h(x» (" - UO)(x)]dx 2: 0 Y" E C.

In fact, the variational inequality is attached to tbe following variational problem: to find UO E C, which is the critical point of the functional

J(u) =

~

l

[I'I7 u l' - G(x,,,(x»

+ h(x)u(x») dx

with respect to the closed convex set C (ef. Definition 6.4 of Chapter I). In this sense, all the critical point theories, including the Morse inequalities on closed oonvex sets, are suitable for the applications. In contrast with the well-developed variational inequality theory, in which 9 is assumed to be nonincreasing in t so that the solution is a minimum of the functional J, the restriction on 9 is avoided in this subsection. Indeed, one can find minimax points. We are satisfied to study the following two examples mainly by explaining the differences.

Example I. Assume that 9 satisfies (g,), (g,) and (g;,). Let C = P be the positive cone in HJen); then there are at least two solutions of (4.6), if g(x, 0) = 0 and g(x, t) 2: 0 Y (x, t) E n x 111.~, and if 3 to > 0, such that g(x, to) > O. Claim. We follow Lemma 2.2 step by step to verify the (PS) oondition with respect to P. Note that

where ( , ) is the inner product in HJen). It implies Y < > 0 3 ko E Z+ such that (dJ(".),u.):5 <11".11, Y k 2: ko. (WARNING: Tbis is only a one side ineqUality! Not like that in Lemma 2.2 in wbich we got l(dJ(".). u.)1 :5 <11".IIl. This i. enougb to assure the boundedness of 11".11, as shown in Lemma 2.2.

178

Semi/in ..... Elliptic Bound"'11 Value Prob,."..

Now we prove the subconvergence of u•. All shown in Lemma 2.2, we obtain a subsequence, which we still write in 'Uk, such that

Since 9 is positive in I ~ 0, u. E P, and (-.11.)-' preserves tbe positive cone (Maximum Principle), u· E P. Again, from 1- dJ(u.)( •• - 0, it follows thot V e > 0 3 leo E Z+, such that (-u.+ (-.11.)-' 0;° 09(·'''.),1I-u.):;; tllv-u.n, 'Iv E P, V I.: ~ leo. Consequently, 3 1.:, E Z+, such that

In particular, set v = u., this proves Uk --+ u·. 'lb study the mUltiple solutions, it is easily seen thot 9 is a local minimum for J in P. Since the first eigenvector 'P' E P, J is unbounded from helow in P. A weak version of the Mountain Pass tbeorem with respect to P is

applied to obtain the second solution. For the same functional J. but we change to a different closed convex !Jet, one has

Ezample 2. Suppose '" E H'(!l), and C = {u E HJ(!l) I 0 :S u(or) :;; "'(or) a.e.}. Under the same assumptions on 9 and h in the Example I, assume that (4.7)

inf{J(,,) I u E C} < o.

Then the variational inequality (4.7) possesses at least three solutions.

Claim. The (PS) condition with respect to C can he verified as above. local minimum, and J has a global minimum H2. Assumption (4.7) implies '" "I u •. We apply the Morse eqnality which provides the third critical point of J. Now, "1 = 8 is a

R.errutrl.: 4.1. Theorem 4.1 is taken from K.C. Chang (ChaI]. For an extension of it see K.C. Chang ICha2]. Theorem 4.2 is an extension of the results due to K.J. Brown and H. Budin [BrBI] and P. Hess (Hes2], in which only the case 9(0) > 0 was discussed. Section 4.4 was stndied in K.C. Chang leba7].

CHAPTER

IV

Multiple Periodic Solutions of Hamiltonian Systems

O. Introduction In this chapter, we shall apply Morse theory to estimate the numbers of solutions of Hamiltonian systems. Let H(t,.) be a 0' function defined on 111.' X 1II.2n which is 2,,-periodic with respect to the first variable t. We are interested in the existence and multiplicity of the I-periodic solutions of the following Hamiltonian system:

Ii = -H.(t;q,p) { 1> = H.(t; q,p),

(0.1)

where q,p E III.n, • = (q,p). Tbe function H then is called the Hamiltonian function. Letting J be the standard symplectic structure on 1R2n, i.e.,

J=(OIn -In) 0 ' In

where is the n x n identity matrix, the equation (0.1) can be written in a compact version -Ji = H,(t,.). (0.2) Equation (0.2) is very similar to the operator equation considered in Chapter II, Section 5. Indeed, let X = £2 (O,l),R2n), and let

A: .(t) ......... -Ji(t) with domain

D(A) = H! (1o, 2,,], lII.. n )

= {z(t) E H'((O,2,,],1II. 2n )] z(O) = z(2..n.

For the sake of convenience, we make the real spoce ROn complex. Let en = R" + iRn, and let {el,e2,'" ,e2n} be an orthonormal basis in 1l2ft. Let j = 1,2, ... ,n,

ISO

Multiple Periodic Soluti"""

en.

which defines a basis in Z

The linear isomorphism llI.'n _

'n

n

j=1

j=1

en

= LZjej ~ z= L)Zj - iZj+n)V'j,

is called the complexification of llI. 2n , which preserves the inner product. Namely, n

[i, WJ

s

He ~)Zj ;=1

s

~)ZjWj + Zj+nWj+n)

- iZj+n}(Wj - iWj+n)

n

2n

;=1

= L ZjWj = (Z, W) , ;=1

where [ , ) is the inner product on en. We introduoe the complex Hilbert space L2([0,2".),Cn ) to replace the real Hilbert space L2([0, 2..J, R2n ), whose inner product reads as

["

(i, iiJ) = Jo [%(t), iiJ(t»)dt =

[2.

J

o

(z(t), w(t»dt

s

(z, w).

From now on, if there is no confusion, we shall not distinguish between these two. Sometime, we only write z but not z. One important thing is that Jz ..... ii. Thus, if we expand z E L2 ([0, 2".J, en) in FOurier series:

z(t) =

D(A) = {

t C~"" eime-,m,) t m~""

z E L2 ([0,2".1, en) I

(1

'Pj,

+ Imll21cjml2 < +00 } ,

and A is self-adjoint with the following spectral decomposition: £2([0, ".1, en)

= EEl M(m), mE'

where M(m) = span {e-im'IP1, e- imt
Let E·, where. is +, 0, or -, be the positive, zero, and negative subspaces respectively according to the spectral decomposition, and let p., where + is +, 0, or -, be the associated orthogonal projections. Set

1. A811"1Plolical/y Linear S",Iem8

181

Let

H = D (IAI'/') = { z E L' (10,2 ..), en) I

t. mf;~

(1 + Im))lcj.. I'

< +00 }

with tbe following scalar product: (Z,W)H

= (z,wh' + (IAII/2z,IAI'/2w)L"

Tben H is a Hilbert space. There are few simple properties often used:

(1) The embedding H ..... L" (IO,2 ..),C") is compact, V p < +00. Claim. Since H'/2 (10,2 ..), en) ..... V ((0, 2..), en) is compact and D(IAI'/') ..... H'/' ((0, 2..), en) is continuous. (2) 3 a compact linear operator K:V' ((0,2..),en ) ~ H, such that

{2.

(Kf,z)H =

10

(J(t),z(t»dt,

Claim. K = (J + IA))-I, the rest follows from (1). (3) Let H" = P" H, where * is +, -, or 0. Tbere exists a bounded linear operator A E £'(H, H) satisfying (Az, W)H = (A+)'/' Z , (A+)I/' W ) L' - (A-)'/'z, (A-)'/2 w ) . Moreover,

Claim.

AIH. has bounded inverse with ker(A) = H". A=

A(I + IA))-I.

For a given polynomial growth Hamiltonian function H(t, z), the functional

(2'

g(z)=- 10

H(t,z(t»dt

is weD-defined and is in Cl(H,IlI.'). Then tbe Hamiltonian system (0.2) is rewritten as

AZ + dg(z) =

9.

(0.3)

This is the problem we studied in Chapter I, Section 5 and Chapter II except now we assume dim H- = +00. We look for critical points of the functional on H: 1 (0.4) J(z) = 2(Az,z) + g(z),

182

Multiple Periodic Solution.!

where dg = K

0

d,H(t,.) oi : H

~

H is compact, with

1. Asymptotically Linear Systems The Morse index plays an important role in distinguishing critical points. For functionals defined bY Hamiltonian systems (004), 2.

d'J(z)(w,w) = (AW,W)H -

1 0


The Morse index ind(J,z) depends on A or A and the matrix
Definition 1.1. Let Sym(2n,lII.) be the set of all symmetric 2n x 2n matrices. Let BEe ([0, 2"1, Sym(2n, 111.)), with B(O) = B(2,,). We say . that A is a Floquet multiplier of (1.1)

-Jw(t) = B(t)w(t)

if A is an eigenvalue of W(2,,), where Wet) is the fundamental solution matrix of (1.1). By definition, z is nondegenerate for J(z) (see (0.4)) if and only if the linear system (1.2) -JtiJ(t) = d~H (t, z(t» wet) has no nontrivial 27f-periodic solution, or equivalently, 1 is not a Floquet

multiplier of (1.2). In order to classify this continuous loop of symmetric matrices without a Floquet multiplier It we study the associated fundamental solution matrices (Wet)}. Noticing that, if B(t): [0, 2,,1 ~ Sym(2n,lII.) is continuous, then

~ -T

T

(WT(t)JW(t»)

-

= W JW + W T JW -T =-WT BW - W BW = -T

0,

and from W (O)JW(O) = J, it follows that W JW = J. Conversely. if -T W (t)JW(t) = J, then Wet) is invertible. T Let B = -JWW-'; then BT = (W-')TW J = (W-'jT JW-'

-, =

--'

JWW

B. Therefore, BEe ([0, 2,,1, Sym(2n, JR)).

1. AsymptoticallJl Linear Systems

183

Recall that a matrix M is called symplectic if MT J M = J. We denote by Sp(n,llI.) the set of all real 2n X 2n symplectic matrices. LetP = h E C ([0, 2..), Sp(n,llI.)) 11'(0) = id, 1'(2,,)has no eigenvalue I}. Definition 1.2. Let 1'0,1" E P. We say that 1'0 is equivalent to 1'10 denoted by 1'0 ~ 1'10 if 3 6 E C ([0,1) x [0,2"), Sp(n,llI.)) such that 6(i, I) = 1',(1), i = 0,1, 6(s,O) = id, and I ¢ ,,(6(.,2")) It • E [0,1).

The following classifying theorem is applied [COZ2), [LoZI). Theorem 1.1 (Conley-Zehnder-Long). Every path YEP belongs to one and only one equivalent class Pk, k E 7l. Moreover, It k E Z, one finds a distinguished "standard path· /3. (I) 8S follows.

In tbe case n :::: 2, let X. = diag(O. (k - n + 2)/2.1/2, ... ,1/2), Y = diag(ln2, 0, 0, ... ,0),

Z. = diag«k - n + 1)/2,1/2,1/2, ... ,1/2), be n x n matrices, and /3.(t) = exp(IJB.), 0 :S t :S 2", where B = ((;

;J

• (Z.o Z.0)

In Ihe case n = I, set wet) = 4(1 w,(I) = {W(I),

0,

if (_I)nH = -I, if (_I)nH= 1.

+ cost),

O
" ;; t ;; 2" ' and w.(t) =

{O,

w(t),

and let K = ( 01 0I) ,R(O)

= (COSO sinO

-SinO) cosO ,H(a) =

O:St:S" 71' 5 t .5 271';

(a-'0 a0) ,a l' O.

If k is odd, then /3.(t) = R(tk/2), 0 :S t :S 2", and B. = k/2. If k = 0, Ihen {3o(t) = H (2""), 0 :S t :S 2", and B. = .~ (In 2)K; if k even, but not 0, then /3.(t) = H (2""(')) R(-w,(I)h), O:S t:S 2", and k B.(I) = --2 WI (t)id + In2"'(t)K.

2"

Definition 1.3. We define a map j: P -+ Z as follows: If l' E 1\, then j (-r) = k. The number k is called the Maslov index of l' and is denoted by j(-r) or j(B), if 1'(t) is the fundamental solution matrix of (1.1). In the following, we shall show that the Maslov index of.., = W (t), which is associated with cP. H (t, z( t)), is a candidate for the replacement of the Morse index. Let us denote Em

= EeJiI$m M(j), m = 0, 1,2, ... , and let Pm. be the orthogonal projection onto Em. m = 0,1,2, ... , in which dim Em = 2n(2m+ I).

184

Multiple Periodic Solulionr

Theorem 1.2. A&.ume that BEe ([0, 2,,), Sym(2n,IR)) has no F/oquet multiplier 1 and k = j(B). Let Qm = A - PmBPm E .c(E.. Em), and "", • = -, 0 be the Morse index and the nullity of Qm. Then 3 N > 0 BUch that

m-=~dimEm+kand

",0=0,

Vm?N.

(1.3)

Proof. We bave seen the one-to-one correspondence hetween tbe set of loops in Sym(2n, IR) witbout the F10quet multiplier 1 and P: B(t) = -J"I(th(t)-I. In order to transfer the defonnation on 7 to B, we claim tbat tbe above equivalent classification may he restricted to a smaller class

po = hE C'([O, 211"), Sp(n,lR)) I "1(2,,) = "I(Oh(21t),7(0) = id, 1 "- ,,(-y(2.-»)}. Claim. Given 70,11 E p', 70 ~ 71 in P, i.e. 3 6: [0,1) x [0,2,,) ..... Sp(n,JR), continuous, such tbat 6(i,.,.) = 7i(t), i = 0,1, 6(.,0) = id and 1 rt ,,(6(., t)) V. E [0,1). By reparametrization of " we may assume tbat 6(.,t) = 7o(t) forO:S.:s E, and 6(.,t) = 71(t) forl-<:S.:s 1. Moreover, by reparametrizing in the t variable along

8

= 0 and 8 = 1, we can achieve

that 6(., t) is differentiable in t in the neighborhood of tbe boundary of [0,1) x [0,211") aod satisfies, in addition, the condition "1(2,,) = "1(0)· 7(2,,). It remains tosmootb out < in tbe interior. This is readily done by smoothing the generating functions of tbe symplectic matrices in tbe parameter t. Tben we obtain a continuous family B.(t), • E [0,1), such tbat Bo(t) = B(t), B,(t) is the staodard patb in tbe above theorem, aod A - B,(t) is invertible in.c (L2([0, 2,,), 1lI.2n») (or equivalently, A - KB.(t) is invertible in .c(H)) V • E [0,1). Tberefore (by small perturbation), 3 e > 0, 3 N, > 0 such that, for m

> NIl

This enables us to figure out tbe Morse index and nullity of B(t) by the eounterparts of B,(t), the staodard path. Obviously, we bave ",0 = 0, because Qm = APrn - PmB1Pm. is invertible. As to m-, we only want to verify the standard paths, which are constant matrices except for n = 1 and k even but not O. It satisfies us to verify the simplest case and leave tbe other cases as exercises because they are elementary linear algebra, cr. Long-Zehnder [LoZ1). Indeed, the representation of Qm on the invariant subspaces M(j) Ell M(-j) under tbe basis (real) {cosjfe" sinjte, Ii = 1,2, ... ,2n} (and M(O) under the basis {el' e2, ... ,e2n}), reads as follows: -BI ( jJ

-jJ)

-Bl

88

.

.

1 = 1,2, ... 1 m , and - Bl as J =0.

185

I. A'lI"'ptoticolly Lin ..... Syorem.

For the 4nx4n matrix

(j~' =~), we split tbem again according to tbe

(~. :.), n + 1)/2,1/2, ... ,1/2}, where k = i(B,) = i(B). 1b

standard path. In the simplest case (n;:: 2,

n+k even), B,

=

and Z. = diag{(k those invariant subspaces, in which the elements in Z. equal to 1/2, we have ind (-1/2

-j )

-1/2

j

j

-1/2

-j

-2 for j

=

1,2, ... 1m,

-1/2

and

.10d (-1/2 0

0 ) = 2 ',or 1. = 0. -1/2

There are (n - 1) such elements, and the total contribution in the index is 2(m + l)(n - 1). The remaining element in Z. is (k - n + 1)/2. In the case k 2:: n, -(k - n ind

+ 1)/2 -(k - n+ 1)/2 -(k -

j

(

-j

=

{2,

·f . >

']

·f . < '1

4,

and ind

(-(k -

n

+ 1)/2

~ + 1)/2

-j

-(k - n

)

+ 1)/2

k-n±l 2 1o-n±1 ' 2

. ) = 2 if j = O. -(k - n+ 1)/2

The contribution to the index is 2 (m - .;R)+4 (·;n)+2 = 2m+k-n+2. Therefore ind(Qm) = 2(m + l)(n - 1) + 2m + k - n + 2 = (2m + l)n + k, i.e.,

m- =

~dimEm + k.

Similarly, we verify the case k < n. This finishes the proof. We turn to study the existence and multiplicity of solutions of asymptotically linear Hamiltonian systems. A Hamiltonian system is called asymJ>totically linear if (H~) 3B~ E C([O,2"J,Sym(2n,R» such that B~(O) = B..,(2,,) and IId.H(t, z) - B~(t)zll." = o(lIzlI).,"

uniformly in t, as

IIzll.'" .... 00.

186

Mtcltipl. Periodic Solutions

c'

Lemma 1.1. Assume that H E ([0,2"1)( llI.2.,llI.') satisJies (H=), where B= has no FIoquet multiplier 1. Then the functional (0.4) J satisfies the (PS)· condition with respect to {Em I m = 1,2, ... }, and J m = JIE_ satisfies the (P S) condition V m large. Proof. 1. We claim that IIdJm(z) - Q::'.'ZIIH = o(lIzIIH) as IIzll ..... for z E Eml where q: = APm - PmBooPm . Indeed,

IIdJm(z) - Q:ZIIH

= l)Pm (d,H(t, z) -

00,

B=z) IIH o(lIzIIH)'

$ IId,H(t, z) - B=zIIH

=

2. Since KBoo is compact, and Pm strongly converges to the identity, we have IIK(PmB= - B=)II ..... 0 as n ..... 00. The operator K(A - B=) =

A - K B= is invertible, so by the Banach inverse operator theorem, APm PmKBooPm has also a bounded inverse. Moreover, there exists a constant C such that

II (APm -

PmK B=Pm) -'II $ C

for m large enough.

Combining the above two conclusions, we obtain from dJm(zj) ..... IIzjllEm is bounded. Thus the (PS) condition for J m holds.

e that

3. Assume IIdJm (zm)IIH ..... 8, as m ..... 00 for Zm E Em. Then IIzmllH is bounded, provided by the same reason. Noticing that Po is of finite rank, we have a subsequence of Zml which is still written by Zml such that

Kd%H(t,z,.,J

--+

u and POzm

-t

AZm +POzm = dJm(Zm)

V

as m

--+ 00.

Consequently,

+ PmKd,H(t,zm) + POzm ..... u+ v,

i.e.,

Zm ..... (A + pol-leu + v) since

A+ Po is invertible in H.

in H as m .....

00,

This proves the (PS)" condition.

Theorem 1.3. Suppose that H E C' ([0,2,,) X llI.2., llI.') heing 2" periodic in t satisfies (H=), where B= has no FIoquet multiplier 1. Then the Hamiltonian system (0.2) possesses 11 2"-periodic solution.

FUrther, we assume (Ho) 3 Bo E C (10, 2"), Sym(2n, JR)) such that Bo(O) IId,H(t, z) - Bo(t)zll.'. = o(lIzll.'.) uniformly in t, as If Bo has no Floquet multiplier 1, and if

= Bo(2") and

Ilzll.'...... O.

j(Bo) f. j(B=),

(1.4)

then (0.2) possesses at least a nontrivial 2"-periodic solution. Proof. 1. First, we use the Galerkin method and study the restriction Jm of J on Em, m = 1,2, .... For large m, we learn from Theorem 1.2

187

1. A'lf'"Ptotically Linear SY31ems

and Lemma 5.1 in Chapter II that dim H. (Em' (Jm).~) = 6...;;,. for some a"., where m;;. = ind(Boo), so there exists a critical point z... E Em of J m • According to (Ho), H is C' at z = II on 1R2ft; therefore, the functional H(t, z(t»dt is C' at% = e under the topology C ((0, 21f], Rft), and then it is C' at z = e on each finite dimensional subspare Em. One concludes by the fact that B is a nondegenerate critical point of J m with Morse index mO = n(2m + 1) + j(Bo) "m;;' (cf. Theorem 5.2 from Chapter II), due to (1.4). We want to give a lower bound for 11 .... 11, in order to distinguish the limit, if it exists, from B. Since both A - K Boo and A - K Bo have bounded inverse, we have R > 0 and N > 0 such that

f:-

.... " e

II (APm - KPmBoo Pm) -'II'<:(H) < R, 1

for m

~

N.

II (APm-KPmBoPmf II.c(H)
> 0 so small that IIdJ(z) - (A - KBo)zIlH

= IIK(d,H(t, z) - Boz)IIH 1 < 2R IIzllH V IIzllH < 6.

Then, for large m, we have

IIdJ(z) - (AP m - KPmBoPm)zIlH

~ ~1I(APm -

~

1 2RllzIIH

KPmBoPm)-'II£(H)lIzIlH V IlzliH < 6.

But if we write Q~ = APm - K PmBOPm, then

IIdJ(z)1I ~ IIQ~zIIH - ~11(Q~)-'II£(H)lIzIlH

~ 1I(~)-'II£(H)lIzIIH - ~1I(Q::,nl£tH)lIzIIH =

Zm

~1I(Q::')-'II£tH) ·lIzll·

Consequently, if Zm is a critical point of Jm with = B. Thus 1Iz...IIH ~ 6 for m large enough.

IIzmliH <

6, then

2. Applying the (PS)" condition, there exists a limit point z' of {Zm}. Therefore, z' is a critical point of J, with IIz'll ~ 6. This is the nontrivial solution. The proof is finished. If one wishes to extend the above result to the degenerate case the Maslov index should be extended. In this respect we refer the reader to Y. Long [Lon], Li Liu [LiLl) and Ding Liu [DiLi].

188

Multiple Periodic Solution3

2. Reductions and Periodic N onlinearities We have seen that for indefinite functionais, the Morse indices of critical points could he infinite (e.g., the functionals arising from the Hamiltonian systems), the Galerkin method plays an important role. Nevertheless, there is a kind of Lyspunov-8chmidt procedure, called the saddle point reduction, which reduces the infinite dimensional problems to finite dimensional ones. The later method has the advantage of simplicity. We shall introduce tbis method in Section 2.1, and apply it to the study of (0.1) in Section 2.2. We shall also investigate a cl .... of Hamiltonian systems in which the Hamiltonian functions are periodic in some of their variables. It is interesting to note that it causes multiple periodic solutions. This is Section

2.3. 2.1. Saddle Point Reduction Let H be a real Hilbert space, and let A be a self-adjoint operator with domain D(A) C H. Let F be a potential operator with 01> e C'(H,IR'), F = dol>, 01>(8) = O. Assume that (A) There exist real numbers a < f3 such that a, f3 rf. u(A), and that ., utA) n ia, f3] consists of at most finitely many eigenvalues of finite multiplicities. (F) F is Gateaux differentiable in H, which satisfies

The problem is to find the solutions of tbe following equation: Ax = F(x)

x

e D(A).

(2.1)

With no loss of generality, we may assume a = -f3, f3 > O. A Lyapunov-Schmidt procedure is applied for a finite dimensional reduction. Let

Po

=

J_(pp dE"

where {E,} is the spectral resolution of A, and let

According to (A), there exists E > 0 small, such that assume further the following condition:

-E

rf. utA). We

189

2. Reductioru and Periodic Nonlinearitia

(D) ~ E C'(V,llI.l), where V

= D(IAl l /'), with the graph nonn

IIxliv = (1iIAll/'xll~ +<'lIxll~ )

1/'

.

We decompose the space V as follows:

where

--IA,I-l/'no,

"0 .,

For each

U

""±

IA I-l/'H±, =,

and A ,

= EI + A •

E H. we have the decomposition

Xo

= I A £ 1-1/' UO,

x±

= IA e1-1/' u±.

Thus we have

and that V±. Vo are isomorphic to H± and Ho. respectively. Now we define a functional on H as rollows:

where E+ fo= dE" E_ f~oo dE" and ~,(x) = ~lIxll~ F, =tl+F. The Euler equation of this functional is the system

=

=

u±

= ±IA,I- l /' P±F,(x),

E±uo = ±IA,rl/'E±PoF,(x).

+ ~(",).

Let

(2.2) (2.3)

Thus x = ",++xo+x_ is a solution of (2.1) if and only ifu = u+ +uo+u_ is a critical point of f. However, the system (2.2) is reduced to

which is equivalent to

(2.4)

190

Mulliple Periodic Solutio""

By assumption (D), FE C'(V, V), and by assumption (F)

11F.(u) - F.(v)IIH :5 (e + tI)lIu - vllH V U, v E H. Furthermore, there is a 'Y > tI + e such that IIA;'IH+EDH_II:5

~

by assumption (A). We shall prove that the operator F = A;'(P+ + P_)F. E C'(V, V) is contractible with respect to variables in V+ (j) V_. In fact, V z x+ + x_ + z, II 11+ + 11- + z, for fixed Z E Vo,

=

=

1I.1"(x) - F(y)lIv = IIIA.I-'/2(p+ + P_HF,(x) - F.(II»IIH :5III A,I-'/2(p+ - P-)lIs(H)IIF.(x) - F.(y)IIH :5 (e + tllIIIA,I-'/2(p+ + p-lIls(H)II(x+ + x_) - (y+ + II-)IIH . .Since

IIz±IIH IIIA.I-'/2u±IIH:5 and IIIA.r'/2(p+

~lIu±IIH = ~lIx±llv,

+ P-)lIs(H) :5 ~,

we obtain

IIF(z) - .1"(y)lIv :5 <+ tlllz - "lIv. 'Y

The impHcit function theorem can be applied, yielding a solution x±(xo), for fixed Xo E Va, such that x± E C'(Vo, V±). Since dim Vo is finite, all topologies on Vo are equivalent. We have

u±(xo) = IA,I'/2x±(xO) E e'(Ho, H), which solves the system (2.2). Let

a(xo) = J (u+(xo) where "o(xo) = IA.I'/2xo and let

a(z) =

= zoo We have

~ (IIA!/2x+1I2 + IIA!/'E+zlI' -1I(-A,)'/'x_II' -1I(-A.)'/'E_zIl2) - 4>.(x)

=

Z

+ ,,_(xo) + "o(xo» ,

1

"2 (Az(z),x(z» - 4>(x(z»,

191

2. Reductiom and Periodic NonlineariUe.s

where x(z) = {(z)

+ z, {(z)

=x+(z) + x_(z) E D(A). Noticing that

d{(z) = A;'(P+

+ P_)r,(x(z»dx(z)

by (2.4), one sees that d{(z) E D(A) and that

Ad{(z) = (1 - Po)F'(x(z»dx(z). Thus

da(z) = (dx(z))O[Ax(z) - F(x(z))) = Az - PoF(x(z)) = Ax(z) - F(x(z))

(2.5 )

and

tPa(z)

= [A -

F'(x(z»)dx(z)

= AIH, - PoF'(x(z))dx(z).

In summary, we have proven

Theorem 2.1. Under the 8S8umptions (A), (F) and (D), there is a on~one

correspondence:

z .... x

= x(z) = x+(z) + x_(z) + z,

between the critical points of the C'-function a E C'(Ho,IIi.') with the solutions of the operator equation

Ax = F(x)

x E D(A).

Now we turn to the asymptotic behavior of the function a.

Lemma 2.1. Under tbe &ssumptions (A), (F) and (D), we assume furtber that there is a bounded self-adjoint operator F00 satisfying (i) (Foo)

(ii) { (iii)

PoFoo = FooPOi IIF(u) - Fooull = 0(11"11) as Ilull - 00; o ¢ utA - Foo).

Then we have that (1) {(z) = o(llzll) as IIzll _ 00, and (2) the function a(z) is asymptotically quadratic with &symptotics AFooIHo, i.e.,

llda(z) - (A - Foo)zll = o(lIzll) as IIzll -

00.

192

Multiple Periodic Solutions

Proof. By (2.4), we obtain Ae(z) = (I - Po)F(x(z».

Since Po commutes with Fool we have (A - F~){(z) = (I - PollF(x(z» - F~x(z)l.

Hence, V E > 0 there exists R > 0, such that

lIe(z)1I :5

II(A - F~)-lIII1F(x(z» - F~x(z)1I

< eC (lIzll + Ile(z)II), if Ilzll > R, where C

= II(A -

F~)-ll1; it follows that

lIe(zlil

=

oOlzll>·

By (2.5) we have llda(z) - (A - F~)zll = IIAz - PoF(x(z» - (A - F~)zll F~x(z)1I + IIF~x(z) o(II",(z)11> = o(lIzll) as IIzll - 00.

:5 IIF(x(z» =

F~zll

Lemma 2.2. Under the assumptions (A), (F) and (D), we assume that F(O) = O. (1) If there i. a self-adjoint operator Co E C(H, H) which commutes with Po and p_. such that

o

min(u(A) n la,i1J)I:5 C :5 F'(O), then a(z) :5

~ (A -

Co)z, z)

+ 0(lIzIl2) as IIzll - o.

(2) If there i. a self·adjoint operator ct E C(H, H) which commutes with Po and P+. 8tJch that

F'(S) :5 ct :5 max (u(A) n (a, i1J) I, then a(z) ;::

1

"2 (A -

ct)z, z)

+ 001z1l2) as IIzll - o.

Proof. By the definition and (2.5), a(z) =

"21 (Ax(z),x(z» -

1 = "2(Aq,q) - (q)

(",(z»

1

+"2 (Ax+(z),x+(z»

- ((x(z)) - (q)) ,

2. Reduction.J and Periodic Nonlmearitiu

193

where q = x_(z) + z. We shall prove that

that is, 1 a(z) ::; 2(Aq, q) - ~(q).

(2.6)

In fact, 1

~(x(z» - ~(q) - 2(Ax+(z),x+(z»

+ /.' ° (F(tx+(z) + q) - F(x+(z) + q),x+(z)) dt

= 2 1 (Ax+(z),x+(z))

~ ~lIx+(z)II' -

f.'

13(1 - t)dtllx+(z)II' = O.

However,

I~(q) - ~(F'(8)q,q)1 =

If.'

1 :::; -2

(F(tq) - F'(8)t q,q)dt l

sup IIF'(tq) -

O
F'(O)II.C(v,v)lIqll~,

that is, (2.7)

Note that x_ E C'(Vo, V_); this implies that if IIzll - 0, then IIx-(zlllv_ = O(lIzllv.) because x-(O) = O. Thus (2.8) Substituting (2.7) and (2.8) into (2.6), we obtain a(z) ::;

1 2 «A - F'(O))q, q) + 0(lIzIl2)

1

::; 2 (A -

as IIzll -

=

~ (A -

o.

Let

CO)q, q)

+ o(lIzll')

Cil)x_(z),x_(z»)

a_

+ ~ (A -

= min{u(A)

n [a,p]},

and, by the assumption,

'LI::; Cil.

Cil)z, z)

+ 0(lIzIl2)

194

Multiple Periodic Solutitms

This implies

therefore

a(z):O:;

1

"2 (A -

_ Co )z.z)

+ 0(lIzIl2) as z - O.

We prove the second assumption in a similar way. Finally. we apply Theorem 5.2 of Chapter II to solve the operator equation (2.1) under conditions (A). (F). (D) and (Foo). For a symmetric matrix B. let m ± (B) be the dimension of the maximal positive/negative subspace.

Theorem 2.2. Under assumptions (A). (F). (D) and (F 00)' we assume F(9) = 8. If one of the following conditions holds: (1) There exists a bounded self-adjoint commuting with Po and P_, such that min{u(A) n lo,{3]}I:O:; Co :0:; F'(8)

Co.

and m- (A -

CoIHo) > m- (A - FooIHo);

(2) There exists a bounded self-adjoint Cj", commuting with Po and P+, such that F'(8):O:; Cj" :0:; max{u(A) n lo,{3]}I and m+ (A - Cj"IHo) > m+ (A - FooIHo); then there exists at least one nontrivial solution of the equation (2.1). Proof. By Theorem 2.1, problem (2.1) is reduced to finding critical points of tbe function a E C 2 (Ho,1R 1 ). According to Lemma 2.1, a is an asymptotically quadratic function with a nonsingular symmetric matrix A-FooIHo as asymptotics. By Lemma 2.2, condition (1) means that d"a(8) is negative on the subspace Z_ on which A - Co is negative. Thus

Similarly, condition (2) means that

In this case,

m- (A - F""IHo)

= dimHo -

m+ (A - FooIHo)

> dimHo - m+(d2 a(8» = m-(d 2 a(8»

+ dimker(d2 a(8».

2. Reductiot13 and Periodic Nonlinearitiu

195

Both cases imply that

The conclusion follows from Theorem 5.2 of Chapter II.

Remark 2.1. The finite dimensional reduction method presented here is a modification of a method due to Amann [Amal] and Amann and Zehnder [AmZI]. Avoiding the use of monotone operators and a dull verification of the implicit function theorem, we change a few of the assumptions and gain a considerable simplification of the reduction theory. 2.2. A Multiple Solution Theorem We apply the saddle point reduction to Hamiltonian systems. Let H = L2 ([0, 2".], 11I.2n) , A = -J1;, with D(A) given in the preliminary. For HE C"(11I.' X 11I.2n ,11I.') being 2".-periodic in t, we define

F(z) = d,H(t,z(t». Suppose that there is a constant C

°

> such that (2.9)

then

4>(z)

=

/.2. H(t,z(t»dt EC'(H,11I.').

Again, the derivative F(z) = d4>(z) is Gateaux differentiable with

IIdF(z)IIC(H)

~

C Ii z

E

H,

so conditions (A) and (F) are satisfied. By observing the continuous imbeddings

condition (D) is also easily verified. When we study Hamiltonian systems under condition (2.9), the equation is reduced to

da(z) =0, where

I

a(z) = "2 (Ax(z),x(z» - 4>(x, (z».

196

Mullipl. Periodic Solulion8

Lemma 2.3. Suppose that %0 is a nondegenerate 27r-periodic solution of (0.2), i.e., the linearized equation -Ji =
z(O) = Z(27T),

(2.10)

has no Floquet multiplier 1, then the correspondence Zo E Ho is a nondegenerate critical point of a(z).

Proof. Since

dF(xo)z =
o¢

,,(A - dF(xo)) , because 1 is not a Floquet multiplier of (2.10). And since" (A - dF(xo)) consists of eigenvalues, (A - dF(Uo))-1 exists and is bounded. However,

<Pa(zo)

= (A -

dF(xo)]dx(ZO)

where

dx(zo) = idH,

+ d{(ZO),

hence <Pa(Zo) must be invertible, i.e., Zo is nondegenerate. Now we turn to a result which is concerned with the existence of at least two nontrivial periodic solutions.

Theorem 2.3. Suppose that H E e'(]R1 x]R'n,ll1.t) satisfies the follow· ing conditions:

(1) There exist constants n < (3 such that

nl:S
X

]R.n.

(2) Let jo,jo + 1, ... ,j, be all integers within (n,(3] (without loss of generality, we may assume n, (3 ¢ il.). Suppose that there exist l' and C, such that it < l' < (3 and

H(t,z) ~ (3) H.(t,9)

= 9.

1 21'lIzll' -

C V (t,z) E]R1

X

]R.n.

3j E [io,it) nil. such that

jl < d~H(t,9) < (j + 1)1 Vt E ]Rl.

(4) H is 21r·periodic with respect to t.

2. Reductimu tlfld Periodic. Nonlineoritiu

197

Then the Hami/toniau system (0.2) possesses at least two nontrivial periodic solutions. Proof. According to the finite dimensional reduction, we turn to the function

°

1 (Au(z), u(z» - /." H(t, u(z»dt, a(z) = 2

where u(z)

= z + u+(z) + u_(z), z E Z ~ Ho, and u±(z) E Hr..

Since

°

a(z) = 2(Aw,w) 1 - /." H(t,w)dt

+ {~[(AU(Z)'U(Z» where w = z

- (Aw, w)[ - l'[H(t,u(Z» - H(t,W)]dt},

+ u_(z), and the terms in the bracket are equal to

by condition (1), we obtain a(z) :5

~(Aw, w) 1

.

-1"

H(t, w)dt

:5 2(2"31 - -y)lIwll' + c

-

-00

as Ilzl[ -

00

using condition (2). Thus the function -a( z) is bounded from below and satisfies the (PS) condition. In order to apply the three critical paints theorem we claim that 9 is neither a minimum nor degenerate. In fact, using condition (3), it follows from Lemma 2.2 that

~ (A - X/)z, z) + o(lIzll') :5 a(z) :5 ~ (A where (X, >:) C (j,j

>:1)z,

z) + o(lIzll'),

+ 1), as IIzll - O. The theorem is proved.

Remark 2.2. Saddle point reduction was first applied to Hamiltonian systems by Amann Zehnder [AmZlJ. Theorem 2.3 is due to Chang [Chal].

198

Multiple Periodic Soluliom

2.3. Periodic Nonlinearity A special class of Hamiltonian systems (0.2), in which the function H(t, q; p), q = (qI,tJ2, ... ,qn), P = (PIIP2, ... ,Pn), is periodic in some of its variables, say ql,·.· ,qiJ' PI,'" ,Prl P.+), ... I 1 :5 r :5 8 ~ T. provides a possible way of gaining more solutions. This is due to the fact that the Hamiltonian function H, and then the functional J, is invariant under certain translation groups; therefore, the quotient space contains certain tori.

We start with an abstract theorem: Theorem 2.4. Under &ssumptions (A), (F) and (D), we assume that (P) 3 e" ... ,er E ker A, they are linearly independent, and 3 (T" ... ,Tr) E R', such that

II> (x+tm;T;e;) =1I>(x),

I/(m" ... ,mr)EZ',

I/xEH.

J=1

(LL) lI>(x) - ±oo if Ixl Then the equation

00

1/ x E ker An span{e" ... ,er

}-'-.

Ax -1I>'{x) = 0 has at least r + 1 distinct solutions; and jf all solutions are nondegener8te, then there are at least 2r distinct solutions.

Proof. A saddle point reduction procedure is applied. Consider the function on the finite dimensional space Vo defined below: a(z) =

1

:I (Ax(z),x(z)) -1I>(x{z)).

We shall prove that 1. x±

(z + E;=I T;e;) = x±(z),

In fact,

and therefore

1/

zE Vo.

2. Reductiom and Periodic Nonlinmrities

199

Claim.

a (z+ tT;e;) = ,""'1

~(A"(Z+ tT;e;),,,(z+ tT;e;)) ,=1 ,=1 -~(,,(z+ t.T;e;))

=

~ (Ax(Z),X(Z) + tT;e;) - ~(X(Z) + tT;e;) ,=1

1

=

"2 (Ax(z),x(z)) -

=

a(z).

,=1

~(x(z))

3. a satisfies the (PS) condition on T'" x (Y $ N(A).1) n Vo where Y = N(A) nspan{e ..... ,e,}.1, and T'" is the r-torus defined by

R'/(TIZ 1

X ••• X

T,ZI).

Claim. Suppose that {z'} is a sequence along which (a(z')} is hounded, and IlOO(z')11 = 0(1). According to (2.5),

IAx(z') - F(x(z'))IH = 0(1). Let Q be the orthogonal projection onto Y, which is considered to be a subspace of the Hilbert space /C = Y $ N(A).1. Thus on the space /C,

(I - Q)x(z') = A-1(I - Q)F(x(z'» + 0(1), and since F is hounded, 11(1 - Q)x(z') II is bounded. Noticing

~(Qx(z')) = ~(x(z')) - [ (F(x,(z')), (I = a(z') -

1

"2 (Ax(z'),x(z'») -

where

x,(z)

=

fl

10

Q)x(z'))dt (F(x,(z')), (I - Q)x(z'))dt,

«1 - t)I + tQ) x(z),

and

(Ax(z'),x(z'») = (Ax(z'), (I - Q)x(z'») = (F(x(z'» + 0(1), (/ - Q)x(z')),

200

Multiple.

P~ic

Solutiom

~(Qx(ZO)) must be bounded.

According to condition (LL), Qx(zk) is bounded. The compactness of Zk now follows from the boundedness of x(ZO) and the finiteness of the dimension of Vo. 4. If we decompose Vo into span{ e', ... ,er} z

= v + w,

and let g(w, v)

(v, w) E span{_" ... ,er }

E!)

E!)

(Y E!) N(A).l) n Vo,

(Y E!) N(A).l) n Vo,

= "21 (A{(w + v),{(w + v)) -

~(x(w

+ v)),

where

{(z) = x+(z) r

then 9 is well-defined on T x (Y dg(w,v)

E!)

+ x_(z),

N(A).l) n Vo, and

= PoF(x(w + v)),

which is bounded and then is compact on finite dimensional manifold. The function a(z) now is written in the following form: a(w,v)

1 = 2(Aw,w) -

g(w,v).

Noticing that F is bounded, iI~(z HI is always bounded. If we denote by y the projection of w onto Y, we have g(y, v)

= 21 (A{(y + v), {(v + v)) -

~(y) - [~({(y + v)

+ y + v) -

~(Y)l.

The first term and the third term are bounded, therefore g(y, v)

~

±oo

as

Ilyil ~ 00.

The function a(w, v) satisfies all the assumptions of Theorem 5.3 of Chapter II. Theorem 2.4 is proved, provided cuplength (r) = r, and the sum of the Betti numbers of r is 2r. Now we study the periodic solutions of the Hamiltonian systems in which the Hamiltonian functions are periodic in some of the variables. We use the following notations: p, q E R.",

P = (PI!··· ,Pr).

p= (Pr+l,.·· ,Ps), p= (P.+lt ... ,PT),

Il + (!'TH, ... ,Pn),

q = (qt, ... ,qr), ij= (qr+l,'" ,q.). q= (q,+h ... ,IJ'T), tj = ('IT+t, ... ,qn).

201

2. Reductiom and Periodic Nonlinearities

We make the following assumptions: (I) A(t), B(t), C(t) and D(t) are symmetric continnous matrix functions on S', of order (8 - r) )( (8 - r), (T - 8) )( (T - 8), (n - T) )( (n - T) and (n - T) )( (n - T) respectively. Let A = Is> A(t), and fj = Is> B(t) be invertible. (II) il E CO(S' )( R'n,R') is periodic in the following variables p,li, ii, q, and d'il is bounded. (III) Let span{'P ..... ,'Pm} = ker (-if. - (C(t) eD(t») where

i

=

(0 In-T

-In_T) , 0

and CPt, ... I CPm are linearly independent, a.nd

il

(t, f

Tj'Pj) - ±oo

as

ITI = (Tl + ... + T;')'/' -

00.

J=t

(IV) c,dEC(S"RT

with c= (c ..... ,CT),d= (d ..... ,~) and

),

( o,(t) = ( dj(t) = 0,

lSI

lSI

i = 1, ... ,r, 8 + 1, ... ,T, j = 1, ... IS. We define a Hamiltonian function as follows:

H(t,p, q) =

~A(t)j). jJ + ~B(t)q. q + ~(C(t)p. P + D(t)ij. q) T

+ L (0, (t)p, + d,(t)q,) + il(t,p,q). i=1

Theorem 2.5. Under conditions (I)-(JV), the Hamiltonian system (0.2) d

-Jdiz

= H,(t, z),

,

t ES ,

has at least r+T+ 1 periodic solutions, and if all solutions have no Floquet multiplier I, then (0.2) has 8t least 2r+T periodic solutions. Proof. Let

o A(t)

A(t) =

o C(t)

o

o B(t) D(t)

202

Mullipl. Periodic SoM""",

and let

where the subscripts on J coincide with those on p. We have (P.q)

e ker ( -J~ #

q= A(t)p . { P=O.

#

{

(A(t)

0))'

~= £~ A(s)ds· c+d.

with q(2lf) = iRQ).

p=c

" (i.e .• with

A· c =

ker

8). According to assumption I,

(-J~

- (A(t) 0))

= {(B.d)

c = 8.

We have

Ide R,-r} ~ R,-r.

Similarly,

Thus

Let

~(z) = fs. {B(t, z(t»

+

t,

I",(t)p.(t) + d. (t)q. (t)] } dt.

Then all the assumptions (A). (F), (D), (P) and (LL) are satisfied. The proof is complete. Example 2.1. If the Hamiltonian function H e CO(SI X R 2 ·,R I ) is

periodic in each variable, then (HS) has at least 2n + 1 periodic solutions. This is the case where r = • = T = n. This result, related to the Arnold conjecture (cf. Section 5), was first obtained by Conley and Zehnder ICoZl], see also Chang ICha5].

3. Singular Pol ...1ia&

203

Example 2.2. If H E O'(S' X R2n, R'), where H is periodic in the components or q, and that there is an R > 0 such that ror IPI ~ R, I

H(t,p,q) = 2Mp.p+a.p where a Ell", and M is a symmetric nonsingular time independent matrix,

then the corresponding (HS) possesses at least N + I distinct periodic solutions. This is the case where r = 0, IJ = T = R. This is a result obtained by Conley and Zehnder [CoZIJ; see also P.H. Rabinowitz (Rab6). Remark 2.3. Perindic nonlinearity has been studied by many authors: Conley-Zehnder [CoZI), Franks [Fral), Mawhin (Maw2), MawhinWillems [MaWI), Li [Lil), Rabinowitz [RAb6J, Pucci-Serrin [PSI-2). Fonda-Mawhin [FoMI) and Chang [Cha9). Theorems 2.4 and 2.5 are due to Chang. The condition H E 0' can be weakened to H E C', cr. Liu [Liu4J. 3. Singular Potentials

Most Hamiltonian systems interested in mechanics have singularities in their potentials. Let 0 he an open subset in llI.n with compact complement C = llI. n \ 0, n "= 2. Find xO E 0'([0,2,,),0) satisfying

x(t) = grad, V(t, x(t», { x(O) = x(21<),

x(O) = x(2,,),

(3.1)

where V E 0' ([0,2".) x O,llI. l ) is assumed to be 2".·periodic in t, with additional conditions: (AI) There exists Ro > 0 such that

sup {1V(t, x)1

+ 11V;(t,x)I!_. I (t, x) E [0,2,,) x (lltn \ BRo)} < +00.

(A2 ) There is a neighborhond U or C in llI.n such that

A

V(t,x) ~ tf'(x,C)

ror (t,x) E [0,2,,) x

u,

where d(x, C) is the distance function to C, and A > 0 is a constant. The condition (A2 ) is called the strong rorce condition, according to W. B. Gordon [Gorl). For the sake or simplicity, rrom now on we shall denote the subset of 0'([0, 2"J, 0), satisfying the 2"'periodic condition, by C'(S"O). Similar notations will he used for other 2"'periodic function spaces.

204

Multiple Periodic Solutions

We shall study the problem (3.1) by critical point theory. Let US introduce an open set of the Hilbert space H'(S' ,111.") as follows:

A'fI =

{x E H'(S"III.") I X(/) E fI, VI E S'}.

This is the loop space on fl. Let J(x) =

US

define

{~lIf(/)1I1. + V(t,X(/))} dt

f'

(3.2)

on A'fI, the Euler equation for J is (3.1). In order to apply critical point theory on the open set A' fI, one should take care of the boundary behavior of J, i.e., we should know what happens if x tends to 8(A 'fI). Lemma 3.1. Assume (A,) and (A2). Let {"'o} C A'fI and Xo ~ x weakly in H'(S', 111."), with x E 8(A'fI). Then J("'o) ..... +00.

Proof. It suffices to prove 2 • V (t, xo(t)) dl ..... +00.

1

Moreover, since Yet,X') is bounded from below, it remains to prove that "there is an interval [a, bJ C [0,2"1 such that V(I, x.(I))dt ..... +00.

f:

By definition, x E 8(A'fI) means that there is I' E [0,2"J such that x(/') E 00. According to (Ad and (A2), there is a constant B > 0 such that

A

V(I, x) ~ d2(x, C) - B; hence

{+6

V(/,x(I))dl

~ {+6 (lIx(I) _~(I')lli. -

B)

dl

'V 8 > O. However, we have IIx(t) _ x(t')lIm.

~ II _ 1'1'/2 (f'lIf(/)lIi. dt) '/2

from the Schwarz's inequality; thus t·+6

1,.

V (t, x(I» dt

Since the embedding H'(S',III.")

'-+

= +00.

(3.3)

C(S',III.") is compact, we have

Max {lIx(I) - x.(t)lIm' I I E S'}

..... 0

as

after ornitting a subsequence. Consequently, t~+6

1,.

V (I, x.(I)) dt .....

provided by Patou's Lemma and (3.3).

+00,

k ..... 00,

3. Singular Potentials

205

Lemma 3.2. A&mme (AI) IJlId (A,); then there is 8 coustaut Co d... pending on the C l norm of the function V on Sl x (RR \ B".,), such that J satisfies the (PS)c condition for c > Co. Proof. Assume that {x.} c A In satisfies

J(x.)

~

c,

(3.4)

and

where

We shall prove the subconvergence of {x.} in Alfl. Since V is bounded below, (3.4) implies a constant C I > 0 such that (3.6)

e.

= ,~ J:' x.(t)dt. If we can prove tbat {e.} is bounded, then {Xk} is bounded in HI(SI,IlI.R). Hence, there is 8 subsequence x. - x (weakly in HI). Applying Lemma 3.1, we have that x E Alfl and that

Let

IIgrad.V(·,x.)II•• is bounded. Hence, the strong convergence of {x.} fol· lows from the compactness of IK and (3.5). It remains to prove the boundedness of {e.}. If not, we may assume that Ie. I ~ 00; then for large k, we have

which implies

11"

V(t, x.(t»dtl :5 2" sup {lV(t, x)ll (t, x) E Sl x (Ill." \ B".,)}. (3.7)

From (3.5), we obtain

where Y. =

Xk - ~.,

for k large.

206

Multiple Periodic S.luti ....

Since fo""lI_(t)dt = 0, we have

IIII_IIH' = IIx_IIL"

and

lIy_IIL'

$

11l,_IIH';

hence

[WII:i:.(t)lIi. dt $lIx_IIL' + IIV;(t,x.(tnllL'II:i:.IIL" It follows that

IIx.IIL'

$ 1 + 11V;(·,x.O)IIL' $ 1 + 2.. sup

IIV;(t, x)IIa..

(3.8)

(t,Z}ESI x(I"\BIlo)

Substituting (3.7) and (3.8) into (3.2), and letting

co =

2 1 -2 (1 + 2" (t.z)eS sup IIV;(t,x) II•• ) x (- .. ,Silo) I

+ 2"

sup (t,z)ESl )(1\'"\8110)

Wet, x)l,

we have J(x.) $ CO. This is a contradiction. Lemma 3.3. There exists

8

sequence of integers

o

Proof. Pick a point Po E C and choose R > 0 such that C c B R, we have llI.n \ BR C fl C llI.n \ {Po}. Since llI. n \ BR is a deformation retract of llI. n \ {Po}, A'(Rn \ B R) is a deformation retract of A'(llI. n \ {Po}), and then A'(llI. n \ BR) is a retract of A' fl. We obtain H.(A'fl)

Q<

H. (A '(llI.n \ BR»)

Q<

H.(A',sn-')

(j)

(j) H. (A 'fl, A'(JRn \ Bn» H.(A'fl,A',sn-')

from Chapler I, Section 10. According to Bott [Bot), the Poincare series of A1 sn-l is written as

with il2 coefficients. Our conclusion fonows.

207

3. Singular PotentiaLt

Lemma 3.4. libr each b > 0, there exists a Brute dimensional singular complex M = M. such that the l.ve/ set J. = {x e A'(} I J(x) :5 b} is deformed into M. Proof. According to (A,) and (A,), we have b, > 0 such that

From Lemma 3.1, there exists EO = E(b, b,) > 0 such that

d(x(t), C) ~ EO V x Let us choose an integer N

e J.

'I t

e 5'.

60 1 / 2

= Nb

> 211' ~, 8Ild let

21ri

. t=O,I,2, ... ,N.

ti=N' Define a broken line

'I t = [t,_"t,l, i = 1,2, ... ,N, for any x e J., and let M = {x(t) I x e J.}. The correspondence:;; ..... (x(t,),X(t2), ... ,X(tN» defines a homeomorphism hetween M and a certain open subset of the N-fold product () x () x ... x (). We shall verify the following. (1) Me A'(}. Indeed, 'I x e J., 'I t, > t2,

Therefore, d(:;;(t), C)

~ d(x(t.), C) -

(1 -;" N(t - t,_,») IIx(t,) - x(t,_,)II••

~ EO - 2..N-'b:/ 2 > 0 Vte It,_"t,),i=0,1,2, ... ,N.

208

Mullipl. Periodic Soluliom

(2) There exists '1 E C '1(1, J.) = M. We define 'I as follows:

'1(" x)(t) =

x(t) 1

I

0°, 1] X J.,A'O)

such that '1(0,·) = id, and

for t

~

2...

t

(-I._I

( - 2•• ,._.) x( ,-,)

+ 2'11"_-"_1 '-';-' x(211"')

for

z(l)

1,_, < t < 211"'

for I:S 1,_, :S 2... :S I,

then 'I is the required deformation. We bave proven that J. is deformed into M in the loop space A' O. The proof is finished. Lemma 3.S. For each q > nN, where N = N", is as defined in Lemmas 3.2 and 3.4, set c = inf max J(x), .lEo xelzl where Q E H,(A '0) is nontrivial. Then c ·ofJ. ,

> eo and tben c is a critical value

Proof. If not, c :S eo, then tbere is a Iz) E a such tbat Izl C J...+,. According to Lemma. 3.4, there exists a deformation 7]: (0, I} x Jco + 1 -+ A10, such that '1(1, J...+l) C Moo+l, with dimMoo+l :S nN.... This implies that '1(I,lzll C M ...+l. But '1(I,lzll E a, and a E H,(A'O), with q > nNoo. This is impossible.

Theorem 3.1. Under assumptions (A,) and (A.), (3.1) possesses infinitely many 211"-periodic solutions. Proof. We prove the theorem by contradiction. Assume that there are only finitely many solutions: K = {x"x., ... ,x,}. Noticing that the nullity dimker(d'J(x;)):S 2n, V j, let

q' > max {nNoo , ind(J,x;)

+ dim ker(d' J(x;» 11 :S j :S I} ,

and

b> max {eo, J(x;)

11 :S j

:S I}.

It follows that

C,(J,x;)=O

Vq~q·,j=I,2,

... ,I,

(3.9)

and

H.(A'O, J ... ) = H.(J., J... ). Consequently,

(3.10)

4. The Multiple Pendulum Equation

209

provided by the Morse inequalities. But

i.:H.(AIO) _ H.(AIO,J",) is an injection for q 2::: q-, and the conclusion of Lemma 3.3 contradicts

(3.10). The proof is finished. For autonomous systems, i.e., the potential V is independent of t, in order to single out nonconstant 21r-periodic solutions, we have Corollary 3.1. Under the assumptions of Theorem 3.1, if, further, V is independent oft, then (3.1) pD" ESSes inJinite1y many 211"-periodic nonconstant solutions, if V" is bounded from below on the critical set K of V. Proof. For Rny constant solution x(t) = xo, the Hessian of J at Xo reads as cP J(xo)x = -x + V"(xo)'" with periodic boundary conditions, and hence, the Morse index and the nul· Iity must he bounded by a constant depending on a, where V"(x) ~ aInxn V x E K. We conclude that all constant solutions have a bounded order of critical groups. Therefore there must be infinitely many nonconstant solutions. Remark 3.1. Problem (3.1) was studied by Gordon [Gorl). The critical point approach was given by Ambrosetti-Coti-Zelati [AmZI-2) and CotiZelati [CotI). Theorem 3.1 improves the results in [AmZI-2) considerably, where assumption (Ad was replaced by much stronger conditions:

)V(t,x)I,lIgradzV(t,xlll- 0 uniformly in t, as

IIxll -

+00; and there exists RI

Vet, x) > 0 V x,

> 0 such that

IIxll ~ RI.

Theorem 3.1 is due to M.Y. Jiang [Jial-2). Some related problems of the three body type were recently studied by A. Bahri and P.H. Rabinowitz [BaRI). By Rvoiding condition (A,), Bahri aod Rabinowitz introduced the concept of generalized solutions. The existence and multiplicity results for generalized solutions were studied in [BaRI). A most important problem is to ask when the generalized solution is a regular solution. 4. The Multiple Pendulum Equation The Problem. The simple mechanical system consists of double mathematical pendula having lengths tt,l2 > 0 and masses ml,m2 > 0, as illustrated in the following figure.

210

Multiple Periodic Solution.o

The positions of the system are described by two angle variables
L(
~(m, + m2}1~.p~ + (m, + m2}l,t2 cos('P' +

'P2}.p,.p2

~m2e..p~ + V(
where