Progress in Nonlinear Differential Equations and Their Applications Volume 6
Editor
Haim Brezis Universih! Pierre et Marie Curie Paris and Rutgers University New Brunswick, NJ.
Editorial Board A. Bahri;"Rutgers University. New Brunswick Jobn Ball. Heriot-Watt University. Edinburgh Lui, Caf=IIi.lnstitute for Advanced Study. Princeton Michael Crandall. University of California. Santa Barbara Mariano Giaquinta. University of Aorence David Kinderlehrer. Carnegie-Mellon University. Pittsburgh Robert Kobn. New York University P. L. Lions. University of Paris IX Luuis Nirenberg. New York University Lambertus Peletier. University of Leiden Paul Rabinowitz. University of Wisconsin. Madison
Kung-ching Chang
Infinite Dimensional Morse Theory and Multiple Solution Problems
Birkhiiuser Boston· Basel • Berlin
Kung-'p: pq:
to
D..p D..q
----+
-
D.. p + q A p+q
be
and then define [fT,cud]
= [fT.\p,c). [fTp"d].
The cup product is bilinear, associative, and possesses the unit, i.e., the ().cochain I, which is defined by [x, 1) = e If x E X. We may easily prove that
c(cu d) = ccu d + (-I)PcU cd
If c E CP(X,G), If dE C'(X, G).
Hence, Z'(X, G) is a suOOlgebra of C'(X, G) and B' (X, G) is an ideal of Z'(X, G). The cup product U is well defined on H'(X,G), and makes it a graded algebra. Furthermore, if f: X Y is continuous, then 1": H'(Y) - H'(X) is a ring homomorphism: l"(cUd) = l"(c)U/'(rI), which satisfies /,/i = c/,.
1.
A Review of Algebraic TopolO91/
9
The cap product is defined as the dual operator of the cup product, i.e., ~ C,(X),
n: Cp+.(X) x cP(X)
VeE cP(X), V dE C'(X), VuE Cp+.(X), (une,d) = (u,eUd), Of,
equivalently,
une= (u~p,e]up•. The boundary operators relate the cap product as follows:
8(u n c) = (-I)P(8u) nc - u noel. VuE Cp+.(X), VeE CP(X). If J : X -----t Y is continuous, then we have
f.(unr(e)] = f.(u)nc. Since VuE Zp+.(X), VeE ZP(X), we have u neE Z,(X), and VuE Bp+.(X), VeE ZP(X), we have un c E B.(X), the cap product is well-defined on homology groups:
n : Hp+.(X) x HP(X)
~
H,(X).
The definition of cup product and cap product can he extended to toplogical pairs. In fact, we have
n: Hp+o(X, Y;G) x HP(X, Y;G) ~ H.(X, G) n: Hp+.(X, Y;G) x HP(X, G) ~ H,(X,Y;G), and U: HP(X, Y,; G) x H'(X, Y2 : G) ~ HP+'(X, Y, U Y2; G),
if (Y11 Y2) is an excisive couple in X. The cup length of a topological space X is defined as CL(X) = max {I E Z+] 3 e" ... ,e, E W(X,G),
dim(",)
> 0, i
= I, ... ,I,
such that
e, U .. · U e, ¥ O} .
This is a topological invariant which is very useful in critical point theory. More generally, we define the cup length for a topological pair (X, Y).
CL(X, Y) = max {I E Z+ 13 CO E W(X, Y), 3 e"e2, ... ,c, E W(X),
with dim(",)
> 0,
i = 1,2, ... , I, such that CO U
c, U· .. U c, ¥ O}.
10
Infinite Dimen.rionol Mone Theory
In the case where Y = 8, we just take co E H"(X). These two definitions are the same. We may characterize CL(X, Y) by its dual.
Definition 1.1. Let (X. Y) be a pair of topological spaces and Y eX. For two nontrivial singular homology classes [17,). [172) E H.(X. Y). we say that [17,) is subordinate to [1721. denoted by [17,1 < [1721. if there exists c E W(X). with dimc > 0 such that
[17,) = [1721 n c. where n is the cap product. Let us define L(X. Y) = max {I E Z+
13 nontrivial classes [17jl E H.(X. Y). 1 :: r
c Yand I/>(r-') = p.
(X,Y,p),
i.e.,
-
X,I/>(Br-')
(Denote by rln(X,p) the set of all continuous maps I/> : (In,BIn-') _ (X,pl). The n-relative homotopy group "n(X, Y;p) is defined as the set of aU components of rln(X, Y;p) (the n homotopy group "n(X,p) for those of rln(X,p». In fact, "n(X,p) = "n(X,p;p). One may define a multiplication on rln(X, Y,p) : Fbr n ~ 2 (and, on rln(X,p), for n ~ 1) : V 1/>, '" e rIn(X, Y,p),
(I/>' "')(t) = { I/> (2t" to,··· ,tn ) '" (2t, - 1, t2,.·. ,tn ) The homotopy class
0$ t, $
!$
!
t, $ 1
Vter.
[I/>. "'I clearly depends only on [I/>I and ["'I. Hence
we may define a multiplication on "n(X, Y,p) by taking
[1/>1* ["'I = [I/> * "'I· According to the multiplication, which is obviously associative, the iden-
tity element [el is the class which contains the unique constant map
An inverse element of [I/>I is the class [I/> Q 01, where 0 : r the map defined by Ott) = (1 - t" t2, ... ,tn )
_
In denotes
for every t e In. With the mnltiplication structure, "n(X, Y,p) is a group for n ~ 2, and "n(X,p) is a group for n ~ 1. Fbr n = 0, we define ".(X,p) to be the set of path-connected components of X with the path component of p as a distinguished element. So,1ro(X,p) is only a set, without group structure, as is 7fl(X,y,p). Moreover, one caD show that 71'n(X,p), for n ~ 3, and 1r'R(X, Y,P), for n ~ 2, are sheHan groups.
There are alternative definitions of homotopy groups. Since In-l = sion map
ar\In-l
is contractible, let
Zo =
9 in In, and the inclu-
12
Infinite Dimen.rional Mor$e Theory
is a homotopy equivalence. Hence "n(X, Y,p) is defined as equivalent to
the set of homotopy classes of maps: "': ([",Il[",zo) - (X, Y,p). Again, let Xo = (1,0, ... ,0) E IlI.n, Dn = the unit ball in IlI.n, and
sn-' = IlD", 1f"(X, Y,p) as the set of homotopy classes of maps
The homotopy groups enjoy analogues of all the axioms of homology groups except the excision axiom. (1) Suppose I: (X,Y,p) - (X',Y',P') is a continuous pointed pair ma.p, then there is a reduced homomorphism
I. : ... (X, Y, p) -
... (X', Y' ,p').
(a) If 1= id, then I. = id. (b) If 9 : (X', Y',p') ---t (X", Y",p") is another continuous pointed pair map, then (gil. = g• . I .. (c), Let Il: 1f.(X, Y,p) - 1f._,(Y,p) be defined by restriction, i.e., for a given",: (I",llln,lll"v n -') _ (X, Y,p), we define Il[",] = ["']81"], We have the following commutative diagram: 8
1f.(X, Y,p)
~
1,·
1f. (X', Y' ,P')
-
8
1f._l(Y,P) l(fly). 1I"k_l(Y',P').
(2) (Homotopy invariance). Suppose that I is pointed pair homotopic to g, where I,g: (X,Y,p) - (X',Y',p'). Then
I.
= g•.
Thus, if (X, Y,p) and (X', Y',P') are pointed pair homotopic equivalent, then ".(X, Y,p) = (X', Y',p').
1f.
(3) (Exactness). Let p E Z eYe X, then the following sequence is exact:
... - "'+t(X, Y,p)
8'
~ ".(Y, Z,p)
~ '1rt(X, Z,P) ~ 7rk(X, Y,P) - ...
1.
A Review
0/ Algebrtric Topology
13
where i : (Y, Z) -+ (X, Z), j : (X, Z) -+ (X, Y) are inclusions, and define 8' : "k+'(X, Y,I') -+ "k(Y,Z,I'), k , I: U, - 4>,(U,) c X, homeomorphism, i E A},
2.
A Review of the BafUlch-Firuler Manifold
15
(3) I/>i 0 1/>-;:' : I/>..(Ui n Ui ,) - I/>i(Ui n Ui ,) is a C'" -diffeomorphism, 'V i,i' E A. Each pair (I/>i, Ui ) is called a chart. The set {(I/>i, Ui ) lie A} is called an atlas. In a similar way, we define C'"(e'-O) maps between two C' Banach manifolds, and vector bundles over Banach manifolds (we omit tbe defini· tions and basic properties), in particnlar, the tangent bundle T( M) and cotangent bundle reM). Let :=: = (E,,,, M) be a vector bundle. { : M - E is called a section, if ,,0 {= id M • A section { is called C'" (or e'-O) continuous, if it i. a C' (or e'-O) map from M to E. A section of the tangent bundle is called a vector field, and a section of the cotangent bundle is called a «>·vector field. For a given C 1 - O vector field e, and a given point p E M, there exists an unique maximal semiflow a : [0, T) _ M satisfying a(t) = {(a(t)), 0(0) = p.
A Riemannian manifold (M,g) is metrizable. The metric d is defined by the arclength of the geodesics, and thus it is defined by tbe Riemannian metric g:
d(x,y) = inf
{1'
g(a(t),a(t))i dt
I a(fJ)(OfJ) c X, and let
{o~
hfJ(x)
then hfJ
e C'-O(X, R~)
Claim. \I x"x.
= inf {lix - ylll y rf. VfJ},
and VfJ
= {x e X I hfJ(x) > O}.
e X, \I E > 0, 3 y E VfJ
such that
and so
hp (x,)
s IIx, - yll S IIxl - x.1I + IIx. - yll
s IIx, - x.1I + h" (X2) + E, i.e., hfJ(x,) - h,,(x.) S IIx, - x.1I + e. We may change tbe positions of x, and X2, Rnd since f. > 0 is arbitrary, we obtain
2.
A &view of the BIDl4Ch-Fi...ler Manifold
17
Defin. and define
)'1/JCp -
//JCp) /J3Cp)" /JEB
E
Then we have
sUP'1/J C
0/J C Uo(/J), V
fJ e B,
and
o Theorem 2.1. Suppose that M is a paracompact Banach m811i[old and that .. : E --+ M is a B8JJach vector bundle, then there exists a FinsIer structure on the vector bundl•. Proof. Choose an open covering {(T"U,) I; e A} which trivializes the vector bundle. On each ,,-'(U,), we have a natural Finsler structure 11·11•. Since M is paracompact, there is a locally finite refinement {o/J I fJ e B} of the above open covering and the corresponding C,-o-partition of unity {O/J, '113) I fJ e B}, say, V fJ e B, 3 t> = o(fJ) e A such that 0/J C Ua (J3). Let
111·111 = 2:>13 0 "'OIl·lIa(/J)' J3EB
This is a Finsler structure on E.
Claim. The continuity of 111·111 is trivial. V'Ve M,3 finitely many fJ e B such that '1J3Cp) i' 0, since IIxll.,o(/J) = IICp,x)lIo(/J) is a norm V fJ e B. Therefore
IlIxlli. = L PeB
'113 ° ,,(x)IICp,x)lIa(1I)
= L'1/JCp)ll xll.,o(J3) l1eB
is also a norm, which is an equivalent norm on Ep . A>; to (3), V Po e M, V k > I, by definition, 3 a neighborhood U of Po
such that there are at most finitely many {fJ;}Y such that '1"Cp) '" 0, V fJ ¢ {fJ;} y, V V e U. Since II . II, is a Finsler structure, 3 V c U, a smaller neighborhood of PO such that
and
18
Infinite Dimensional Morse Theory
'rip E U, i = 1,2, ... ,n, we obtain
o Definition 2.3. A regular C'-Banach manifold M, together with a Finsler structure on its tangent bundle T(M), is called a Finsler manifold. Example 3. Any paracompact Banach manifold possesses a Finsler structure on its tangent bundle, making it to be a Finsler manifold.
Example 4. If M is a Finsler manifold, with Finsler structure
II . II, then
we can define
II(P, x')11 = sup {(x' ,x)llIxli p :5 1 \I x E Tp(M)) . \I x· E Tp(M)', it is a Finsler structure on T'(M). Particularly, if f : M ~ Rl. where M is a paracompact Banach manifold, then lI(p, dJ(p»1I is well defined and p ~ II(P, dJ(p}}1I is a continuous function. In t!'e following, we omit pin (P, df(P», and denote it by Ildf(p)lI. For 8. Finsler manifold M 1 we may define a metric d as follows:
d(x,y)
= inf {L(u)lu E C ' ([O,I],M),
where L(u) =
1.'
u(O)
= x,
u(l)
= y},
II(u(t),u'(t»lIdt.
One can prove the following:
Theorem 2.2. Suppose that M is a Finsler manifold, and that the 8 metric on M, and the reduced topology is equivalent to the topology on the manifold.
function d is defined above. Then d is
For a proof we refer to Palais [Pa!4]. Finally, we come to the main subject of our theory. Definition 2.4. Let M be a C'-Finsler manifold. Let J E C'(M,JlI.I); a point p E M is called a critical point of J if dJ (P) = 9. The set K = {p E Mldf(P) = 9} is called the critical set. A rea! number c is called a critical value if (c) n K # 0. A real number, if it is not a critical value of I, is called a regular value. The complement set of K, i.e., M- = M\K, is called regular set. A point in M- is caUed a regular point.
r '
3.
A Pseudo Gmdient Vector Field and De/ormation Theorems
19
3. A Pseudo Gradient Vector Field and Deformation Tbeorems The basic idea in critical point theory is to investigate the variations of topological structures of the level sets of a given function I. Certain flows, depending on tbe gradient of the function I, are .-I to deform these level sets. In finite dimensional manifolds, or more generally, in Hilbert Riemannian manifolds, gradient How is a natural candidate. However, there are two disadvantages to gradient How: (1) It needs more smoothness, say c>-o, and (2) it only works on Hilbert Riemannian manifolds, because, for a Banach Finsler manifold M,I : M _ RI, dl E TO(M) rather than T(M), tbe gradient flow is not well·defined. We are introduced to
Definition 3.1. (Pseudo gradient vector field). Let M be a Finsler man· ifold, and let I: M - RI be differentiable at p EM. X E Tp(M) is called a pseudo gradient if
IIXII::; 211d/(P)1I, (2) (df(P),X):::: 11dI(P)1I 2 , (1)
where ( , )p is the duality on Tp(M), and II II is the Finsler structure. Let ScM be a subset, and, if I is differentiable on S, X is called II pseudo gradient vector field on S (p.g.v.f. in short), if V pES, Xp is a pseudo gradient of I at p. Lemma 3.1. Suppose that M is II Finsler manifold, and that I : M E:.. RI. Let MO = M\K, where K = {p E M I dl(P) = OJ is the critical set; then V p E MO, there exists a pseudo gradient of I at p. Proof. Since 11dI(P)1I '" 0 V P E MO, by definition, 3 X E Tp(M) such tbat IIXII = I and (dl(P),X) > ~11dI(P)II. Let Y = ~lIdl(p)IIX, then IIYII = ~1Id/(p)1I < 211d1(P)II, and (df(P), Y)p > Ildl(p)1I2. 0 Theorem 3.1. Suppoee that M is a C>·Finsler manifold and that There exists a C'-O p.g.v.f. of Ion MO.
f :
ME:.. RI.
Proof. According to the lemma, V pO) E MO, 3 Xpo E Tp(M) such that
IIXPOII < 2I1d/ IIdI(pO)112.
The continuity of the Finsler structure and the continuity of dl(p) imply that there is a neighborhood VPO of pO) , VPO C MO, such that
IIXPOII < 211d1(P)1I,
20
Infinite Dimensional Morse Theory
and Since M· is metrizable, it is paracompact. There is a locally finite C 1 - O partition of unity {'III I [3 E B}, with supp 'III C V.. ' for some Po = Po([3) E M'. Let
x=
Xl!') = L'III(P)XPO(P)' lIeB
This is the p.g.v.f we need. Claim.
IIX(!')II ~ L'Ip(P)lIx""(II)1I < 211 dl(P)II, (dl(P), X(P)) =
L 'III (p)(dl(p), XPO(II)l > 1Id/(p)1I2.
The local finiteness of the supports of {'III I [3 E B} implies the C 1 - 0 smoothness. Once we have a p.g. v.f. of a function J, we get a decreasing flow by solving the following ordinary differential equation (ODE):
u(t) = -X(cr(t)) 0'(0) = Xo E M' . The equation is locally solvable. Along the How, the function t ..... is decreasing: d d/
0
cr(t)
10 cr(t)
.
= (/'(cr(t)), cr(t)) = -(f'(O'(t)),X(cr(t))
~ -1I1'(cr(t»1I2.
r
=
Our first goal is to prove that if 1 la, b)nK 0, i.e., I has no critical value in the interval la, b), then la is a deformation retract of lb. The deformation will be realized by the flow. Now the problem is that the pseudo gradient flow so far is only defined locally; we do not know if it could exist as long as it arrives at J0.' A condition on the function I is needed. I
Definition 3.2. Given I : M S Rl &ad c E Rl, we say that I satisfies tbe (PS)c condition if any sequence {Xn} C M along which I(x n ) - c and d/(xn) - 6 (strongly) possesses a convergent subsequence. We say that I satisfies the (PS) condition if it satisfies (PS)c for all c E Rl. We have the following facts:
3.
A Pseudo Gradient Vee/or Field and De/ormation Thoorems
21
S
(1) If F : M RI satisfies (PS)e Vee [a, b), and if K n I-I)a, b) = 0, then 3 fO, 00 > 0 such that
Claim. Ifnot, 3x n e 1-I[a-~,b+Hn = 1,2, ... ,satisfying «f(xn) ~ 9. According to the (PS)e Vee [a, b), there exists a convergent subsequence x ni - x· , which implies x· E K n [a, bJ. This is a. contradiction. (2) If I : M
S
I
,-I
RI satisfies (PS)e, then Ke := K n I-I(c) is compact.
Claim. If {xn} eKe, then I(xn) = c and «f(xn) condition implies a convergent subsequence.
= 9. The (PS)e
Lemma 3.2. (Deformation). If I e CI(M, RI) satisfies (PS)e, Vee [a, b), and if K n I-I (a, b) = 0, then I. is 8 strong deformation retract of
J.. Proof. 1. We consider the pseudo-gradient (semi) Bow on I-I [a, b) :
u(t) { u(O)
=
-X(u(t»/IIX(u(t»112
=Xo e I-I[a,b).
We want to show that the maximal solvable half interval [0, T.,) satisfies (1) (2)
T:3:0 < +00,
f(u(T•• -
0»
= a.
Claim. Since
I(u(t» - f(xo) = - ] . ' < d/(u(T», U(T) > dT each initial point Xo e I-I [a, b) arrives at time, i.e., T~o
< +00.
I. along the flow in a finite
Noticing
,...
(to
dT
II it u(T)dT 11:$ it II X(U(T) II :$ this implies u' =
lim
1, fO
It - tl,
u(t) exists. If f(u') > a, then by fact(I), the Bow
t--Tzo-O
O"(t) can be extended beyond T.ro. This is in contradiction with maximality. We call Tzu the arriving time. 2. The arriving time function x _ T,r: 1-1 [a, bl _ Rl is continuous. Claim. t
= Txo
is the solution of the equation
f(u(t, xo»
= a,
lnfinj~
22
Dimensioool MorM! Th 0 such that I_T._O lim 10 u(t,x) = 1. We shall prove that the
lim u(t,x) does exist and then
t-+T",-O
l(u(T. - O,x)) = a. Claim. (PS). implies K. is compact. Either one of the following cases occurs:
inf dist(u(t,x), K.) > 0, e (O,T.:a:) inf dist(u(t, x), K.) = O. t e IO,Tz )
(a)
t
(b)
In case (a), again by (PS). VeE [a, bJ, 3 inf
IE [O,Tz }
> 0 such that
[W(u(t,x))11
~
.
Thus dist(u(t"x), u(t"x))
Thus we have two sequences
ti
0 and ttl -+ T;ro - 0, Xn -+ Xo such that dist(a(tn,X n ), a(T., - O,xo» ~ 0 such that dist(,,(t,xo), ,,(T•• - O,xo»
0 such that T%>T:ro-o
if x E B(xo,.,). For t E [T., - .,T.,) n [T•• - .,T.), x E B(xo,.,), 36, E (0,.) such that dist(,,(t,x), ,,(t,xo»
O.
II'!; IIdt::;
~(t. -
t;.1
~ O.
This is a contradiction. The proof is completed.
o
Remark 3.1. Tbeorem 3.2 is due to E. Rothe IRotlJ, K.C. Chang [Cball and Z.Q. Wang (WaZ4J. Another deformation lemma is also often used.
Theorem 3.3. (First deformation lemma). Let M be a C' Fiosler manifold. SUPJX16e that I E C 1(M,1R1) satisfies the (PS)c condition. Assume that N is a closed neigbborhood of Kc = K n 1-1 (c). Then there exist a continuous map ~: [O,lJ x M ~ M and constaots f > • > 0 such that
(1)
~(O,·)
= id,
(2) ~(t, ')lcf-'[C-'.C+~ = idlcf-'[c-l,c+~' (3) ~(t,·): M ~ M is a homeomorphism If t E [O,IJ,
(4) 7)(I,/c+. \N) C Ic-" (5) 10 7)(t,x) is nonincreasing in t If(t,x) E [O,IJ x M. We would father prove a more general version of Theorem 3.3, which
indicates the fact that f depends on constants derived from the Palais-Smale condition. Namely,
InJinire Di....fI6ionol Mors. 'I'Ioemy
30
Theorem 3.4. £etM beaC'-FinsJerm8Jlifold. Suppose/ E C1(M.IR'). Let N' c N be two closed neighborhoods satisfying
dist(N'.aN) ?
~6.
6> O.
Suppose that there are constants b and • positive, such that Ild/(x)1I ? b V x E /,+. \(f,-. UN').
o < • < Min {~6b2, ~6b} Then for any 0 < < < (1)-(5) in Theorem 3.3.
!.
.
there exists ~ E C([O.I] x M. M) satisfying
We assume Theorem 3.4 at this point; then we give the Proof of Theorem 3.3. According to (PS)c) Kc is compact. Hence, for 6 > 0 sufficiently small. N(6) = {x E M I dist(x. Kd < 6} c N. Again. by (PS)" there exist constants b• • > O. such that (3.1)
Since (3.1) remains valid. if e is decreased. we may assume
Theorem 3.3 follows from Theorem 3.4. if we take N' = N (~). Proof of Theorem 3.4. Define a smooth function:
p(s)={~
for s rf. [c - e. c + (8 - 8 - "4)6 = 0, so 9 ° '1(t, x) = 1. Now,
;/ ° '1(t,x) =
(d/('1(t,x»,Ij(t,x»
= -q(l[V('1(t,x»IIl(df('1(t,x», V('1(t,x)))
I $ -2 q(IIV('1(t, x»II)IW('1(t, x»I1'·
If IIV('1(t, x»11 ::; 1, then we have
d 1 I d/°'1(t,x) $ -21Id/('1(t,x»II' $ -2 b'.
32
Infiniu Dimensional Morse Theory
Otherwise,
In summary,
Thus,
This is a contradiction.
Remark 3.2. Theorem 3.3 is due to Palais [Pall); see also P.H. Rabinowitz [Rabl). For Theorem 3.4, cf. J.Q. Liu [Liu2), and K.C. Chang, J.Q. Liu [Cl,tL2).
4. Critical Groups and Morse Type Numbers Basic Morse theory is set up in two steps: (1) Locally, we define a sequence of groups, which we call critical groups, to describe the local behavior of a function / near its critical point. (2) Globally, we define a sequence of numbers, which count the critical points in accordance with the critical groups. These numbers are called Morse type numbers. We study the relationship between these Morse type numhers and the deformation property of the underlying manifold via the deformation property.
Definition 4.1. Let p be an isolated critical point of!, and let c = !(p). We call
C.(/,p) = 11. (/, n Up, (!c\{p)) n Up, G) the qth critical group, with coefficient group G of ! at p, q = 0, 1, 2, ... , where Up is a neighborhood of p such that K n (/, n Up) = {p}, and H.(X, Y;G) stands for the singular relative homology groups with the abelian coefficient group G. According to the excision property of the singular homology theory, the critical groups are well-defined; i.e., they do not depend on a special choice
of the neighborhood Up. Now we give some examples.
4. Crit.ical GroUP6 and Mor6e 7yPe Numbers
33
Example 1. Letting p be an isolated local minimum of! € C(M, RI), then
C.(/,p) = { : In a finite dimensional manifold MR, if p is an isolated local maximum of ! € C(M",RI), then
C.(/,p) = { :
q=n q#n.
Example 2. Let M be a I-dimensional manifold. If p is an isolated critical point of ! € C(M, RI), which is neither a local maximum nor a local minimum, then we have Co(/,p) = C1(/,p) = O.
Example 3. (Monkey saddle.) Let ! be the function x 3 - 3xy2 defined on R2. Then we have Co(/, 0)
= C2 (/,0) = 0,
and
C1(/,0)
= G (f)G.
Let M be a Hilbert-Riemannian manifold. ! € C'(M, RI), p € K is called a nondegenemle critical point, if d' ! (P) has a bounded inverse. Since A = rP !(P) is a self-adjoint operator which possesses a resolution of identity, we call the dimension of tbe negative space corresponding to the spectral decomposing, the Morse index of p, denoted by ind(/,p). (It can be 00). Now we are in a position to compute the critical groups of a nondegenerate critical point via its Morse index. The following Morse lemma is a cornerstone in studying the local behavior of a nondegenerate critical point. The proof is postponed to the next section in which a general splitting lemma will be proved. Morse Lemma 4.1. Suppose that! € C 2 (M, RI) and that p is a nondegenerate critical point; then there exists a neighborhood U. of p and a local diffeomorphism oI> : U. ~ T.(M) with oI>(p) = 8, such that ! 0 oI>-l({) = !(P) + ~ (d'!(p){,{) , V { E oI> (U.) ,
where ( , ) is the inner product of the Hilbert space H, on which the Riemannian manifold M is modeled.
34
InfiniU Dim"""iona/ M"..". Th,: B (0, 0,) .... B(Z;,o) (into)
=
satisfying
f
0
'1>,
(Y') = ~ (Mil' -IIY~II') + c,
where 11' E B(O, 0;), the ball centered at 8, with radius 0, > 0, in the tangent space T z , (M), and y' = 1J~ + 1J~ is the orthogonal decomposition according to d' f(z,), i,j, = 1,2, ... ,t.
40
Infinite DimemionGl Mon- Theory
Let N = ut=I4>;B(IJ. 6;). We apply tbe first defonnatioD theorem to retract T/ : '1(/e+, \N) C 1._•. Since T/ decreases tbe wlue of I. T/ (I.)
N.
yielding a defonnation
c (I. n N) u 1.-,.
It suffices to show (/e n N) u 1._. We choose fl E (0. f) satisfying
'" Uf:lh;(Bm;) U le-.·
v'2f1 <mio{6;.
i= 1.2•... •i},
and deSne
=
= + V!. according
wbere t E [0.1). y; 4>-I(X) as x E B(z;.6). and y; Y~ to the orthogonal decomposition, i = 1,2, ... ,t. Thus
The restriction 4tIB(9,yl2;),>np,-T::1(M) yields the bomeomorphism hi. where P;- is the orthogonal projection onto the negative eigenspace of d' fez;). i=1,2, ... ,t. Combining T/ with ~. we obtain the defonnation retract. (4.3) is easily verified. 0 This completes the proof.
The references for Theorems 4.1 through 4.4 are Milnor (Mill). Schwartz [ScJI). Rothe (Rotl). Palais (Pall). Pitcher (Pitl) and Marino Prodi (MaPI). In Theorem 4.4. the handle body theorem is established on Hilbert Riemannian manifolds. wbere the Morse Lemma holds. and the local behavior of a nondegenerate critical point is quite clear. In order to extend this theorem to Finsler manifolds, or to Banach spaces, new difficulties arise in two ways. One way is conceptual: The above definition of nondegenerot:.y does not mal<e sense in Banach spaces because the Hessian d' f( x) is a bounded linear operator from the space to its dual, so that one cannot say that d' fIx) has a bounded ioverse except when the spot:.C is isomorphic to its dual. The other difficulty is technical. The Morse lemma is no longer useful because it is oot compatible with the Palais-Smale condition. e.g .• the quadratic functional feu) = u 2 (t) dt does not satisfy the (PS) conditioo on the spot:.C V(O.IJ. for p > 2.
Jd
4. Critical Group. and Mor•• 7\Ipe Number.
41
The theory can be extended as follows. An operator L E B(X), the Banach algebra of all bounded linear operators from a Banach space X into itself, is said to be hyperbolic, if the spectrum u(L) is contained in two compact subsets separated by the imag_ inary axis. Definition 4.3. Let M be a C' Finsler manifold,! E C'(M,IlI.I). Let Po be an isolated critical point. We say that Po is nondegenerate if there is a neighborhood U of Po on which T(M) is trivialized to a be U x X, such that there is a hyperbolic operator L E B(X) satisfying
(1) tP !!.Po)(Lx, 11) = tP !(po)(x, LII), V x, 11 E X, (2) tP !!.Po)(Lx, x) > 0 V x E X\{9}. (3) (df(P), Lx) > 0, It p E U, P = Po + x in the local coordinates. Tbe dimension of the negative invariant subspace of L is called the index ofp. According to the new 'definition (which coincides with the old if M is Hilbert-Riemannian by taking L = tP !(Po)-I), Theorem 4.4 does hold for Finsler manifolds modeled on Banach spaces with differentiahle norms. The reader is referred to Chang ICha2] and K. Uhlenbeck /Uhll]. A different statement was also given hy T. Tromha ITrol]. Remark 4.1. The above theorems as well as the corollary can be extended to functions defined on a manifold M with boundary 8M. The same proof works and the same conclusion bolds for functions! under the foUuwing assumptions: (1) Kn8M=0; (2) 3 a p.g.v.f. V of! such that - VI8M\I-I(o)
° Vx E direct inward (i.e., the negative pseudo-gradient Buw '1(t, x) E M, 8M\f-I(a), V t > 0 small). For a more general houndary condition, see Section 6.1. Now we are interested in defining a reparameterized flow in order to make the flow have in6nite arriving time if it passes throngh a critical point. Let p(-r) = -X(p(T» { p(O) = Xo E rl(a,b]\K•. Then p = p( T, xo) is a reparametrization of the p.g. flow u defined in Theorem 3.2. We shall prove Theorem 4.5. If! E C 2 - O(M,IlI.I), and jf u(Too - O,xo) = z E K o, lim p( T, xo) = z.
then p( +00, xo) =
t_+oc
42
Infinite Dimen.rioMl M"...e Theory
Proof. Let (4.4)
• = / 0p(T,XO).
We have
dB dT =-(df,X)Op(T,Xo)·
Suppose that of is the arriving time in the new parameter that of < +00; we ohtain
-
1.
4
T= lim
and lISSume
6
+ -dB dr
6-+0 /('0)
d.
d.
/('0)
=
T,
1
6~~0.+6
(d/,X)op(T,xol"
Let 4>(8) = (d/, X) 0 p(T), where. and T are related in (4.4). We have /('0)
1 •
d. 4>(') < 00.
From
4>(8) :5
IIX 0 p(T)II'
:5 4 IId/ 0 P (T,Xo) _ d/(z)II' :5 4C
lit
:5 4C ( [
p(T',XO) dTf
lip (T', xo)11 1:::1
< 4C (l'"XOP(T',xo),, 2
-.
l'
My dB' ,) (.-a)
14> (.')1
dB'
:5 4C' • 4>(.') (. - a) :5 M(. - a) for some constants, C, C' and M > 0, it follows
-=-
1 1/('0) ds .. ( ) :;;. M
/('0) d.
1 0+6.,,8
a+68a
1 = +M (In(f (xo) - a) -In.) ~ +00
as 6 ---. +0. This is a contradiction.
The new reparametri2ed ftow RI x (M\K).
pet, xo)
is well-defined for all (t, xo) E
5.
GromolI-M<JI 0 where c = 1(P).
Let U+ = U,~op(t, U), where p is the reparametrized How defined above. Then we have
C.(f,p)
~
H. (fe n U+, (fe\{P)) n U+)
'" H. (fe+.
n U+, (/e\{P}) n U+)
because Ie n U+ is a strong deformation retract of fe+. n U+ (Deformation Theorem). Since fe+. n U+ is path connected, and (fe\{P}) n U+ ¥ 0. :. Co(f,p) =
o.
As aD extension of Example 2, we have Example 4. If / E co-O(Mn, Rn), and if p is an isolated critical point of J, which is neither a local maximum nor a local minimum, then Co(f,p) = Cn(f,p) = O.
5. Gromall-Meyer Theory The contributions of Gromoll and Meyer to isolated critical point theory are threefold: (1) a splitting lemma, which is a generalization of the Morse lemma (cf. the previous section); (2) an alternative definition of critical groups; (3) a shifting theorem which reduces the critical groups of a degenerate isolated point to the critical groups of the function restricted to
Infinite Dim.nsion4l Mar•• Theury
its degenerate sub-manifold. In this section, we shall rewrite their theory with slight improvements and prove the equivalence of the two definitions of critical groups. Theorem 5.1. Suppa;e that U is a neighborhood 018 in a Hilbert space H and that I E C'(U,R'). Assume that 8 i. the only critical point of I and that A = 4'/(8) with kernel N. 110 i. either an isolated point 01 the spectrum utA) or not in u(A), then there exist a hall B., 6 > 0, centered at 9, an origin-preserving Jocal homeomorphism rf> defined on B., and a C' mapping h : B. n N _ N.L such that (5.1)
1
1 0 rf>(. + y) = 2(A.,.) + !(h(y) + II), 'I x E B.,
where y = PNX, subepace N.
Z
= PNl.X,
and PN is the orthogonal projection onto the
Proof. 1. Decomposing the space H into N $ N.L, we have d,f (9,
+ 82 )
=
8,
and
Because of the implicit function theorem, there is a function h : Bs n N N.L, 6 > 0, such that d,f(y + hey)) = 9,.
Let u = • - h(y), and let (5.2)
F(u, y) =
1(' + y) -/(h(y) + 1/). 1
F2(u) = 2(Au,u).
(5.3) Then we obtain
F(O',II)=O d"F (9" II) = d,/(h(lI) + II)
= 8"
tP.l (0,,0 2 ) = cf./(0) = AIN.L· 2. Define
e: (u, y) ..... un E Fi'
0
F(u, II) n {,,(t, u) I It I < lIull}, where"
is the 80w defined by tbe following ODE
. ,,(s) =
,,(0) = u.
A'l(s) IIA'1(s)1I
5.
Claim.
Gromoll-Meyer 7'heo'1l
'I is well-defined for ItI < lIuli.
45
Since
11'1(.) - ull :5
I~I,
we have 1I'I(t,ulll :::: Ilull-iti. From this, together with '1(t,u) E NJ., it follows that the denominator of the vector field is not zero for It I < lIuli. Claim. {is well-defined on Br x Bt, where Br = B. n N, and B. n NJ. for some 6 > O. In fact, the following inequalities bold: (a) V < > 0, 36, = 6«) > 0 sueb that
Bt
=
IF(u, y) - F.(u)1 = fF(u,y) - F (I", y) - (duF (/I"y) , u) - F.(u)! =
If.'
(1 - t)
((d~F(tu, y) - ~F (/1,,/1,» u, u) dtl
< 0 is a constant determined by the spectrum of A. We conclude that (c) F2 ('1(t, a», as a function of t, is strictly decreasing on (-lIull,lIull); (d) F2 ('1(-t,u)) > F(u,y) > F2 ('1(!,u» holds for (5.4)
(1 -)1 - ~) lIuli :5 t :5 lIuli.
Therefore, there exists a unique f( u, y) with (5.5)
-
It(u,y)l:5
(.CF\ V (j) lIuli 1-
1-
46
Infinite Dimensional Mor•• 'l'heoryJ
such that (5.6)
F2 (Ij (f(u,II),u)) = F(u,y).
Thus the function { is of the fonn:
B' {(u,lI) = {
Ij (f(U,II), u)
u=B, u f B,.
3. Define a map'" : (U,II) .... ({(u,Y),II). We shall verify that '" is a local homeomorphism. That f( u,y) is continuous easily follows from the implicit function theorem for u = Uo f 9, provided (5.7) and for u = 9, provided by (5.5). We have used the path Ij(t,u) to carry a point (u,y) to the point ({(u,II),Y); the same path can he used for the opposite purpose, i.e., to define the inverse map ~ = "'-'. The same reason is provided to verify the continuity of ~. Therefore ~ is a homeomorphism. Th.e equality (5.1) follows directly from (5.6). 0 Note. The function y .... I(y + h(y» is C 2 •
In the case where N = {9}. the Morse Lemma is a consequence of this theorem except the conclusion is that ~ is a diffeomorphism. Proof of the Morse Lemma. We have proved that ~ is a homeomorphism. Now we shall prove it is a dilfeomorphisUt. That f(u), and then {(u), is continuously differentiable for u E B.\{9}, follows from the implicit function theorem and (5.7). It is also easily obtained that d{(9) = id, by using
IIIj (f(u) , u) - ull ::; If(u)1 ::;
(5.8)
(1 - VI - ~)
lIuli
= o(lIull)
(i.e., Ij(f{u), u) = u+ o(lIull) as u .... 8). Hence 1Id{(u) - idll = 0(1) remains to he proved. Since Ij{t, u) = u we
'l 0
AIj(s,u) IIA!j(s, u)1I d.
for
It I < Ilull,
write Ij(t) = ,,(t, u), 'lu(t) = dulj{t, u) for simplicity,
(5.9)
'1.
(t) = ·d I
-1' 0
(AIj.(S) _ A,,(s) ® (AIj.(s»' A,,(.»)) d IIAIj(s)1I IIAIj(s)lIa s,
5.
that is 1I".(t)1I S 1 + C, where C,
(5.10)
>0
47
Gromoll-M"I/er T/oeoTJl
J.'
lI'1t.) II
II". (s) II dB,
is a constant depending on utA). But
Jl-
~lIuli $11'1(')11 S 211ull
as
I_I S li(u)l·
Applying the Gronwall inequality, 1I".(t)1I S
1+ .'-~. 11 $
C2 for 0 < It I $ feu).
Thus . If(ull 11'1_ (I(u),u) - ,dll S C3 = 0(1)
W
as
Ilull-+ o.
Since 1Id{(u) - idll =
II". (f(u), u) -
id
+ duf(u) . '1: (f(u) , u)1
and by (5.6) and (5.10), we have lid -( )11 - IlduF(u) -". (f(u), u) A'1 (f(u), u)1I _I u IIA'1 (f(u), ulll
= IIAu + o(lIull) -
(id + o(I»A(u + o(lIull))1I = 0(1) IIA(u + o(llull)}11 as u -+ (J.
This proves that 1Id{(u)- idll -+ 0 as lIull -+ O. The implicit function theorem is employed, and { is a local diffeomorphism. 0
Remark 5.1 There is no difficulty extending the Morse Lemma as well as Theorem 5.1 to the case where M is a Hilbert Riemannian manifold. The theorem was first obtained by Gromoll and Meyer (GrMl] under the condition / E C3. However, the C'-Morse Lemma was proved by N. H. Kuiper (Kuil], from whom the idea of the above proof was taken. The c"-Morse Lemma was given by Palais (pal2], Nirenberg (Nil] and Schwartz (Schl] (cf. Appendix). As to Theorem 5.1, for a different proof see J. Mawhin and M. Willem (MaW2] based on (Caml], in which the condition that 0 is either isolated or not in u(d' /(0)) is replaced by one in which d' /(O) is a Fredholm operator with index O. However ,our condition seems weaker, because a self-adjoint Fredholm operator A must have a finitely dimensional kernel N(A) and a closed range R(A). In the case where N(A) -F {O}, the induced operator
A: H/N(A) -+ R(A),
48
Infinite DimenaionDl Mor5f! Theory
defined by A[x) = Ax
V x E [x), [x) E H/N(A),
is invertible. Because of the Banach theorem, A-I i. bounded, which implies that 0 is isolated in (7(A). On the other hand, the condition dim N(A) < 00 is not assumed in Theorem 5.1. Let M be a (j2 Finsler manifold, and let I E C l (M, Ill") be a function satisfying the (PS) condition. Suppose that V : M\K -> T(M) is a p.g.v.f. of I. Definition 5.1. Let p be an isolated critical point of I, c = 1(P). A pair of topological spaces (W, W_) is called a Gromoll-Meyer pair with respect to V, if (1) W is a closed neighborhood of p possessing the mean value property, i.e., V tl < to, '1(t;) E W, i = 1,2, implies '1(t) E W for all t E [tl' to), where 'I( t) is the decreasing flow with respect to V. And there exists. > 0 sucb that W n Ic-. = l-l[C - .,c) n K = 0, W n K = {p};
(2) W_:= {x E W I 'I(t,x) ¢ W Vt > O} is closed in W; (3) W_ is a piecewise submanifold, and the flow '1 is transversal to W_, i.e., '1m W_.
At this point, the existence of a Gromoll-Meyer pair is assumed. The following theorem claims the motivation of the definition. Theorem 5.2. Let (W, W _) be a Gromoll-Meyer pair with respect to a p.g.v.f. V of an isolated critical point p of the function I E Cl(M,JR1) satisfying the (P, S) condition. Then we have H.(W, W_;G) '" C.(!,p).
Proof. We now introduce two sets U+ ~ Uo 0 'I x E B,/ B'I"
o
50
Due to the (PS) condition, there exists f3 = inf.EBsIBS/2IW(x)1I > O. Let m = SUP'EB. II.q(x)ll. ~, p and 7 ore then determined consecutively: A>
~, 0 < 7 < min{.,
*},
and
s: + A7 OJ. Obviously we see W_ C W-. Now we prove W- C W_. By definition, W- CoW and oW = W_ u U- 1C"Y) nO~) u (g-1(1') n (W\W_». If x E 1- 1(7) nO., then x ¢ W-. For x E g-1(1') n (W\W_), according to (5.14) and (5.15), we have (g 0 11)'(0, x) < 0 and I(x) > -7. These imply that 3 T > 0 such that go ,,(T, x) (U n N) the characteristic submanifold of M for I at p with respect to the parametrization 4». The following theorem sets up the relationship between the critical groups of I and those of i := I IN' Theorem 5.4. (Shifting theorem). Assume that the Morse index of I at p is j, then we have
Cq(f,p) = Cq-i (i,p) , q = 0, 1, ....
5.
Gromoll-Mever Th""'ll
51
First we need: Lemma 5.1. Suppose that H = H. E!l H" g, E C'(H"R'), IJ, is an isolated critical point of g" i = 1,2, where H., H, are Hilbert spaces. Assume that (W" W,_) is a Gromoll-Meyer pair of IJ, with respect to the gradient vector field ofg, , i = 1,2; then (W.XW2, (W._ xW2)u(W. x W._)) is a Gromoll-Meyer pair of the function 1 = g. + g. at 9 = 9. + 9, with respect to the gradient vector field dI, if II is an isolated critical point of I. This is easy to check. We omit the proof. Theorem 5.5. Under the assumptions of Lemma 5.1, we have
Proor. This is a combination of Theorem 5.2, Lemma 5.1 and the Kiinnetb formula. Proor or Theorem 5.4. This is a combination of Theorems 5.1, 5.5. Remark 5.2. Theorem 5.5 was conjectured by Gromoll and Meyer IGrM1) and was solved by G. Tian ITis1). In IDan1), Dancer independently proved the conjecture in the finite dimensional case. In order to verify that the pair «W._ x W2), (W. x W,_)) is an excision couple. we use the notations: U+ 1 U+, V+ 1 V+ 1 defined as in Theorem 5.2 where U+ = LIt;:;:O!).(t, W.), lh = U'>O!).(t, W._), V+ = U,;:;:O'12(t, W2 ), V+ = u.>o'12(T, W,_), and !).. '12 are the decreasing Hows with respect to dg. and dUo respectively. Also U., {j > 0, stand for the same meaning. Thus
v.,
H. ((W._
x W2) u (W.
W,_), w.
£!! H.
((U+
£!! H.
(Us X V+,U. x V+)
£!! H.
(W._
x
X
V+)
X
U
(U+
X
X
W,_)
V+) ,U+
x
V+)
(excision)
W" W,_ x W2-).
Corollary 5.1. Suppose that N is linite dimensional with dimension k and pis (1) a local minimum of j, then
(2) a local maximum of j, then
52 (3) neither a Jocal maximum nor a Joca1 minimum of
C.(f,p) = 0 (or q 5 j,
and
q~ j
i, then
+ k.
Finally, we prove the homotOPY invariance of the critical groups. It is similar to the Leray Schauder index for isolated zeroes of vector fields. The perturbation should preserve the isolatedness of the critical points.
Lemma 5.2. Let (W, W _) be" GmmoJJ·Meyer pajr of an iaoJated critical point p of a 0>-0 function I, defined on a Hilbert space H, satisfying the (PS) condition, with respect to -41(x). Then there exists • > 0 such tbat (W, W _) is also a Gromoll·Meyer pajr of9 satisfying the (PS) condition with respect to certain p.g. v.1. of 9, provided that 9 has a unique critical point q in W, and III - gllct(w) < •. o
Proo£ 3 r > 0 such that B(p. r) c W. Due to (PS) {3 = inf {W(x)1I1 x E W\B
(p,~)} > O.
Define p E C·-o(H,IR'). satisfying
pIx) =
{Io xx ¢EB(p,r). B (P.~) •
with 0 5 pIx) :S 1. and a vector field
3 Vex) = 2 ip(x)dg(x)
+ (1 - p(x»df(x»).
Choosing 0 < • < ~. we obtain
IIV(x)1I :S 2I1dg(x)ll. and for
(V(x). dg(x» ~ IIdg(x)11 2
IIg -
Vx E
W.
IlIc'(w) < •.
Claim. V x '" B(p.~) IIdg(x)1I ~ IId/(x)1I - • ~ {3 - . > 3 .. We have
(V(x). dg(x» ~
3
2 [Il dg(x)1I 2 -
~~
2 [lldg(x)1I -
3 IIV(x)1I :S 2(lIdg(x)1I
'lIdg(x)lI]
~lIdg(x)1I2] =
+.) 5 2I1dg(x)lI.
IIdg(x)II'
5.
Since V x E B(P. ~). VeX)
Gromoll-Meyer Tha>ry
53
=dg(x). the verification is trivial. o
Notice that Vex) = -dJ(x) outside a ball B(P. r) C W. and q is the only critical point of 9 in W. It is not difficult to verify that (W. W _) is a GM pair of 9 with respect to V. (The mean value property holds because the Bow is the same negative gradient Bow of I outside B(P. r). particularly in a neighborhood of 8W. Similarly. for other properties assumed on W_.) Theorem 5.6. Suppose that {/. E C'(H.JRI) 117 E [0. I]} is a family of functions satisfying the (PS) condition. Suppose that there exists an open set N such that I. has a unique critical point P. in N, Vue [0.1), and that u _ I. is continuous in CI (N) topology. Then
c. (f.,P.)
is independent 01 u.
Proof. Due to the (PS) condition. u - P. is continuous. Applying Theorem 5.3, we may construct a GM pair (W•• W._) for P., V (f E 10.1)' Combining Theorem 5.2, Lemma 5.2. and the finite covering, we obtain our conclusion directly. Theorem 5.7. (Marino Prodi IMaP2)) Suppose that I E C'(M.JRI) has a critical value c with Kc {Ph". .... ,pt}. Assume that cP 1(P,) are Fredholm operators, i = 1,2, ... ,t. Then V £ > 0, there exists a function 9 E a'(M.JRI) such that
=
0-1
(1) 9 = I in M\ B(Pj, 0
VxEM,
VxEE.
Proof. The proof is an extension of the standard proof of the existence of a pseudo gradient vector field for manifolds without boundaries. We shall pay more attention to the construction of such a vector field near the boundary E. First, we note that 3 d :> 0, such that IIi'(x)II ~ 2d V x E E--. Claim. If not, 3 Xn E E-- such that i'(x n ) - O. According to the (PS) condition for the function j, 3 x· E B-, with l(x·) = 9, Le., x· E K-; this is a contradiction.
Next, let {Uo I Q E A} bean open covering of M' satisfyingT(M) luo £; Uo x H, V Q E A. If either x E int(M'), or x E E+ := {x EEl (J'(x) , n(x» > OJ, then we choose V. = /,(x)/II/,(x)II E T.(M) '" H, and a neighborhood D. C Uo of x, for some Q, such that (V.. /,(y)) > 1/2 II/'(Y)II V Y E 0%, and
0.nE=0 (n(y), V.) > 0
VyEO.nE
if x E int (M'), ifxE E+.
In the remaining case, i.e., x E B-, one ma.y choose W;t E TAEL i.e., (n(x), W.) = 0, satisfying IIW.II = 1, and (1V.,1'(x» > d, provided by
6. Ertensio7U of Morse Theory
57
11i'(x)1I ~ 2d.
After a small perturbation, we bave V. e T.(M), satisfying IIV.II = 1, ( V.. /'(x)) > d, and (n(x), V.) > O. Again, we have a small neighborhood O. C UQ for some 0, such that (V.. f'(y» > d
(n(y) , V.)
>0
"fly E Oz, and
'IV eO. c E.
Finally, to the open covering {O. I x eM}, we have a locally linite relinement covering {c~ I {3 e B}, and an associated C l - O partition of unity {4>" 1{3 e B}, i.e., supp 4>13 C C", 0 ~ 4>13 ~ 1, and EpeB 4>,,(x) = 1, V x e M. For each (3, 3 x = x({3) e M· such that supp 4>13 COo . Let us define Xix) = EpeB 4>p(x)V.(p). This is just wbat we need. In fact, (1) is trivial. By the definition, (X(x).J'(x)) =
E
4>p(x)(V.(p).J'(x)) PeB ~ Min {d, 1/211/,(x)lI}
"fIxEM-,
=
(2) follows. And (n(x),X(x» E"eB(n(x), V.(p) . 4>p(x) > 0, V x e E, because V x e E, x e Supp "'13 C Oo(~) implies x({3) e E. 0 In the case K- = 0, both the Morse inequalities and the Morse handle body theorem hold under the following assumption:
feCI (M,IltI) (6.1)
~d both the functions J and , J satisfy the (PS) conditions
on M and E respectively. Now, we are going to reduce the GBC problem to case K- = 0, by perturbations. Assuming that {V} e K-, we choose a chart (U,4», 4>: U ~ H_ := {{ e II I ({, e) ~ O} for some fixed e e H\{O}, satisfying K n U = 0, K- nu = {v}, 4>(Y) = 0, n(y) = e, and En U = 4>-1 (4)(U) n lID), where Ho = span {e}.l. In other words, if we write 4>(x) = te+ z, then t ~ 0, and t = 0 if and only if x e E. Define (6.2)
w
= J 0 4>-I(te + z) -
J 0 4>-I(Z)
then
/' 0 4>-1 (9) . (,p-I), (8) =
(I' (V), n(y» < 0,
bccause i'(y) = 0, and f has no critical point on E, so /'(y) F O. According to the Implicit Function Theorem, 3 T,f > 0 and 8 C 1 function t = t(w, z), which solves (6.2) in (-T,OI x (B(O,.) n Hu) c 4>(U), with t(O,z) = o. Now, (w,z) is regarded to be" new local coordinate of x = ",(w,z) =
58
Infinite
Di~
MON. Theo<J/
-'(t(w,z)e+z), for (w,z) E [O,T,} x (B(9, f) nHo), where T, on T. Let us define pew) = { where 0 < 5
0 depends
~5 _ w)3 f6'
T, is to be detennined later; and let {~
xes) = where X E C~, 0
=" X ="
«(x) = pew) . xUzJ)
I, and for
s
~ f,
f/2 ~. ~ 0,
Ix'i =" 3/f.
We define
x E U, := .p ([0, T,) x (B(9, f) n Ho»
and equal to zero elsewhere. Then we define
f,(x) = f(x)
+ «(x).
Lemma 6.2. SUPpo6e the assumption (6.1), K n U = 0, and K- nU = {Y}. Then we have y. E U, and the foJ/owing: (1) K" the critical set of I., equals K U {y.}, (K.)-, the set K- for the function It. equals K- \Iv}, (2) I. and po6SeSS the (PS) condition; (3) C.(f"V·) = C.(f, 11).
i,
wherever 5 > 0 smw/.
i,
Proof. (1) Both functions I. and bave no critical point in f/2 =" Izl In fact, provided by the (PS) condition, 3 d > 0 such that 1Ii'(x)11 dV x E U nE, with f/2 =" Izl =" f. Let g(z) = fo-'(z); then f.
IIg'(z}1I > d
for
f/2
=" >
=" Izl =" f.
Thus,
IIll,f,(xlil =
11g'(z) + p(w)x'(lzl)z/lzllI ;:>: d - 36/f > d/2
V (w,z) E [O,5} x [(B(9,f)\B(II,f/2»n H o), if we choose 5 < 0 = (1
thus. II '" (Ktl-· (5) Since" is of separate variables in a neighborhood of II'. and a minimum of w + p(w) we have
c. (f,. y')
wo
is
= C. (i. II) ® C,(id + P. wo)
=c.{i.II). o
provided by Theorem 5.5. The lemma is proved.
Proof of Theorem 6.1. According to the isolatedness of the critical points of I and and the (PS) conditions. we conclude that both I and have only finitely many critical points. say. {Z'.Z2 ..... z.} and {y" 1/2 ..... II,} respectively. Now we apply Lemma 6.2 I times. and the new function" satisfies the condition (K,)- = 0. Both possess the (PS) condition, and It has the critical set {Zl,Z21'" ,zk,yi,yi.· .. ,y:}.
i.
i
",i,
The Morse inequalities are applied for It. Since a. b are regular values. 3 0 > O. such that
Kn/-'Ib.b+ 2aJ
= 0.
a- a
K n I-'Ia If we choose 0 so small that
< 1('" (10.6J
x (B(8. 0
such that x
+ fV E C},
then
T.(C) = T.(C), \I x E C Claim. T.(C) c T.(C) is trivial, and since T.(C) is closed, T.(C) c T.(C). It remains to verify that \I v E T.(C) 3 fn > 0, Vn E X such that x + enVn E C, and Vn --+ V. By Definition 6.3, we have Yn E C, h n
! 0 such that
h;;' IIx + hnv - Ynll ~ O. Set
Vn
= h;;l(Yn -
x), and En = hr., n = 1,2, ... i these are what we need.
Lemma 6.3. Assume that S is a locally convex closed set with respect to (M,A). Then T.(S) is a closed convex cone in T.M \I XES.
Claim. In fact, if (V,4» is a chart in A at x,4>: U ~ X, then
According to (6.4), T~.)(4)(U n S)) is convex, and since 4>'(x) is linear, the convexity of the cone T.(S) follows from (6.4). Therefore T.(S) is convex. By Definition 6.3, if v E T.(S), then tv E T.(S), so that T.(S) is A
r-nnp
62
Infinite Dimeruional Mor.. Theory
Definition 6.4. Let S be a locally convex closed set with respect to (M,A), and let I E C'(O, lit') where is an open neighborhood of S. We say that Xo E S is a critical point of I with respect to S if
°
If v E T.o(S),
where ( , ) is the duality between T;oM and T'oM. Or, equivalently, we say that Xo is critical with respect to S if (df(xo), 4>'(xo)-' (y - 4> (xo))) ;? 0 Ify E 4>(u n S).
Ix·l.o
= Sup{(x·,v) I v E T.o(S)
with
114>'(xo)vll x :O; 1}.
Therefore: Xo E S
is a critical point of
if and only if
'.
I
with respect to
S
1- dl (xo) 1'0 = o.
The Palais-Smale condition (PS), with respect to S is extended as follows: Any sequence {Xn} C S, along which (xn) ~ e and I - df (xn) I.n ~ 0, implies a convergent subsequence.
I
Applying the same argument employed in Theorem 3.3, we have
Lemma 6.4. Suppose that / E C'(O,IIt') satisfies (PS), with respect to a locally convex closed set S. Then the critical set K, = Kn/-' (e) with respect to S is compact, and for any closed neighborhood N (if K, = 0, then N = 0) of Kco 3 constants b, < > 0 such that
1- dl(x)!.
;? b,
If x E
snr'lc -
~I- 0 and T(t) satisfying f(t) = V(T(t» { T(O) = x E B;
if and only if
V t E 10; 6) T(t) E B
lim h-1d(x + hV(x), B) hID
i.e.• V(x) E T.(B)
=0
V x E B.
Lemma 6.5. Let S be 8 closed subset of M. Then S i, a semi-invariant set with respect to a locally Lipscbitzian v.f. V on M. if and only if (6.5)
lim h-1d (4)(x) hID
+ h4>'(x) . V(x); 4>(U n S» =
0
V XES V chart (4). U). Proof. Assume that (6.5) holds. The foJlowing ODE in X
y(t) = 4>'(x)V(x)I.=.-I(.(.» { 11(0) = 4>(x) has a local solution II•. Since 4>(UnS) is a closed set in 4>(U) and (4)'. V)o4>-1 is tangent to 4>(U n S). by the Brezis-Martin Theorem. (U n S) is locally invariant with respect to 4>' 0 V. Setting x.(t) = 4>-1 (1I.(t». it foUows that x.(t) E unS. It is not difficult to verify that does not depend on the special choice of 4> in a neighborhood of t = O. and it is easy to extend the solution to a maximal solution which remains in S via the method of continuation. Conversely, if S is a semi·invariant set with respect to V, then for any chart (4).U). 4>(UnS) is semi-invariant with respect to (4)'. V)o4>-I. (6.5) foJlows directly from the Brezis-Martin Theorem.
x.
3. Defonnation theorems and critical groups With the above preparatory work, the first and second deformation theorems do hold for any 10caJly convex closed subset S of a c" paracompact Banach manifold. The proofs are just the same as in Section 3 and hence are omitted. Similarly, the critical groups for isolated critical points with respect to S are weJl-defined.
7. Equivariant Morse Theory
65
4. Morse relations lor functions with isolated critiooi points Let S be a locally convex closed subset S of a (fl paracompact Banach manifold. Let I E CI(O,RI), where 0 is a neighborhood of S. Assume that I satisfies the (PS) condition with respect to S and has only finitely many critical points {XI,X" ... ,xn } with respect to S in 1-I[a,b[nS, with critical value {Cl,~t ... ,em} satisfying a < Cj < b, i = 1,2, ... ,m. Relative to any coefficient field for homology, set m
M.
= M.(/; [a,b)) = L rank C.(/,Xj) , j=1
and
n. = n.(/; [a, b)) = rank H. (/r./.) , We have
~
q
= 0,1,2, ....
~
L M,t' L R,t' + (1 + t)Q(t), =
q=O
q=O
where Q(t) is a formal series with nonnegative coefficients. Remark 6.2. Readers might be puzzled about the different conditions in the two subsections. The functions in the first subsection are under strong boundary conditions, but in the second subsection there are no restrictions.
The underlying spaces in the first are manifolds with smooth boundaries, but in the second are locally convex closed sets. After all, we have the same conclusion, i.e., the Morse inequalities.
The point is that the meanings of the critical points in Sections 6.1 and 6.2 are different. In Section 6.1 they correspond to the variational equation, i.e., the Euler-Lagrange equation, while in Section 6.2 they correspond to variational inequality.
Remark 6.3. Morse theory on convex sets was initially studied by M. Strowe [Strl] and K. C. Chang, J. Eells [ChEll in the Plateau problem. It was developed in K. C. Chang [Cha7] to study the variational inequality problems. The introduction of local convexity in critical point theory first appeared in T. Q. Wang (WaTl).
7. Equivarlant Morse Theory Let us assume a compact Lie group G and a smooth manifold on which the group G acts. Equivariant Morse theory studies the Morse relations and the Morse handle body theorem for G-invariant functions. Noticing that for a G~invariant function, if x is a critical point, then the points on the G orbit containing x are also critical points. It is hard to say regarding the
66
Infinite
Di~
Morse Theory
isolatedness of critical points if G is continuous. In this section, we shall introduce the notions of isolatedness and nondegeneracy for critical orbits.
7.1. Preliminaries Let G be a compact Lie group. A G space (or manifold) M is a topological space (or manifold resp.) with a continuous G action, i.e., '" : G x M _ M with ",(g, x), written as 9 . x, sucb that e . x = x, and (g, . 92) . x = 9"(92'X)"'9,,92 EG, "'xEM. V x E M, we call O(x) = {g . x 1 9 E G} a G orbit. The set of all G orbits is called the orbit space. Endowed with the quotient topology, it is denoted by MIG or simply M. The subgroup of G, defined by G. {g E Gig· x = x}, is called the isotropy subgroup at x. If x has the isotropy subgroup H, then 9 . x has the isotropy subgroup 9 . H . g-1. Thus a conjugacy class of isotropy subgroups is attached to
=
each orbit.
G is called a free action if G. = e V x EM. Let H he a subgroup of G, and M be a G space, we denote MH = {x E M
1 h·
x = x V h E H},
i.e., the set of points fixed under H. MG is called the fixed point set of G, whicb is also denoted by Fixe. A set A eM is called G-invariant, if g. x E A, '" x E A, '" 9 E G. A G pair (X, Y) is a pair of G-invariant spaces (X, Y) with Y C X. A function I : M - JR1 is called G-invariant, if I(g· x) = I(x);'" x E M,"'gEG. A map F: (X, Y) _ (X', Y') between two G pairs is called G equivariant, if F(g. x) = g. F(x) V x E X, V 9 E G. Thus,
8
G equivariant map F induces
8
map
F= FIG: (X,V) -
(X,Y').
Let 1f : E _ B he a fiber bundle. (E, 1f, B) is called a G bundle, if'" 9 E G, 9 : E _ E is a differentiable bundle map, such that gE. = E g .• V x E B. Thus, if M is a G-manifold, the tangent bundle T M is a G bundle, with g. X = d.",(g,x) . (X), V X E T.(M), V x E M. (E, 1f, B) is called a Riemannian G vector bundle, if the G vector bundle possesses a Riemannian metric, and the G-action is isometric. In the following, we always assume that M is a Hilbert Riemannian manifold, with a Riemann metric on T M. Let E c M be a compact connected submanifold; then TE, the tangent bundle of E, is a subbundle of T M , and then the normal bundle NE, which is the orthogonal complement to TE, is also a subbundle of T M.
7. Equi1l4ri4nt Morse Theory
67
If in addition M is a G manifold and E in G invariant, then both TE and NE are all G-bundles. Let I E Cl(M,Btl) be a G invariant. It gives rise to a G-equivariant gradient vector field df (df(g. x),g' X)
= (df(x),X)
\I (g, x) E G x M, If X E T.(M), i.e.,
g'. dl· 9 = df. Since the action 9 on T.(M) is unitary, dg' = g-l, we obtain
dl·g=g·df. Analogously, the Hessian d'1 is also G-equivariant, if I E C" It is obviously seen that Ihe level sets I .. I-l(c), and the critical sets K, Kc = Knrl(c) are all G-invariant. And a critical orbit 0 = O(x) is a G submanifold of M. It follows that T.(O) !; ker d'1(x) and that the induced bounded selfadjoint operator d'1(x) : N.(O) ..... N.(O), satisfies
g' . d 2 /(g· x) . 9
= d 2 /(x).
'7.2. Equlvariant Deformation For a G-invariant function, we shall improve the first and second deformation theorems to make the deformations equivariant. Namely, we shall prove
Theorem '7.1. (First G-equivariant deformation theorem) Let M be a C"G-Hilbert ruemannian manifold. Suppose that I E Cl(M,1lI.l) is Ginvariant and satisfies the (PS)c condition. Assume that N is a G-invariant clOBed neighborhood of K c for some c E 1lI. l . Then there exist constants 1" > f > and a G-equivariant continuous map 'I : (0, 1J x M ~ M such that
°
(1) '1(t,·) Icrllc-"c+~ = id Icrllc-"c+~ (2) .,(0,,) = id, (3) .,(I,/c+< \N) C Ic-, (4) If t E (O,IJ, '1(t,·) : M ..... M is a G-homeomorphism, (5) t"'" I 0 '1(t, x) is nonincrcasing \I t.
Infinite DimemionaI M.,..' Theory
68
TheOrem 7.2. (Second G-equivari8l1t deformation theorem). Let M be 8 C>G-Hilbert Riemannian manifold. Suppose that f E C1(M,1II. 1) i. G invarUJnt and 88tisfies the (PS)c condition, VeE [a, bl. Assume that a is the only critical value of f in [a, b), and that any connected component of K. is always a part of a certain critical orbit. Then f. i. a G-equivariant strong deformation retract of f. \K•. The proofs of these theorems are just modifications of those given in Sec· tion 3. A new ingredient is to construct a G-equivariJJnt locally Lipschitzian p.g.v.f. We carry this out as follows:
Lemma 7.1. Assume that f E C'(M, 111. 1 ) is G-invari8l1t. Then there exists a CI-oG-equivariaut p.g. v.£. Proof. We have already a CI-O p.g.v.f. X(x) from Theorem 3.1. Let us define
X(x)
=
fa
g-I . X(g· x) d",
where dl' is the right invariant Haar measure on G. Since
x (g' . x) = Lg-I X (g . g' . x) d" =
9'
L
(g. g,)-I X (g. g' . x) d"
= g'X(x) V g' E G,
X is G equivariant.
We shall prove X E CI-O. It is only required to prove that V orbits G(xo), 3 a neighborhood U, such that X is uniformly Lipschitzian on it. Actually, V x E G(xo) 3 an open ball B(x,o.) and a constant C. > 0 such that IIX(y) - X(z)1I ~ C.d(y, z) V y, z E B(x, 0.) where 11·11 is the Finsler structure on TM. Since G(xo) is compact we find a finite covering U = Ur~lB(x"o.,) ~ G(xo). Let
6= Min {6Zi,15i.$n}, C=
Max {C."
and
~ !~e IIX(y)iJ.! ~ i ~ n} .
We conclude
IIX(y) - X(z)1I ~ Cd(y,z)
Vy,z E U.
7. Equiooriant MM"Se Theory
In fact, if dey, z)
~
69
6/2, then IIX(y) - X(z)11 $ IIX(y)1I + IIX(z)1I $ 2 sup IIX(y)1I .EU
6
2 .C $
$
Cd(y, z).
Otherwise, d(y, z) < 6/2, y and z must fall into a certain neighborhood B(x., 6.,), and hence IIX(y) - X(z)1I $ C.,d(y,z) $ Cd(y,z). Finally, we prove that
X is again a p.g.v.f.
fa fa = fa
Since 9 is isometric
IIX(g· x)1I dl'
IIK(x)1I $
$ 2
IIdf(g· x) II dl'
2
IIg· df(x) II dl'
= 2I1d/(x)II,
and
fa = fa ~ fa
(df(x),K(x» =
(df(X),g-IX(g· x» dl' (d/(g· x),X(g· x» dl' IId/(g· x)II' dl'
= IIdf(x)II'· By virtue of Lemma 7.1, the flow defined by the G-equivariant p.g. v.f.
K
must be G-equivariant. (The uniqueness of the solution of ODE). The rest of the proofs of Theorems 7.1 and 7.2 is exactly the same as in Section 3. 7.3. The Splitting Theorem and Handle Body Theorem for Critical Manifolds
Definition 7.1. A connected submanifold 0 C M is called a critical manifold, if d/(x) = e II x E 0 (may not have group action) and flo = const. A critical manifold 0 C M is called isolated if there is a neighborhood U of 0 such that un K = O.
70
Infinite Dimen.oionol M ..... Theory
In the following, we always assume that 0 is compact. Now we shall study the local behavior of the function I near its isolated critical manifold. The restriction of the tangent bundle T M on 0 now is split into a direct sum TO Ell NO, the tangent bundle and the normal bundle of the manifold O. The Riemannian metric on T M reduces a metric on the normal bundle NO. The exponential map exp, regarding tbe geodesic sprays on M, yields a diffeomorphism from the normal disk bundle NO(f) = {(x, v) E NOIliall :s f} to the tuhular neighborhood N(f) = {x E MI dist (x, 0) :s f} of 0, for some f > 0 because 0 is compact. In this sense, we shall not distinguish NO(f) from N(f). In order to split the normal bundle NO into mutually orthogonal bundles, we need
Lemma 7.2. Let H be a Hilbert space and Q, R be two orthogonal projections. IfllQ - RII < I, then QR: RH _ QH is an isomorphism. Proof. QR is an injection. Otherwise, 3 v E RH\{9} with Qv = O. Therefore II(Q - R)vD = IIvll, which contradicts IIQ - RII < 1. QR is also a surjection. Otherwise, 3 v E QHn(RH)J.\{9}, then (v, QRw) = (Rv, w) = oVw e H, which implies Rv = 9. Again, we get IIQ - R)vll = IIvll; this is impossible.
Lemma 7.3. Let { = (E,,,,N) be a Hilbert vector hundle on a connected manifold N, with a Riemannian metric ( , ). Let P be an orthogonal bundle projection with respect to ( , ). Let I be a section satisfying I. = id in C(E.. E.) It x E N. Assume Q = 1- P. Then P{ = (PE, "lpE, N) and Q{ = (QE, "IOE, N) are Hilbert vector bundles with {= P{EIlQ{.
Proof. It suffices to prove that It x E N 3 a bundle chart", : U x H ,,-'(U), x E U, and an orthogonal projection P E C(H,H) such that It y E U, "'.: H - E., ",.(v) = "'(v, v) maps PH, QH isometrically onto PEr and QE. resp. where H is a Hilbert space, and Q= I - P. From the definition, we have a bundle chart 4>: UxH _ "-'(U), x E U, and then define p. = 4>;' P4>. It y E U. For small U,
liP. - p.1I < 1 'ty E U.
By Lemma 7.2
p.p. : P.H - P.H
is an isomorphism. Similarly, set Q. =
4>;'Q4>., we have
Q.Q. : Q." _ Q.H is an isomorphism.
7. Equi'lJ4ricJnt Morae Theory
~. ........
E.
~,
H
---
and h are all G-equiWlIiant.
o is an isolated critical orbit of I.
Proof. Since
d" I(g· x)
= g-'d" I(x)g
V x E M,
the spectrum u(d" l(x)I{.) is G-invariant V x E 0, so Assumption (7.1) holds. It remwns to verify G equivariance. Since z = h.(y) solves the equation uniquely:
d,1 0 exp.(y + z) =
9,
we conclude that gh.(y) = h•.• (gy), provided hy the equivariance of dl. Similarly, the map 1Jx solves the equation uniquely:
1
2 (A(x)1).( u, Y),1).( u, v)) = f oexp. (u + h.(y) + y) - I oexp. (h.(y)
+ y),
where Y E {~(f), U E ({t ${;)(f), A(x) = d" I(z), and 1).(u, y) E ({t .f(:r;) + dist (x, 0)'. In particular, if f is G-invariant, 0 is an isolated critical orbit; then the Gromoll-Meyer pair (W, W _) may be chosen G-invariant, and the flow in tbe definition is G-equivariant. 7.4_ G Cohomology and G Critical Groups Tbe extension of nondegeneracy of critical manifolds has tbe following advantage: Suppose that E ~ M is a fibring, and ! is a nondegenerate function on M in the sense of Definition 7.2. Then it is easy to see that tbe pull back ... f on E is again nondegenerate. Further I the index of 0 as a nondegenerate criiical manifold of M equals the index of ,,-'0 as a critical manifold of E, i.e., ind (j,0) = ind (". f, ,,-'0). Let us consider a compact Lie group action G. I£ the G action is free, then the orhit space M MIG is also a manifold, the projection
=
,,:M-MIG is a smooth fibration, with fiber G, and there is not any difficulty in carrying
out the Morse theory on the orbit space M. However, if the action is not free, then M possesses singularities, and one cannot do the same as above. Consider any smooth principal G bundle E over a base manifold B, and the following diagram: E
1, EIG
-• "
ExM
1-
EXGM
-" "
M
1,
MIG
of the G-actions on M and E, and G operates diagonally on Ex M, i.e., g(O',x) = (90',9X) V (0', x) E Ex M, where E XG M = (E x M)/G. Since the action on E is free, this diagonal action is also free. On the other band, a G-invariant f on M clearly lifts to a G-invariant f on Ex M, and hence to a smooth function IE on E XG M. Now E XG M is itself a fiber space over the base B = EIG with fiber M and structure group G. Noticing that
7. EquitJoriant Morse TheDr'JI
for any critical manifold 0 of manifold of /E. We have
I,
75
E x a 0 is the corresponding critical
ind (f,O) = ind (fE, E Xa 0). There are many principal G-bundles E we may choose; among them we single out a universal G-bundle that is unique up to homotopy which p0ssesses the following important property: The total space E is contractible. Such bundles always exist cf. Housmoller [Houl]. We are satis6ed ourselves to give the following few examples: E=Ea=
G=
B=Ba=
Z.
SOO
SI
S"" S""
SU(2)
IIlp oo cpoo
real complex
projective
JIlIP'" quatemion spaoe
=
In the universal case, we shall write Ma E Xa M, Ba = EIG, Ea = E, and la = /E. Ma is called the homotopy quotient of M by G. The advantage of this choice is that the map 1I: M a-MIG
is a homotopy equivalence, if G acts freely on M. In summary, we consider the function la on Ma, the homotopy quotient, which is G-free, instead of Ion MIG. However, there is an (equivalently) alternative way to consider the problem, that is the concepts of G cohomology.
Definition 7.4. Given a G pair (X, Y) and a coefficient field OC, let He(X,Y;OC) = H' (Xa,Ya , IK), where Xa and Ya are the homotopy quotients of X and Y by the group G. We call He the G cohomology. It was proved by A. Borel that the G oohomology enjoys m06t of the properties of the cohomology. More precisely, the exactness, the homotopy, and the excision axioms hold, but not the dimension axiom:
He(Pt) = H' (Ba). FUrthermore if F: (X, Y) _ (X', Y') is a G-equivariant map, then F xl: (X, Y) x Ea - (X', Y') x Ea induces a G-equivariant map on the homotopy quotient: Fa: (Xa, Ya) - (Xc, Y
a),
and hence a homomorphism: Fa : He (X', Y') - He(X, Y)
76
Infinite Dimensional Morse Theory
cf. T. Tom Dieck [Die1). Now we are able to define the G-critical groups, and present some com· putations. At the end, the Morse inequalities are established. Definition 7.5. Suppose that U is a G-invariant neighborhood of an 0, where c 1(0). isolated critical orbit 0, such that K n (Ie n U) Then for any coefficient ring 1K,
=
Ca(l, 0) = Ha (Ie
=
n U, (leo \0) n U,IK)
=
qth G critical group of 0, q 0,1,2, .... And by the same proof as Theorem 5.2, we have
is called the
(7.4)
Ca(l,O) = Ha(W, W_;IK)
where (W, W_) is a G-Gromoll-Meyer pair of O. Example 1. Suppose that 0 is an isolated critical orbit, corresponding to a local minimum of a G·invariant function f. Then
(7.5) "
Ca(l,O)
= Ha(O) = W(Oo).
Example 2. Suppose that the normal bundle { = NO is a trivial bundle. Then we have the following formula:
(7.6)
~(I,O) =
• El)8l,-;(f, 0) ® Hb(O) ;=0
where (:Lie group of G. By the assumption, we can modify the function lis to " function gl5 such that gls satisfies (1) - (4) with M Sand G G,. Then we apply Lemma 7.5, and the function gls can be extended to = NO(f) and then to M as a G-invansnt function 9 with only nondegenerate critical orbits, Le., 9 satisfies (1)-(4). If x E FiXe, then 0 = {x}, G, = G, and NO«) is an Max{f(x) I XED}.
Proof. Since D c F•• the class [aD] which includes the map idl8D is trivial in where k = dimD. On the other hand. aD c f.; but
".-,U.).
Max.EDf 0 ",,(x)
>a
V""E C(D.M)
provided that ",,(D) n S # 0. This means ",,(D) is not included in [aD] is nontrivial in ".-,U.).
f.; hence
We observe the exact sequence
where i :
." -. ". U.,J.) ~ ".-,U.) ~ ".-,(/.) -. ... fa - /b is the inclusion, and fJ is the boundary operator.
It
foU?WS
[aD] E ker i. = 1m a•. Consequently ".U•.!.) is nontrivial. Similarly, we obtain
Definition 1.2. Let D be a k-topological ball in M. and let S be a subset in M. We say that aD and S homologically link. if aD n S = 0 and 11"1 n S # 0. for each singular k chain 1" with 8T = aD where 11"1 is the support of 1". In the same way, we prove
Theorem 1.1'. Assume that aD and S homologically link. If f E G(M.lli.') satisfies (1.1) and (1.2). then [IoU•• f.) # o. There are many linking examples:
Example 1. Let n be an open neighborhood of a point zo in a Banach space. Set Zl ¢ fi, S = 8D, and D = ZOZi", the segment joining zo and ZI.
Then S and aD = {'o. "} homotopically link. Claim. By connectedness, S n t and
I- 0, where t
is any path-connecting Zo
Z1_
Example 2. Let X be a Banach space. X decomposition where dim Xl < +00. Set
S=X••
and
= X, Ell X.
D=B,nX,
is a direct sum
85
1. Topological Link
where B, is the unit ball centered at B. Tben aD and Slink. Claim. Obviously S n aD = 0. We only want to show V", E C(D,X) and '1'180 = id 180 ~ ",(D) n S = 0.
(1.3)
Define a projection P : X - X,. It is equivalent to showing that 3 Xo E D such tbat Po ",(xo) = B. Define a defonnation
F(t,x) = tP 0 ",(x)
+ (1- t)x
V (t,x) E [0, I] x D.
Since B ¢ aD = F([O, I] x aD),
deg(F(I,'),D,B)
= deg(F(O,·),D,B) = 1.
The equation Po ",(x) = 9 is solvable, using Brouwer degree theory. Example 3. Let X = X, E!) X. he defined in Example 2. Let e EX., = 1, and let R R" p > witb p < R , . Set S = X. naB" and " n B " s E [O,R,J}. Then aD and S homotopically D = {x + se I x E X, R link.
°
Ilell
Claim. Obviously S n aD = 0. It remains to prove (1.3). Again we use the same projection P in Example 2 and define a new deformation:
F(t, x
+ se) =
[(1 - t)x + tP 0 ",(x + se)]+ [(1 - t). + tll(I - P)
0
",(x + .e)1I - pie.
It is easily seen tbat
+ se) = x + (s - p)e, F(I, x + se) = Po ",(x + se) + 1II(l F(O, x
P) 0 ",(x + 8e)lI- pie,
and
F(t,x + se) = x
+ (8 -
p). '" B, 'It E [0, I], 'Ix + se E aD.
Thus (1.3) is equivalent to finding Yo E D satisfying PO"'(Yo) = B, { 1I"'(Yo)1I = p;
i.e., F(I, Yo) = B. Since deg(F(I,·), D, 9) = deg(F(O, '),D,B) = deg (idx l ,X, n BR"B) . deg (id - p, (0, Rtl, 0) = 1.
86
CritiC41 Point Theory
Consequently, (1.3) is solvable, i.e., S and
aD homotopically link.
Actually the above three examples are of homological linking as well. We have
Theorem 1.2. Suppose that the boundary aD of the k-topologicaJ ball D and S homotopically link. Assume that (1) S n D = single point, (2) S is a path-connected orientable submanifold with codimension k, (3) there exms a tubular neighborhood N of S such that N n D is homeomorphic to D. Then aD and S homologicaJly link. Proof. (N, aN) can be regarded as an orientable sphere bundle over the base space S with fibre (D·,S·-I), i.e.,
(D',S·-I)
_
(N,aN)
/
S
According to Thorn's isomorphism theorem (see Chapter 1, Section 7), "
H.(N,aN) E!!! H.(S) ® H. (D·,S·-I). Since S is path-connected, we have Ho(S)
H.(N,aN)
~
~
G. It follows that
G.
Let IT) be the generator of H.(N,aN). We choose a singular chain such that (}T = aN n D, according to (3). Since N is a tubular neighborhood of S, we have
T
E IT],
H.(M,M\S) E!!! H.(N,N\S) provided by the excision property. Let us apply a deformation retract along the exponential map. It follows that
1l.(N, N\S)
~
H.(N, aN).
From assumptions (1) and (3), we see that aD and (}T are homologous in H.(M, M\S). Therefore ID] = IT] is nontrivial in H.(M, M\S), where ID) is the relative singular homology class containing D. The nontriviality of ID] implies that aD and S link homologically. Thus the above three examples are also homological links. Remark 1.1. In order to prove the homological links for the above three examples. only the Kiinneth formula is needed.
1. Topological Link
87
Theorem 1.3. (Minimax Principle). Suppose that :F is a family of subsets of M. Set c = inf sup fIx). FeF zeP
Assume that (1) c is !inite, (2) f satisJies the (PS)e condition, and (3) 3 0, such that the famiJy :F is invariant under the family of maps ~ a (and c' > a resp.), and that f satis!ies the (PS)e (and (PS)e') condition. Then c (and c· resp.) is a critical value of f. Moreover, we have c·
:5 c.
Proof. It suffices for us to verify that the families :F = {lZI I Z E a} (and:F" = (lrll rEa}) are invariant with respect to ~'o' where eo E (0, c - a) (and eo E (0, c· - a) resp.). This is just the homotopy invarianee of the homotopy (and homology) classes. The inequality e' :5 c follows from :F C :F". In particular, let a he the class in ". (f. , f.) (or H.(f.,J.» with = [aD) in Theorem 1.2. Then
a.a
c = inf sup f
(1.4)
0
'P(x)
£Er zED
where (1.5)
r
= {'P E C(D,M) I 'PlaD = idaD} (and
c' = inf {sup f(x) I r is a k reI. singular chain in (f.,J.) with :Z:EI"TI
(}r
=
aD}).
88
Ct-itiroI Point Th a (and c' 2: d > a), under which c and c' are critical values.
Remark 1.2. The separation conditions (1.2) and (1.6) may be weakened to (1.2) and (1.7)
J(x)
0 and a homeomorphism 71 such that
K.
'I: (- J)-.... 03 Z E '" with support IZI c fe'+ 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 =
-'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 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
: 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 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' 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 cohmology 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 : (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) 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
-
c
U6ij2,~6b},
Iudeed, if not, 3 Xn
IIdfn (Xn) II
and
-
E
xn \(Nn)'
0,
then hy (PS)', 3 x' 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
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 toplogical 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]
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+ 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+,}
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- 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 such that a, b ¢ f'(K(f')) II e E 10,6]. We claim that (f 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)
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» 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 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
X},
AI),
H _ = span {'I'j I Aj
. 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
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 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.,,) +! 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 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
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'- 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. 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: 0 small, and ~ > ~, large enough. The function W = W6 is just what we need. One may assume~, :.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 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 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' 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~ + 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 .(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 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 = '{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 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
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 Y'
u :> Z'.
If Z is a strong deformation retract of X, and if Z' is a strong deformation retract of X', then the inclusion map j: (Z, Z') - (Y, Y') induces a monomorphism j.: H.(Z, Z'} _ H.(Y, Y'} (4.5) in homology and an epimorphism j":H'(Y} - W(Z}
(4.6)
in the cohomology ring.
Proof. We consider the commutative diagram
H.+.{X,Z)
-
H.(Z, Z'}
i·l
H.(Y, Y'}
W(Y',Z'}
--L ,,/"
--->
"1 H.(X,Z'} ,1
--->
H.{X,Y'}
~
--->
H.(X',Z'} (
H.(X,Z}
{4.7}
214
Multiple Periodic Solutiom
where the longest row is the exact sequence of the triple (X. z. Z') and tbe longest column is tbe exact sequence for the triple (X. Y'. Z'). The indicated maps are induced by inclusions. By the assumptions
H.(X. Z)
=0
and
H.(X'. Z')
= o.
Therefore 13 is an isomorphism and ~ is a zero map. To prove that j. is injective. assume a E H.(Z. Z') satisfies j.(a) = O. Then by the commutativity of the rectangle in diagram (4.7). 7013(0) = 60 j.(a) = O. Therefore. by tbe exactness of the longest column in (4.7) there exists an a E H.(Y'. Z') such tbat '1(0) = 13(0). By tbe commutativity of the triangle in (4.7) and the property of ~. we have '1(0) = , 0 {(a) = O. and since 13 is an isomorphism. we conclude a = 13-· 0 'I(a) = 0 as claimed. In order to prove (4.6) we consider tbe commutative diagram W(X.Z)
-
,_____..!..
W(X)
W(Z)
Ti"
_
W+l(X.Z)
(4.8)
--- W(Y) . where the longest row is tbe exact sequence for tbe pair (X. Z) and 13 aod '7 are bomomorphisms induced by inclusions. Since H·(X. Z) = o. 13 is an isomorphism. If a E H·(Z). then by the commutativity of the triangle in (4.8) j'(-y 0 P-·(a» = a. so that j" is indeed surjective.
Proof of Theorem 4.1. 1. The first conclusion follows directly from Corollary 3.4 of Chapter II. because CL(T" x ii) = 2. 2. As to tbe second conclusion, we consider two separate pairs: (M,J-.,.T) and (J-,.T.0). and we want to prove that there are at least two distinct critical points in each pair. For the pair (J_,.T.0). Lemmas 4,3 and 4,4 yield
(S· \ {O})
X
S· )(
ii:::> J_,.T:::> {1r}
)( S· )( {e}.
Construct a strong deformation retraction
'I: [O.lJ x (S· \ {OJ)
X
S· )(
ii -+ {1r} X S· x {II}.
by
Apply Lemma 4.5. Then tbere are a monomorphismj.: H.(S·) and an epimorphism j':H'(L"r) -+ H·(S·).
-+
H.(J-"r)
215
5. Some RuulI& on Arnold Conj
Cpn is the projection z ..... [z], the equivalence class
under the group action S1, and i: S2n+l -+ C n+ 1 is the imbedding, we define a symplectic fonn w on Cpn as follows: 1\"·w = i·wo. It is well defined, because Wo is equivariant under the group action 8 1 •
Looking at the symplectic manifold (cpn,w) in this way, the submanifold L = {(z] E cpn I z E [zl, z = x + iy, y = 9} is diffeomorphic to the real projective space \ll.pn, and is easily verified to be a Lagrangian submanifold. 5,2, The Fixed Point Conjecture on (T"n,wo) Theorem 5,}, (Ae!) is true for (T 2n , Wo), i.e., there are at least 2n+ 1 fixed points for 'Po., and at least 22 • fixed points if 'Po. (T"n) is transversal to T 2• at all its fixed points. Proor. As we mentioned before, the problem is reduced to finding the number of 21f-periodic solutions of equation {5.5}. Since Wo is canonical, and lR 2n is the covering space of T'ln ,one may extend the Hamiltonian function h(I, -) from T'n = \ll.2n/2"zn to \11.2. by H E C2(111.' X \ll.2n,\II.'),
satisfying H(t, z) { H(I,z)
= H(t, z + 2"ej), = h(t,z)
j = 1,2, ... ,2n, V (t,z) E \11.' X T 2 . ,
5. Scmae Ruu/Is on Arnold Conjeduru
219
where {ej I 1 :5 j :5 2n} is the orthonormal basis in JR2n. Noticing that the Hamiltonian system induced by H and the canonical symplectic form Wo reads as -Ji = H.(t,z),
(5.7)
this is the equation we have studied so far. Each solution of (5.7) with the boundary condition z(2.. ) = z(O)
+ 2..ko
for some ko E Z2",
corresponds to a 2"-periodic solution [z) of (5.5) on ']"l". Moreover, two such solutions z" Z2 are in the same class [z) if and only if there exists i< E Z2" such that z,(t) = z,(t) + 2.. i 0, and (P"p,) > 0 for 1 - e :5 Iv,l :5 1, i = 1,2 and lv, - p,1 < e, and that Ih.(t,V,q)1 < 2~ for 1 - 25 :5 Ivl < 1 and (t,q) E 8' x T". Let us define a Hamiltonian HE C 2 (R' X R 2 ",JR') heing 211"-periodic in t and q, as follows:
H(t,V,q) = (1- p([v[)h(t,v, q)
+ P(lv[)alvI 2 ,
where p E C"" (JR~) satisfies 0 :5 p :5 1,
p(s) = {
~
if s ~ 1 ifs:51-5,
220
MuUiple Periodic Solutio ...
and 0 $ p'(s) $ Consequently,
i, and ¢ Z is chosen such that > 6Maxh(t,p,q).
(P" Hp(t,p" q))
= (p"
(1 - p(lPollhp(t,p"q)
+ [P'(IPoIl(1Po1 2 - h(t,p"q) +21p,I'p(Ip,I)) ~I) > 0 '11- E $ 11',1 $ 1, j = 1,2, 11', - p,1 < E and 'I (t,q) E to consider the new iIaIniltonian system -Ji(t) = H. (t,z(t)) ,
s' x '1"'.
We tum
z = (p,q),
(5.9)
and claim that (5.8) and (5.9) have the same 21r-periodic solutions. Indeed, we conclude that 1. (5.9) has no 21r-periodic solutions zIt) (P(t),q(tll such that zIt) ¢ 1f' x llI.n for some t. If not, with no loss of generality, we may assume p(O) ¢ 1f'. In a neighborhood of (P(O),q(O)) we have
=
{
P=-H,=O q = Hp = 2ap'
so the solution must be (P(O) , q(O) + 2apt), which cannot be a periodic solution. Moreover,
2. (5.9) has no 21r-periodic solution zIt) = (P(t),q(t)) such that zIt) ¢ B'_6 x JRn for some t. Otherwise, from 1- 6 $ Ip(O)1 $ 1 aod Ipi = IH.I $ 2~' it follows that 1- 26 $ w(t)1 $ 1 and W(t,) - P(t.lI < E. Thus 0< (p(O),
l'
Hp(t,P(t),q(tlldt) = (P(0),q(21r) - q(O)) = O.
This is a contradiction. However, according to Example 2.2, (5.9) has at least n+ 1 (or 2n) distinct 21r-periodic solutions (if all solutions do not have Floquet multiplier 1 respectively). This proves the theorem. 5.3. The Lagrange Intersections for (CP",IRpn) We turn to (AC.), where the symplectic manifold M = CPR and the Lagrange submanifold L is taken to be IRP", as in Example 5.3. Since CL(IRpn) = SB(llI.P") = n
+ 1,
it is not necessary to consider the transversa) case. We have
221
5. Some R..uIJo on Arnold Con;'''''''"'
Theorem 5.3. (ACo) is true for (eP",RP"), i.e., there are at least (n + 1) intersections of <po.(RP") n RP". We reduce the intersection problem to a critical point problem by several steps: Step 1. Reduction to a boundary value problem. By definition, PI E )RP"n<po.(IRP") if and only if3Po E IRP" such thatp, = <po.CPo) E 11.1"', i.e., the equation
wIt) = X,(w(t)) { w(O) = Po, w(2,,) poss
= p"
po,p, E llI.P"
(5.10)
.eo a solution wIt).
Obviously, there is a one-to-one correspondence between the intersection points and the solutions of (5.10): wIt) = ""Po = "', 0 "'2~P'.
The problem is transferred to find the number of distinct solutions of the BVP (5.10). Step 2. Reduction to Hamiltonian .ystems on C n +!. Note that
cn+ 1 2....
82n + 1 ~ SJn+l/sl = cpn I
where" is the Hopf fibration. We can associate with h:IR' x eP" -llI.' a function H: llI.' x iCn +! _ R' satisfying
(1) H(t,e;#z) = H(t,z) V z = (Z"Z2, ... ,Zn+!) E en+!; (2) H(t, = h(t,.) 0 "; (3) H(t,·) is positively 2-homogeneous in a neighborhood of the unit ball; (4) H(t,·) is C' and C'-bounded. With no loss of generality, we may assume that h(t,·) > 0, so that H(t, z) > o V z ~ 9, and also that H(t, 9) = 0 and H.(t, 9) = 9. We turn to consider a new Hamiltonian system:
·lI......
-Ji(t) = H,(t,z(t)) + oXz(t) { z(0),z(2,,) ERn+! n S"n+!, where
J~
(0
In+!
(5.11)
-In+!) , 0
and oX is a Lagrange multiplier. It plays a role here as an eigenvalue. Lemma 5.1. Let z be a solution of (5.11). Then zIt) E S'n+', and ,,(e;>lz(t)) solves (5.10).
Proof. We consider the derivative of the norm:
!
Iz(t)12 = 2(z(t), zIt)) = 2(z(t), J H.(t, z(t))),
222
Multiple Periodic Soluti ....
where 1·1 and ( , ) are the norm and the inner product on en+, = JlI.2(n+') respectively. But 0 = I,;H(t,.'.z)I.=o = (H,(t),z(t»,-Jz(t» because H is 8' invariant. Therefore Iz(t)l = const. Here and in the following, we write either z = (x,lI) E Rn+' x Itn+' or z = x + ill E iC"+', irthere is no confusion. Al; to the second conclusion, we observe that the symplectic structure w on CP" is defined by 1f·W = i·wo, where Wo is the canonical symplectic structure on cn+'. Thus w(·, X,)
=
dh" and dh,
0".
= dH, ° i = i·wo(·, Yi) = ",·w(·, Yi) = w(·, ".Yi),
where Yi = JH,(t,·). Since d(h, 0") = w(·, X,),
. therefore
Xt = On the other hand, letting z(t)
1r.Yi.
= .'>lz(t), (5.11) is rewritten as
-Ji'(t) = H, (t, z(t».
By the uniqueness of initial value problems, wet)
we
have
= "z(t) = ?TIe'" z(t»),
where wet) is defined by (5.10). Therefore ,,(e'''z(t» solves (5.10). The proof is finished.
Lemma 5.2. Let (,',A,) and (,2,A2) be two solutions oi(5.1l). Then ,,(e"",'(t») =" (e"",2(t») implies A, = A2 (mod ,,). Proof. First, we claim that if z:t ,Z2 solves -Ji = H,(t,z)
(5.12)
and ,,%, = "ZO, then 3 /l E]R' /2,,"§., such that
z' (t) = e'·ZO(t). Indeed, by definition, 3 a function /l(t) such that %'(1) = .'.(')ZO(I). Substituting into equation (5.12), we have
s.
223
Some Ruu/ts on Amoltl Conjectu,..
which implies either tbat 3 to such tbat ro(to) = 6, so that ro(t) = 6, V t, and we can choose p(t) to be constant, or that Mt) = 0 V t. And again we have p(t) = const. Next, we have
e,
p
z2(0) = z'(O),
"",+>,).2(2.. ) = e'z'(2..).
+'
Since z'(2j ..) E an n ,y2n+' (i = 1,2, j = 0,1), must be real. Consequently,
,'P and tben ,,(>.->,)
A2 = A,(mod ..). Let us define an operator on L2([O, 2..],IC"+!), d
A= -J dt' witb domain
Lemma 5.3. Th, operator A is self adjoint, with spectrum utA) has multiplicity (n + 1).
each eigenvalue
lk
= lZ;
Proof. Indeed, we have the following spectral decomposition:
L2([O, 2 .. ],C n+!) = E!1span{cos ~kt';+iSin ~ktej+n+' ] j = 1,2, ... ,n+1}, kEZ
where {e;
+ iej+n+! Ii =
1,2, ... ,n + 1} is tbe basis of en+!.
Step 3. Reduction to a variational problem. According to the spectral decomposition, we decompose A into the positive, zero and negative parts: A = A+ + AO - A-, where A+ and A- are positive operators on their associated subspaces. Let us introduce a Hilbert space E = D(IAI'/2), the
domain of the square root of IAI, with norm
liz II "
= ( Izll'
+ IIIAI'/2,"i, )
'/2
.
In the following, if there is no confusion, we denote by ( , ) and by L 2 inner product and norm, respectively. Let us define
J(z) =
~ (I(A+)'/2," 2 -1I(A-)'/2'11 2 )
on the manifold S
= {. E E IlIzIIL' = 1}.
-12'
H(t,.(t»dt
II· II the
224
Multiple Periodic Soiutiom
Lemma 5.4. Suppose that zo is a critical point of J on S with Lagrange multiplier Av. Then (ZO, Av) solves (5.11), and J(ZO) = Av. Proof. Letting (ZO, '\0) be the critical point and the Lagrange multiplier, we have
(5.13)
v z E E, and it follows that Po(H,(t, ZO)
+ .\zo) = 6,
where Po is the orthogonal projection onto the space associated with eigenvalue O. Consequently,
and therefore
zo E D(lAI)
C D(A).
Then the weak solution equation becomes Azo
= H,(t, ZO) + '\ozo,
so that Zo solves (5.11). In particular, if we choose z = ZO in (5.13), then
(1IA+)I/'zoll' -1I(A-)'/'ZoIl')
- 2('\ozo, zo)
-1"
(H,(t, Zo),zo)dt
= O.
Since ZO(t) E S'n+1 (Lemma 5.1) V t E [0,2"J, and H(t, z) is positively 2-homogeneous in a neighhorhood of the unit hall, we have (H,(t,Zo),zo) = 2H(t,zo).
Thus
Remark 5.1. We take the working space E = D(lAII/') with norm
rather than the space HI/2([0, 2"1, C n+ l ) since the trace operator is not well-defined on Hl/2, so that the boundary value condition cannot be formulated in Hl/'l.
IIzllE = (lIzlll, + IIIAI'/'zlll')'/
225
5. Some Results on Amold Conjectures
Finally, we observe that there is a natural symmetry for the functional. Namely, J( -z) = J(z). Indeed, the function H(t, z) is SI-invariant, so that H(t,-z) = H(t,e"z) = H(t,z). Moreover the boundary value condition D(A) is also invariant with respect to this group action. Consequently, J is well defined on the space P = SI'§." where S is the unit L' sphere in E. Returning to the original problem, we point out that we Me not concerned with bow many distinct critical values of J there are, but how many distinct critical points there are associated with critical values in an interoal with length < ... After the preparation, we shall give a proof of Theorem 5.3. The Galerkin approximation will he applied. Let E. =
$
span
{cos ~ltei + i sin ~ltei+n+1 Ii = 1,2, ... ,n + I},
1'1:5'
p.
=
pnE.,
and
J. = Jlp•. Then dim E. = (2k + l)(n + I), dim p. = (2k + l)(n + 1) - 1.
Lemma 5.5. J sati"1ies (PS)" with respect to {Po I k = 1,2, ... }. Proof. SUPP05C that z. E p. is a sequence satisfying IJ(z.)1 =
I~(AZ"Z') -
J.2. H(t,z.)dtl:o; C.
(5.14)
and
(dJ.(Zk),
W)E = (Az., w) - J.2. (H,(t, Zk) + AkZ., w)dt = o(llwIlE)
"W
E E.
(5.15)
zt
We decompose Zk into +z:+z;, according to the spectral decomposition, to positive, zero, negative eigenvalues, respectively. First, setting w = in (5.15), we have
zt
IIz;lI~ :0; IAkl
+ o(llz;IIE) + C 2
where C2 is a bound for IH,(t, z)l. Set w = z. in (5.15). Then by (5.14), 1.1..1 = I(AZ"z,) :0; 2C.
2 J. ·(H,(t,z,),Zk) dt l +o(lIz.IIE)
+ 2"M + O(llz.IIE),
226
Multiple Periodic Solutioru
where M is a bound for
IHI on the sphere 8"+'.
Since
IIz~IIE = Ilz~IIL' : 2. A severe problem occurs: The Palais-Smale condition is missing! Carefully analyzing the role of the Palais-Smale condition in the proof of the deformation theorems, we find out that it is strongly tied to the gradient How. We observe, however, that the heat equation for harmonic maps Ot!(t,') = [)./(t,·) J(O,·) = cp(.) (1.3)
{
l(t")18M = ",(.)
produces a How I(t, .), which depends on the initial data cp and reduces the energy:
L =- L
:tE(J(t,.») =
(V/(t,'), VfJ,J(t,·»dV. ([)./(t, ·),fJ,J(t,·»dV.
+ (
J8M
= -
(1.4)
g'j h~/Jo.l{1(t, ')/.': . nj dS.
L
1[)./(t,·)I'dV.
~ O.
One expects to replace the gradient How by the heat How I(t,·) (= to indicate the initial data). This is possible if one can prove the following conclusions:
I~(t,.),
(1) the global existence of the heat How, i.e., l(t,·) is defined on the whole half-axis t ;:>: 0; (2) the limit of I(t, .), as t --0 00, should in some sense be a harmonic mapi (3) the How I~(t, .), as a map depending on t and the initial value cp, is continuous. Before going into these statements, we introduce some notation:
C!+'(M, N)
= {u E C'+>(M, N) I ul 8M =
"'},
0
< 'Y
0; C1+b/')"+>(QT,lII.k) equals the completion of GOO(QT,lII.k ) functions under the norm
II1II c>+'·'+'
=
II1II
c
+
su P 0 0 such that for a solution of the system (E) in a domain 10, Tl x D, where D = Bp(zo) nM, for some Xo E M, and p > 0, if sup
j
IVI(t, -)l'dV. < Eo
telto,ttl D
for some to, t1 E (0, T), then for any p' E (0, p) and (to, f,) c (to, t1), we have some Q = 1 - ~ > 0 and a constant C depending on £'0, 0, p, p, and to, tl J ~J t~ only such that sup tE{ta,t.]
II/(t,·)IIc>+o(D'):O:
e[1 + 111/l1lC'+'(OMnD) +
VP
1.
IVfl2 dtdV.
e~(QT)'
satisfying
( (to.hJxD
p
)
]
for p > 4, where D' = Bp'(xo) n M. Proof. Define a cutoff function
and
For F
"'I E
(t,x) E It~,t;l x D' (t,x) ¢ Ito, til x D.
= 'PI . I,
a,F - AMF = r(f)(VF, V f) - r(f)(fv",,, V f) - 2V/· v"" F(to,') = 0,
+ I(a, -
AM )'Pl,
F(t")IOM = 'PI' 1/l According to the Sobolev embedding theorem (cf. Nikol'ski INiklJ) and linear V theory, we have" = I - 4jp > 0 such that sup
tel~.tU
III(t, ')lIc"'(O')
:0: sup IIF(t")lIc'+O(D) telto.tt)
:0: eallFllw:·,u,•.•• lxD) :0: e[1 + 111/l1lC'+'(OMnD) + IIVIIIL.U..... lxD)
+ IIVF· V/IIL.([..... lxD»).
(1.7)
1. Harmonic Map8 and the Hmt Flow
235
However, provided by tbe Sobolev imbedding theorem together with Lemma 1.1, letting PI = 2p/(P + I), we have
Jr" •• IIVFII~'(D)dt
lto IIFII~."I (D)dt
::; f"
::; e[I + 1I"'"~h(8MnD) + fto IIV11Ii':'(D)dt
(1.8)
J••
+ f.·IIVF. VIIIi':. (D)dt) Applying the Holder inequality,
(1.9)
For sufficiently small eo > 0, we put inequalities (1.8) and (1.9) together and obtain
f.·IIVF(t")II~'(D)dt::; C•• [1 + 1I"'"~h(8MnD) (1.10)
+{
IIV/(t")IIi':'(D)dt].
Again by the Holder inequality,
IIVF· V IIIL'U',,'.lxD) ::; IIVFilL" . IIVIliL" ::; C•• [1 + 1I"'"co+'(8MnD) + IIVIIIL•• U..... )XD») x IIVIIIL.. (I..... )XD) Returning to (1.7), we have sup II/(t, ')IIC'+O(D') ::; e[l + II!J>IIco+'(8MnD)
telco,til
+ IIVIIIh(I..... )XD») Lemma 1.3. Let w > 0 be finite or infinite. Assume that 'IT < w, IE W':"(QT, N), P > 4, is a solution of (1.3). If there is a relatively open set D c M and a sequence of intervals I; C [O,w) with mes(Ij) ~ 6> 0 such that sup f IV/(t")I'dVg < eo. tEIJ
1D
236
Application! to HafTJ10nic Maps and Minimal Surjaa:J
Then for any opeD subset D' cc D, for any sequence It;} with t; E I j and t; - w, there is a subsequence t;, such that I(t;.,.) is C'(l5',N) convezgent to some U E Wi(M, N). Proof. Since
LIV
2
I(t, ')1 dV9 :5 E(",),
the family of maps {J(t;, ')Ii ~ 1,2, ... } is weakly compact in Wi(M, IRk), so that there is a subsequence {t;.} along which I(t;.,.) - u weakly in Wi(M,lR k ) . Starting from (1.10) with p ~ 2, we obtain a constant, which depends on N +00
ul 8M =.p} harmonic, and nonconstant}
if there is no such map.
Lemma 1.6. Suppose that DT = m8X('.o)EQT Then E('P) ~ m+b. Proof. We may find sequences T. / w and
IV/(T., a.)! =
IV/(t,x)! is not bounded.
a.
E M such that
maxi'" !(T.,x)1 =
oEM
DT"
k = 1,2, .... From now ont we write iJ.r. simply as fJk. Neglecting subsequences, we may only consider the following two possibilities:
(1) 6. dist(a., 8M) (2) 6. dist(a., 8M)
-> ->
+00, < +00;
p
I. Harmonic M,..,. and !he Heal flow
in both cases, we may assume 4k Take a local chart U of B. Let
-+
239
a E Il.
DO={YER'IBo+%. EU} 8Jld
10 = [-(/~To,(/~(",-To»). Define a function on lox Do as follows:
k = 1,2, .... Then we see (1.11)
k= 1,2, ....
(1.12)
Let
Then holT)
~
0, and 'lie> 0
I" hO(T)dT
J-e
$l
T dt { • T,,-e/fJ: 1M
=
10.1(1, x)l'dVg
E (I (To -:~,)}- EU(T.,·») -
0
as k -+ 00. Thus, neglecting a subsequence, we may assume h.(T) - 0
i.e., for almost aU
T
a.e.
T E [-e, 0),
E [-e, 0), (1.13) o
In case (I), B E M, 8Jld Do - R' in the sense that V R > 0, 3 the ball BR centered at (/ in R' is included in Do for k ~ ko. On the one hand, by (1.13) ,
VR>O,
ko > 0,
(1.14)
Applications to Harmonic Map$ and Minimal Sur/aces
240
for almost all r" E [-e, OJ. On the other hand, hy Lemma 1.2, we have IIV.(T, ')IIC'+O(BR) :-;: C[l
sup
+ (e4"R2)'/P).
(1.15)
rE[-e,O)
This implies a. subsequence, where we do not change the subscripts, so that
v.(r",y)
-+
Cl+ a ' (R2)
ii(y),
for some r" E [-e, OJ (actually in a countable dense subset of [-e, 0]). We conclude that ~ii=O in R2. According to the singularity removahle theorem due to Sacks-Uhlenbeck (cf. [SaUl]), ii is extendible to a harmonic map ii: 8 2 -+ N. We are going to show that v is nonconstant. Indeed,
v.
since satisfies (1.11) on I. x D. with the condition (1.12). The Schauder estimate applies to obtain an estimate: IIv.(r,y)lIc'+l','h 0 small enough so that
E(J(T~,.J)
=
L
IV'!('r.,x)1 2 dV. I
=
{
JM\U~""IB4(%J)
+L
{
j=l JBJJ(%j)
IV' !('r.,x)1 2 dVg •
1. Harmonic Mapa and the Heat Flow
241
Since
and there exists at least one io such that a =
XjDI
we have
and
f
IV/(7k,x)I'dVg ?:
f
IVvk(rO,y)l'dy
} 8 1 / 2 ,,.,(8)
} B,(zjo)
for k large. First let k - 00; by definition
and then, because 6 > 0 is arbitrary, E(rp)?: lim
f
k-OO}M
?:
IVI(T., x}l'dVg
L
(1.18)
IVu(x)I'dVg
+ b?: m + b.
This is the desired conclusion. In case (2), a E 8M n U. We choose a suitable coordinate (YI,!/2) in R', such that the !/2-axis is parallel to the tangent at a of 8M, and the YI-axis points to the interior of U. Thus Dk tends to the half plane R~ = ((YI,!/2) I!/I > -pI, and for each point on the boundary,!/I = -p,
As in the proof of (1.15), now we have \I R > 0, sup IIvk(r, ·)IIC'+-(BRnD.) ::; rE(-e,O]
C[l + (e47rR')I/. +
II~ (a
k
+ :.)
t"'(8D.nBR)]·
Since on the right hand side, there is a constant control independent of k, we find a function ~ on R! and a subsequence Vk(T·,·) such that Vk(rO, y) _ ii" (V)
242
Applicatiom to Hannonk Maps and Minimal Surfacu
and then a~ =0
inR~,
V'18R' = .p(a). +
On the one band, similar to the proofs of (1.16) and (1.17), we see that i? is nonconstant; and on the other hand, let us define 8 romplex function '1(z) = h(V;,V;)
where h is the Riemannian metric on N, and
v; = ~ (8"1 Z
= Yl
i81n )v\
+ iY2.
Therefore,
The harmonics of ii' implies the analyticity of the function '1. The boundary condition on ii implies that the function 'I can be analytically extended to the whole complex plane. From the condition 'I( -p + i1l2) = 0,
we conclude that '1(z) == 0, and hence that ii' is a constant map. This is a contradiction, SO Lemma 1.6 is proved. In the following, we assume '" (N) = O. We shall expand the conclusion of Lemma 1.6 to the following:
E(rp)
~ mT
+ b,
where :F is the homotopy class of '1', and mT
= inf{E(u) I u E :F}.
Only the inequality (1.18) should be fixed. It is known that f(T.,·) u(·) in CHa' (M \ Lf;=l B.(x;),R k ). We only want to show u E:F. Let 6> 0 be small enough so that B.(x,)nB.(x;) = 0, if i # j. Combine u(x,) with the map f(T., ,)188,(,,) by the following map:
v x ~ U~=l B.(x,), V x E B.(x,),
1. Harmonic Maps and the Held Flow
243
where '1 E C""(R') satisfies
'1(r) = {
~
and exp is the exponential map. Since 7r2(N) = 0, and
1.
we see that remains in the same homotopy c1ass:1'. And from C(M, N), we conclude ii E :1'.
1. _ ii
Step 6. Continuous dependence. From the point of view of
POE,
the continuous dependence
'1' ..... I",(t, x)
from C~+'(M,N) to C~+P+'(QT,N) does hold. The proof depends on the locally uniform houndedness of the heat ftow I"" i.e, V"", E Ec = {u E C!+'(M,N) I E(u) : 0 such that
'I' .....
Sup(.,z)EIO.oo»)xMIV/",(t,x)l: (M, N). Indeed, if (1.19) does Dot hold, then 3 '1" 3 Tk with T· = limT.' satisfying IIV I •• (Tk' ')IIL~(M)
-
(1.19)
'1'0 in C;+'(M, N) and
00
and V T < TO, 3 C2(T) < +00 such that IIV I",. (t, ·)IIL~((o.T(XM) : 0, and we shall prove II VI",. (TO - O")IIL~(M) = 00, which contradicts Lemma 1.6 because '1'0 E Ec. For simplicity, we write Jk = / "'.' k = 0, 1, 2 . . . . It is sufficient to prove that {It} is a Cauchy sequence in W':·2(QT), VT < T". Since IIVI>(t, ')IIL~(M) : 0 satisfying {J.P. : -y. In the following, we shall employ the heat flow I~(t,·) as deformations, under a weaker topology W:(M, N), on the incomplete manifold C!+' (M, N). For details, cf. Chang ICba).
W;,
Step 7. The First Deformation Lemma, The critical set } Kc= {UEC",2+, (M,N)lllu=O
is compact in C;+' as well as in tbe W;-topology. In extending critical point theory, we have the modified first deformation lemma: For a closed neigbborhood U of K, in the W; topology, 3 • > 0 and a
W:
continuous deformation '1: [0, I) x Ee+~ -
Er:+~ satisfying
,,(0, .) = idE,+.,
,,(1, Ec+< \ U) c E,_ 0
VteIR~.
246
ApplicatiOfl8 to Harmonic Map. and Minimal Surface!l
Then we have e = e( 6) >
°such that
1I1l./",(t, ·)IIL'(M.H) 2: e.
"
- N) of K e , under the (6) For any cloeed neighborhood U C Col (M, W:-topo/ogy, where c < mF + b, 3 e > 0, a cJoeed neighborhood V C U, and a W: - (p> 1~7) stroag deformation retract 1): [0, I] x Ee+< .... Ec+ .. satisfyiag 1)(1, Ee n V) 1)(1, Ee+< \ V) where E. =
C C
Ee n U, Ee_ co
and
{u E :F I E(u) :5 a} is the level set, V a E Jli.~.
Remark 1.1. The heat flow method was first used by J. Eells and Sampson [EeSl] in proving the existence of harmonic maps, where m is arbitrary and N has nonpositive sectional curvature. See also Hamilton [Haml]. Without the restriction on curvatures, but with m = 2, see M. Strowe [Str4] and K.C. Chang [ChalO].
2: Morse Inequalities
In this section, we establish Morse inequalities for harmonic maps under the assumption that all harmonic maps are isolated. As shown in Chapter I, the crucial step in the proof is to prove the following deformation lemma: Lemma 2.1. Let:F be a component of C!+'(M, N). Suppose that there is no harmonic map with energy in the interval (c, d], where d < mF + b, and that there are at most finitely many harmonic maps on the Jevel E-l(C). Assume that ",(N) = 0. Then Ee is a strong deformation retract of Ed. In order to give the proof, first we must improve conclusion (2) of Section 1, under the condition that tbe set of smooth harmonic maps is isolated. Namely, Lemma 2.2. Let E(IP)
< mF + b, and let
c= lim
t-+<XI
E(!",(t,.»).
If Ke is isolated, then /",(t,·) .... u E Ke in the W:-topology, If p > 1~" as t - +00. Proof. According to Theorem 1.1, conclusion (2), combined with a bootstrap iteration, shows that 3 E Ke and tj t +00 such that
u
/",(tj,') ....
u,
C""(M N)
'"
"
V'Y' E (0, 'Y).
2. Mon. lnequaliti..
247
If our conclusion were not correct, there would be a 6 > 0 such that the neighborhood U. = {u E C!"(M,N) I distw:(u.U):5 6} contains the single element ii in K c , and a sequence fj 1 +00 such that I~(fj,·) ~ U•. Therefore 3 (t" ti') satisfying
( 1) f' f" - +00 .' 1 ,
(2) I",(f,,·) E BU,., Mfi',·) E BU•• and (3) I",(f,·) EU.. \U. VfE(fi.tt'). On the One hand, we had 6
:5 II/",(f;,.) - I",(f;', ·)IIw: :5 C.lf; - fi'I',2,
provided by the embedding theorem. On tbe other hand, according to Theorem 1.1, conclusion (5) states
E(!,,(f;',.)) - E(!,,(f;,.») =
1't:;' J. 1;" L1~/(t,
1lJ,I(f,·)I'dV.df
M
=
·)I'dV.df
> «6)lf;' - til· Since tbe left band side of the inequality tends to zero as i _
00,
this is a
contradiction. Now we return to the proof of Lemma 2. 1. The basic idea is to reparametrize the beat flow I",(f,.). Let -r = p(t), where p(f) = (E(",) - c)-' if E( "")
l' IIM~(o,
·)IIi,do,
> c, and let g(-r,.) = I(f, .).
Then we have the following relations:
(E(",,) - e) df (1) 8T g(T,·) = dTlJ,I(f,.) = II~g(-r,.)IIi, ~g(T,·), (2) ::.,. E(g(T,·)) = -
L
(8T g(T, .), ~g(T,· ))dVg
= -(E(",,) -
e).
Therefore
E(g(T, .») = (1- T)E(",,)
+ Te,
V T E [0, 1J.
(3) The functionp: [0, 00) -!R'is continuous and monotone increas-
248
Application.! to Harmonic Mops and Minimal Surfaces
ing which satisfies the following properties:
p(O) = 0, p(+oo)=l
p(+oo)
>1
if f~(t,·) ~ ii e K, as t ~ +00, if
lim E(f~(t,·J)
t_+oo
< c.
Let us define a function 'I: 10, 1) x Ed ~ Ed as follows: 1} (r,f{) )
={
g~(T,·)
cP
if (T, cp) e 10,1) x (Ed \ E,), if (T, cp) e 10,1) x E,.
In order to show that Ec is a deformation retract of Ed, only continuity at the following sets is needed:
(1) {I} x A, where A = {cp e Ed \ E, I f~(oo,·) e K,} (2) 10,1) x E-'(c). Verification lor cose (1). V CPo e A, Ve > 0, we want to find 0 > 0 such that distw', (cp,1 CPo)c- < 0 }.ImpI·les d·IStw3( ) ,U -) < 91{J T,' T> -u P
E',
where ii = I~.(oo, .). Choose eo = eo(o.) as in conclusion (5), Le., lIt.f~(t, ·)IIL' ~ eo
if distw:(f~(t, .), K) ~ 0, V t,
and choose
e 0 0 such that, for a Wi-ball B. with radius 6, centered at the zero section of C 2"(u oTN), we have 1
-"_ < 1-"
IR(u)1 < 211is-Coron (BrCI), Jost (Joal». Suppoee that N =
8', and that", E Q2"(8M, 8') is not a constant. Then there exist at least two homotopically different harmonic maps. Proof, By the argument used in Theorem 4.1, we obtain a minimal energy harmonic map ii among all homotopy classes E(ii) = m. The second harmonic map will be obtained by constructing a map v homotopically different from u having energy
E(v) < m+b.
(4.1)
The construction of the map v is as foUows: Cboose a small disc Do on M, take an isometric copy D" and identify Do and D, along their boundary to obtain a 2-sphere 8'. Take a map w: 8' -< N = 8', wbich
Applicoti.... to Hannoni O. Thus
I" 0 il(z) - 'il( .. 0 ii)(O)zl = O(lzl'). We denote V( .. 0 ii)(O) hy a, which is a nonzero complex number, and write z = rei9 .
Letting
., (r- - -I-e) - +a (1- - -r) ee
t(z) = (.-oii)(ee')
e2
e
£2
£
is ,
we have
I(ee") = .. 0 u(ee;') and t(e - e')e;") = aee". Then we define a function rp: C - C as follows:
Izl$e-eZ e -e' < Izl $ e Izl ~ e. If e > 0 is small, rp is continuous and surjective. The map w is defined to be 1('-1 otp01r. Noticing that 1r is conformal, and that the energy is conformal invariant, we may compute the energy of w by the energy of .. -lrp.
Now E(v) =
~
I.
M\B(Z"o.~)
IVill 2 + ~
J,
B(zo.t-t 2 )
IVwl' +! 2
J,
IV( .. -'I)I'·
8.\8.. _ .. 2
Since (a
-F 0)
4. &i81ence and Multiplicity fur H...,.on;'; Map.
259
and
1
B.\B. ___ ,
~cl'
~_e2
IV(1r-'t)I'
/.. .
[I(1rOii)(Eeil )-aee;·I·
0
+ia = 0(E3), where C
Therefore, for
E
~ + 1("Oii)'(Ee;8)(r. _I-E) £ E
E
G-;.)e;'I}drdO
= Maxlld,,-'U'.
> 0 small enough,
we have
E(v) < m+b. The remaining part of the proof is the same as tbat in Theorem 4.1. Theorem 4.3. Given a Riemann surface M ",ith boundary 8M, if'" E C>+'(8M,9"), .., > 0, n 1 is continuous, we have
1 la' 2.
110' 0 uIlC(T.(X),H'''):S
(
0
u(8l1'dB
)'/r
for
1 2 - = 1--. r p
Therefore, the C(T~(X), H'/2)-continuity of a' 0 u with respect to u in
X follows from the Lebesgue dominance theorem.
5.
~
PIaI...u Probl.... lur Minimal SurJ-
263
It remains to verify the .c(Tu(X),C)-continuity of a' ou with respect to " in X. Since 1(0' 0 u)· vi S la' 0 "le((o,2'I ••') 'Ivle,
the CI-continuity of a implies that 110' 0 " - a' 0 Uon.c(T.(X).e) - 0 as
II" - Uolle - O.
This is just what we need. (2) Suppose that 'P = R(-y). with., E HI/2(SI,It"), i.e., A'P=O in D {
'P18D
=.,.
According to the Poisson fonnulas, we have
"" :l:
cp(r,8) = Re
c".rme;m'
m=-oo
= Re
"" :l:
c".zm.
m=-oo
where
00
.,(8) = Re
:l:
c".e;m'.
m=-oo
(3) 'I X,II E HI/2(SI,IIl"), suppose that 'P,,p are corresponding solutions of the Dirichlet problem, with boundary data x and II respectively. Then we have (X,lIlt/2 = Re
m~oo Iml c".dm = Iv V'P' V,pdxdy
= ]." ('Pro ,p)d8. (4) 'I X,II E HI/'nC(SI,R"), we have (X,II) E HI/. nC(SI,Jli.I) and
I(x, 11)11/2 S IIxlle ·llIh/. + 1IIIIIe 'lxll/2'
Claim. We only verify that
J1{s.J( ls.
2 l(x,II)(O - (.1:,Y)('1)1 d{ d'l
xs.
I{ - 'II'
I(:o({),y({) - 11('1» + (:oW - x('1),Y('1»Id{d'l xS' I{ - '112 S IIxlle . Illh/2 + IIYlle 'l x it/2'
=
Applicatiom to Harmonic Mop' and Minimal Sur/acu
264
(5) If p E CI(III.' ,III.R) and" E HI/2(S"1II.2), thon po" E HI/'(SI,lII.n), 8lld Ip 0 ,,11/. :5 11'1P 0 "IIL~ 100h/,· Claim.
Integrating both sides of the inequality, we obtain the desired conclusion. Next, we need to verify that critical points of J with respect to M satisfy the conformal condition. Letting 'P = ~(", 0 ,,) 8lld F(z) = {),'P = 'P. - Up,: D
->
en,
we have IF(z)I' = ('P. - i'P., 'P. - i'P.) = I'P.I' -I'P.I' - 2i('P., 'P.).
Therefore,
IF(z)i'
= 0
if 8lld only if 'P is conformal.
Howeverl we observe that
(),IF(z)l' = 2({),F(z), F(z» = (l>.'P(z), F(z» = 0; therefore IF(z)I' is 8llalytic in D. In polar coordinates,
Lemma 5.1. V" E M, letting 'P = ~(a 0 u),
we
have
('Pro'P.>I.=1 E CI({)D)" C 'D', the Schwartz distribution space. Proof. First, we assume u E C""({)D). In this case
Then (Hu,O') =
['W
Jo
('P~,'P') ·0'(8)d8,
V" E CI({)D),
deSnes a linear continuous functional on CI({)D).
5. TM P"",",u Problem lor Minimal SurftJCU
265
In order to extend this functional continuously to M, V u, v E C""({JD),
we let 'I' = R(o 0 u), '" = R(o 0 tJ), and we make the following estimates: /."- ('1'.,'1'0) - (';"'''',)judO = /."" [«'I' =
"')., '1',) + ("'" ('I' - "'M J" dO
1[('1('1' - "'), V(rpou»
+ (V"" '1«'1' - ",),u)jdzdll,
where the function u in the l..t integral is understood to he an extension of the same function de6ned on {JD. The last integral is split into two terms:
1 «'1('1' - "'), Vrp.)u + (V"" '1('1' - "')o)ujdzdll
+ 1[('1('1' -
",),rp,Vu)
+ (V""
('I' - "').Vu)jdzdll.
Noticing that
1 u(V"', '1('1' - ",).)dzdll
= - 1 ('1('1' -
"'), U8V", + uV",,)dzdll,
we have
11,,('1('1' - "'), '1('1' - "')8)dzdlll
= ~11 !IV(rp =
~11IV(rp -
1 ~ 211'1' -
",)I' . Udzdlll
"')1'· U8dzdll l
, "'"H'(D)
·lI u llc'(8D).
The remaining three terms are estimated by
In summary,
I(H. - H., ,,)I
:5 c [11'1' -
"'"~'(D) + 11'1' - "'"H'(D) (211"'IIH'(D) + IIrpIIH'(D»)]
lI u llc'(8D) :5 C(lI" - vIlH"" lIuIlH"" IIvIlH''') II" - tJIIH"'(8D) ·lIullc'(8D).
266
App/icalioru 10 Honnoni< Map. and Minimol Surfacu
Since C""(lJD) is dense in M the domain of H can be extended to M, such that Vue M, H. E (C'(lJD»)", the dual of C'(lJD). The lemma is proved. We turn now to finding out tbe derivatives of J. V CT E C'(lJD), with 10"'(11)1 < I, define
p.(II) = II + £11(11),
for
lei < 1.
By definition, Po( II) = II, and d de P.(8) = 0"(11).
For any u E C'(lJD),
Generally speaking, however, Vue M, u 0 p;' does not satisfy the three point condition, so we do not know if it is in M. In order to find the derivative of J with respect to M, we need more work. Note that there is a conformal mapping W,: D - D satisfying
[.
wr::exp tUOp;l
(2i"'311')] -exp [2ii"] -3- ,
j = 0, 1,2,3.
If we define T,(II) = -ilnw,(e") Yr:
and
= Tr: 0 U 0 p;l,
then V,EM.
Lemma 5.2. Ifu E M is 8 critical point of J with respect to M, then the distribution H. = O. Proof. Choose a sequence u· E C'(lJD) such that u· _ u in X = C n H'/·. Observing that J is invariant under T" we have
S. 1'7ul Plateau Problem lor Minimal SurflJl:U
where
u: = T~
0
267
u" 0 p;l. Therefore
(J'('h~I.=o+~ oa)=o. Since
If. EM, and u is a critical point of J with respect to M, we have lim( ]'(u·),
which implies
~I.=o) ~ 0,
-( dU.) lim ]'(u·), d8 a 0
~
o.
Exploiting Green's formula, it follows that
(J'(u.),
~•. a) = (H~ •• a).
Therefore (H.. a) ~ 0 i.e.,
Hu
Va E e'(OD),
= O.
Theorem S.l. SUPJX16" that u E M is a critical point of] with respect to M. Then
O. We want to &how that V 6 E (0,1), 3 p E (6,,/6) such that
f l 0, 3 p E (6,./6) such that the arc length of the curve C. has the following estimate: t('P(C.))' :5
:5
(!c. (!c.
1'P.lds)' 1'P.I'ds) ·27rp
4ffM
:5 1n(1/0J" We choose 6 > 0 such that
and that at least two of the following three inequaliti... hold: V z E BD, and
i
= 0, I, 2.
One may assume c < Min dist;,+I(M)
-'"
.-'f 1
d'
--4
A'(M)
-'"
with p = 0,1,2, ... ,n -1,
and to compute the perturbed Laplacian p.
Ll.
We range the eigenvalues of
0:-:;
1
I-
= til' til' + +I(M)
_'" _
0
e-" !
A"(M)
It is easily verified that (E, (t2/''P;(x»)('P~P 0 'P;(x), j=l
where ('P~); is the j-component of 'P~, which is a vector in AP(M). For t > 0 large, the support of "'~ is concentrated in Uj=, UJ • These vectors are considered to be "approximate eigenvectors" for the operator
6f.
In order to prove (.), the fol\owing Rayleigh-Ritz principle is needed. Theorem (Rayleigh-Ritz]. A• .rume that A is a self-adjoint operator bounded below on a Hilbert space H. If A only possesses discrete spectrum,
consisting of eigenvalue. with finite multiplicities, A, ::; A, ::; ... ::; An ::; "', then (Ax,x) Aft == sup inf ~, ..... ~._,EH
IIxll' .
.ED(A)
.2:Eepan{¥"l •...•IP.. _l}.1.
Proof. According to the spectral decomposition theorem, (Ax,x) = LAiX~, where Xi = (X, ei), and ei is the ortho-normw eigenvector corresponding to Ai, i == 1.2, .... Therefore,
(Ax, x)
inf .ED(A) :l:Espan{el •... ,e ... _l}.1.
::;
ij;jj2
sup
inf
.. " ......"_,EH
.ED(A)
(Ax, x)
ij;jj2'
xEllpan{ "'1, .. - .!{}._l}.l.
On the other hand, V {'P"'" ,IPn-'} 0 such that
Then we have F'_I(t) (8 bounded operator with rBllk ~f ~ t(~
- .W) = t
...11=1 ...."""1 ..... ··· .... _.)' for t>O large,
~~-e
provided we take
1 •
...iWAn
.....~.~~._.
"'I, ... ,"'.-1
("', t~' "')
as a basis of the subspace 1m F'_I(t). Since
e > 0 is arbitrary, we have
lim >.:(t)
t~OQ
(2) ~-d
~_I =~.
t
> e!'. -
A:
We may 8SSurn. that ~ > 0, BIld then 3 d =~. According to case (1), we have
> 1 such that
< ~-d+l = ...
r
t~oo
>.W) > r t
t~oo
>':-d+J (t) > e!' t
-
- e!'
k-d+l -
k"
This proves our conclusion.
Theorem. Suppose that M is a compact, connected, orientable manifold. Then there exists a Riemannian metric 9 such that
C~
4. Morse Inequalities We have defined {Jp, m p , p = 0,1, ... I n in Sections 1 and 2. Now we are going to prove the following inequalities:
Witten', Prool 01 MONe Inequoliti ..
296
or, in a compact form, letting
pM(t) = LP,t', MI(t) we have
MI (t)
= Lm,t',
= pM (t) + (1 + t)Q(t),
where Q(t) is a formal power series with nonnegative coefficients. Let < < < Min{e!:.p+' I p = 0,1, ... ,n}. Fixing t large enough, we define a. new cohomology complex as follows:
°
X'
= Xr = {w E AP(M) I it is an eigenvector of '\~(t)
with eigenvalue
such that
ar,
'\~t(t) < -}. 0
According to the theorem in Section 3, we see that dimXP == m p '
p=O,l, ... ,n,
and we have (i) rtf: X· _ XP+l, rtf-': X' _ X·-l.
Claim. V wE X', we have arw = ar+'rtfw =
(rtf+"
'\~(t)w
rtf+1
with >'::'(t)
< '::'(t) '" 0,
4.
297
Morse Inequalities
and Since ~-l· wE XP-l, we see
W
=
-147. (BaRI) Bahri, A. and Rabinowitz, P., A minimax method for a class of Hamiltonian systems with singular poteutials, J. Ftincl. Anal. 82 (1989), 412--428. [BaR2) Bahri, A. and Rabinowitz, P., Periodic solutions of Hamiltonian systems of 3-body type, CMS Report #90-8, (1989), Uoiv. Wisconsin. (BaCI) Bartsch, T. and Clapp, M., Bifurcation theory for symmetric P'>' tential operators, preprint.
[BCPI) Bartsch, T., Clapp, M., and Puppe, D., A mountain pass theorem for actions of compact Lie groups, preprint. (Ben I) Bend, V., A new approach to Morse theory, Jaunt Advances in Hamillonian &y81em8 (G. Dell' Antonli, ed.), World Sci. Pub!. (Ben2) Bend, V., Some applications of the generalized Morse-Conley index, Con/. Semin. Mal. Uni •. BaTi 218 (1987), 1-32. [Ben31 Bend, V., A geometrical index for the group S' and some applications to the periodic solutions of ordinary differential equations, Comm. Pure AWl. Math. 34 (1981), 393-432. [Ben4) Bend, V., On tbe critical point theory for indefinite functiona1s in the presence of symmetries, TMAS 274 (1982), 533-572. (BeCI) Bend, V. and Coron, J.M., The Diricblet problem for harmonic maps from the disk into the Euclidean n-sphere, Analyse Nonlineaire 2 (1985), 119-141.
300
Rejef'f!nce3
[BePl[ Benci, V. and PaceUa, F., Morse theory for symmetric functionals in the sphere and application to 8 bifurcation problem, Nonlinear Anal.
9 (1985), 763-773. [BeRl] Benci, V. and Rabinowitz, P.H., Critical point theorems for indefinite functionals, Inv. Math. 52 (1979), 241-273. [BeLl] Berestycki, H. and Lions, P.L., Sbarp existence results for a class of semilinear elliptic problems, Bal. Brasil. Mat. 12 (1981), 9--20. [Berl] Berger, M.S., Nonlinearity and FUnctional Analysis, Acad. Press, 1977. (BePl] Berger, M.S. and Podolak, E., On tbe solutions of a nonlinear Diricblet problem, Indiana Univ. Math. J. 24 (1975), 837~6. (Birl] Birkhoff, G.D., Dynamical systems with two degrees of freedom, TAMS 18 (1917), 199--300. (BoFl] Bonic, R. and Frampton, J., Smooth functions on Banach manifolds, J. Math. and Mech. 15 (1966), 877-898. (Boll] Bott, R., Nondegenerate critical manifolds, Ann. of Math. 60 (1954), 24S-261. (Bot2] Bott, R., Lectures on Morse theory, old and new, BulL AMS 7 (1982), 331--M8. ',(BreI] Brezis, H" On a characterization of flow invariant sets, Comm. Pure . Appl. Math. 23 (1970), 261-263. [BrCl] Brezis, H. and Coron, J.M., Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), 203--215. [BrC2] Brezis, H. and Coron, J.M., Multiple solutions of H-systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984), 147-187. [BrBl] Brown, KJ. and Budin, H., On the existence of positive solutions for a class of semilinear elliptic BVP, SIAM J. Math. Anal. 10 (1979), 875-883. [Caml] Cambini, A., Sui lemma di Morse, Boll. Un. Mat. Ital. 10 (1974), 713--723. [CaLl] Castro, A. and Lazer, A.C., Critical point theory and tbe number of solutions of a nonlinear Dirichlet prohlem, Ann. Mat. Pura Appl. 70 (1979), 113--137. [Chal] Chang, KC., Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure AppL Math. 34 (1981), 693--712. [Cha2] Chang, KC., Morse theory on Banach spaces and its applications, Chinese Ann. Math. SeT. b4 (1983), 381-399. [Cha3] Chang, K.C., A variant mountain pass lemma, Sci. Sinioo, SeT A 26 (1983), 1241-1255. [Cha4] Cbang, KC., Variational method and the sub- aDd 5uper- solutions, Sci. Siniea, SeT A 26 (1983), 1256-1265. [Cba5] Chang, KC., Applications of homology theory to seme problems in differential equations, Nonlinear Functional Analysis (F.E. Browder,
ed.), Proe. Symp. Pure Math., AMS, 1986,
Re/err.nces
301
[Cha6j Chang, K.C., On" bifureation theorem due to Rabinowitz, J. Syst. Sci. Math. Sci. 4 (1984), 191-195. [Cha7] Chang, K.C., On the mountain pass lemma, Equadiif. 6, LN Math. 1192 (1986), 203-207. [Cha6) Chang, K.C., An extension on the minimax principle, Proc. Symp. DD3 (Chern, S.S., etc. ed.), Science Press, Beijing, 1986. [Cha9) Chang, K.C., On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Analysis TMA 13 (1989),527-537. [ChalO) Chang, KC., Heat Bow and boundary value problem for harmonic maps, Ana/IIBe Nonlin';ai .. 6 (1989), 363-396. (Chall] Chang, KC., Morse theory for harmonic maps, Variational Methods, Proc. of a Conf. Paris, June 1988, Berestycki, Coron, Ekeland, eds, Birkhliuser (1990), 431-446. [Cha12] Chang, KC., Infinite dimensional Morse theory and it. applications, Univ. de Montreal 97 (1985). (Cha13] Chang, K.C., Critical point tbeory and its applications, Shanghai Sci. Techn. Ptts. (1986), (in Chinese). (Cha14] Chang, K.C., Variational methods for nondilferentiable functional. and its applications to partial differential equations, J. Math. Ana/. Appl. 80 (1981), 102-129. [ChEl) Chang, K.C. and Eells, J., Unstable minimal surf.... coboundaries, Acta Moth. Sinico, New Ber. 2 (1986), 233-247. (ChJ] Chang, KC. and Jiang, M.Y., The Lagrange intersections for (CP",JlI.P"), Manuscripta Math. 68 (1990),89-100. [ChLl) Chang, KC. and Liu, J.Q., Morse theory under general boundary conditions, J. SYBtem Sci. & Math. Sci. 4(1991), 78-83. [ChL2) Chang, K.C. and Liu, J.Q., A strong resonance problem, Chinese Ann. Math. 11, B.2. (1990), 191-210. [CLZl) Chang, KC., Long, Y., and Zehnder, E., Fotced oscillations for the triple pendulum, Analysis et cetera, Rabinowitz, Zehnder, eds, Academic Press, 1990. [CWLl] Chang, KC., Wu, S.P., and Li, S., Multiple periodic solutions for an asymptotically linear wave equation, Indiana Math. J. 31 (1982), 721-731. [Clal] Clark, D.C., A variant of Ljusternik-Schnirelman theory, Indiana Math. J. 22 (1972), 65-74. [Cofl] Coffman, C. V., Ljustemik-Schnirelman theory, Nodal properties and Morse index, Nonlinear diffusion equations and their equilibrium BtateB I, Springer, (1988), 245-266. [Cof2] Coffman, C. V., Ljusternik-Schnirelman theory: complementary principles and the Morse index, Nonlinear AnalYBis, TAM, to appear. [Coni) Conley, C.C., Isolated invariant sets and the Morse index, CBMS Regional Conf,ren"" Serieo 38, AMS, 1978. [CoZl) Conley. C.C. and Zehnder, E., The Birkholf-Lewis fixed point theorem and a conjecture of V. Arnold, Inven~ Math. 73 (1983), 33-49.
302
[COZ2) Conley, C.C. and Zehnder, E., Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Moth. 37 (1984), 207-253. [CoWl) Costa, D. and WiDem, W., Points critique multiples de fonctionnelles invariantes, CR Acod. Sci. Po";" 298 (1984), 381-384. [Cotl} Coti-Zelati, V., Morse theory and periodic solutions of Hamiltonian systems, Ph.D. thesis, 'nieste, 1987. [Coul) Courant, R., Dirichlet Principk, Conformal Mopping and Minimal Surfaces, Wiley, Interscienee, 1950. [Danl} Dancer, E.N., Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math. 350 (1984), 1-22. [Diel) Dieck, T. Thm., Thlnsformation Groups, Walter de Gruyter, 1987. [Dinl) Ding, W.Y., Ljusternik-SchnireJman theory for harmonic maps, Acta Moth. Simco 2 (1986), 105-122. [DiLl) Ding, Y.H. and Liu, J.Q., Periodic solutions of asymptotically linear Hamiltonian systems, J. Sys. Sci. & Moth. Sci. 9 (1989), 30-39. [DoLl) Dong, G.C. and Li, S., On the infinitely many solutions of the Dirichlet prohlems for some nonlinear elliptic equations, Scient. Sinico (1982). [Dpul} Douglas, J., Some new results in the problem of Plateau, J. Moth. 'Phys. 15 (1936), 55-M. [Dul) Dumford, N. and Schwartz, J. T., Linear OperutorB, fIOl. 2, Wiley: interscienee, 1962. [DuLl} DUVl!.ut, G. and Lions, J.L., Lea iniquations en m.canique et en physique, Dunod, 1972. [EeLl) Eells, J. and Lemaire, L., A report on harmonic mops, Bull. LondDn Moth. Soc. 18 (1978), 1-68. [EeL2} Eells, J. and Lemaire, L., Another report on harmonic maps, Bull. London Moth. Soc. 20 (1988), 38!>-524. [EeSl) Eells, J. and Sampson, J.H., Harmonic mappings of Riemannian manifolds, JAMS 88 (1964), 109-160. [Ekel) Ekeland, I., Une thearie de Morse pour les systemes hamiltoniens convexes, Analyse Non/ineaire I, (1984), 19-78. [Eke2) Ekeland, I., Periodic solutions of Hamiltonian equation and a theorem of P. Rabinowitz, J. Dii!. Equo. 34 (1979), 523-534. [EkLl} Ekeland, I. and Lasry, J.M., On the number of closed trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Moth. 112 (1980), 283-319. [Fadl) Fadell, E.R., Cohomological method in non-free G-space with applications to general Borsuk-Ulam theorems and critical point theorems for invariant functionals, Non/inoor FUnc!. Anal. and Its Appl. 1-45 (Singh, S.P., ed.), Reidel Pub!. Co. (1986). [FHR1) Fadell, E.R., Husseini, S.Y. and Rabinowitz, P.H., Borsuk-Ulam theorems for arbitrary S' actions and applications, TAMS 274 (1982), 345-360.
303
lFaRI) Fadell, E.R. and Rabinowitz, P.H., Bifurcation for odd potential operators and an alternative topological index, J. Fbnct. AnaL 26 (1977), 4IHi7. [FaR2) Fadell, E.R. and Rabinowitz, P.H., Generalized cobomological index theories for Lie group actions with an application to hifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139-174. [Fell) Felmer, P.L., Periodic solutions of spatially periodic Hamiltonian systems, CMS Report #9(}.3, Univ. of Wisconsin, (1990). [Flol) Floer, A., A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ery. Thea. and Dvnamic SIP. 7 (1987), 93-103. (FI02) Floor, A., Proof of the Arnold conjecture for surfaces and generalization to certain Kahler manifolds, Duke Math. J. 53 (1986), 1-32. (FI03) Floer, A., A Morse theory for Lagrangian intersections, J. Diff. Geam. 28 (1988), 513-547. (FI04) Floer, A., A cuplength estimate for lagrangian intersections, Comm. Pure Appl. Math, XLII (1989), 335-357. (FIZl) Floor, A. and Zehnder, E., The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems, Ery. and Dvnam. SlPt. (1987). 1F0MI) Fonda, A. and Mawhin, J., Multiple periodic solutions of conservative systems with periodic nonlinearity, preprint, Univ. Catholique de Louvain, 1988. (For I) Fortune, C., A symplectic fixed point theorem for C pn, Invent. Math. 81 (1985), 29-46. (FoWl) Fournier, G. and Willem, M., Multiple solutions of the forced double pendulum equations, Analyse non/ineaire, .upplement au 1101. 6, (1989), 259--281. (FoW2) Fournier, G. &ad Willem, M., Relative category and the calculus of variations, preprint, 1988. (Fral) Franks, J., Generalizations of the Poincare-Birkhoff theorem, p..... print.
(Fuel) FuCik, S., Remarks on a result by A. Ambrosetti and G. Prodi, Boll. Un. Mat. ltal. 11 (1975), 259--267. (Ghol) GhoU8S0ub, N., Location, multiplicity, &ad Morse indices of MinMax critical points, Z. Reine und Angew. Math., in press. (GiTI) Gilharg, D. and Trudinger, N.S., Elliptic Pa71ial Differential Equation. of Second Order, Springer-Verlag, 1977. (Gorl) Gordon, W.B., Conservative dynamical systems involving strong forces, TAMS 204 (1975), 113-135. (Grel) Greenherg, M.J., Lectures on A/gebroic Topology, Benjamin, 1967. (Grml) Gromoll, D. and Meyer, W., On differentiable functions with iso. lated critical points, Topclogy 8 (1969), 361-369. (Haml ( Hamilton, R., Harmonic maps of manifolds with boundary, Lecture Notes 471, Springer, 1975.
304
a
(Helll Helffer, B., Etude du laplacien de Witten associ'; une fonction de Morse degem!ree, (1985) preprint. (Henil Henniart, G., Les inegalites de Morse, Semmaire Bourbaki, 1983/ 84,617, 43-00. (HesI) Hess, P., On a nonlinear elliptic boundary value problem of tbe Ambrosetti-Prodi type, BoiL Un. Mat. ItBl. (5) (1980), IT-A, 187192. (Hes21 Hess, P., On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. PDE 6 (1981), 951-961. (HeKII Hess, P. and Kato, T., On some linear and nonlinear eigenvalue problems with inde6nite weight functions, Comm. Partial DiiJ. Equatio ... 5 (1980), 999-1030. (Hinll Hingston, N., Equivariant Morse theory and closed geodesics, J. DijJ. GeomelnJ 19 (1984), 8:>-116. (HoO) Hofer, H., Variational and topological methods in partially ordered Hilbert spaces, MatA. Ann. 261 (1982), 493-514. (Hof2) Hofer, H., A note on the topological degree at a critical point of mountain pass type, PAMS 90 (1984), 309-315. (Hof3] Hofer, H., Ljustemik-8chnirelman theory for Lagrangian intersections, AnBlyse non/in'aire 5 (1988), 465-500. (Hof41 Hofer, H., A geometric description of the neighbourhood of a critical point given by the mountain pass theorem, J. London Math. Soc. 31 (1985), 566-570. (HusI) Husemoller, D., Fiber Bundle., McGraw-Hill, 1966. (JiWI] Ji, M. and Wang, G.Y., Minimal surfaces in Riemannian manifolds, (1988) preprint. (JiaI] Jiang, M.Y., An existence result for periodic solutions of a class of Hamiltonian systems, Ke:r:ue Tong""o 33 (1988), 1679-1681. (Jia2) Jiang, M.Y., A remark on periodic solutions of singolar Hamiltonian systems, Acto Math. Sinica, in press. (JosI] Jost, J., The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values, J. DifJ. GeomelnJ 19 (1984), 393-401. (Jos2) Jost, J., Confonnal mappings and the Plateau-Douglas problem in Riemannian manifolds, J. Reme Angew. Math. 359 (1985), 37-54. [JoSI) Jost, J. and Struwe, M., Morse Conley theory for minimal surfaces of varying topological types, Invent. Math. 102 (1990), 46:>-499. (KaWI] Kazdan, J.L. and Warner, F.W., Remarks on some quasilinear elliptic equations, Corum. Pure Appl. Math. 28 (1975), 567-597. (Kell] Kelley, J.L., General Topology, D. Van Nostrand, 1955. (Klil] Klingenberg, W., Lectures on Closed Geodesics, Springer, 1978. )KraI] Krasnoselskii, M.A., Topological Method. in the Theory of Nonlinear Integrol EquatioRl, Pergamon, 1984. )Kuil) Kuiper, N., C'-equivalence of functions near isolated critical points, Symp. on Infinite-dimensional topology, Annab of Math. Studies 69,
305
Princeton Univ. Press, 1972. [Lazl] Lazer, A.C., Introduction to multiplicity theory for boundary value problems with asymmetric noulinearities, LN in MBth. 1324 (1988). [LaMl] Lazer, A.C. and McKenna, P.J., On the number of solutions of a nonlinear Dirichlet problem, J. MBth. Anlll. Appl. 84 (1981), 282-294. [LaM2] Lazor, A.C. and McKenna, P.J., Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues, Comm. PDE 10 (1985), 107-150. [LaS1] Lazor, A.C. and Solimini, S., Nontrivial solutions of operator equations and Morse indices of critical points of MinJll8J[ type, Nonlinear Analysis TMA, 12.8 (1988), 161-775. [Leml] Lemaire, L., Boundary value problems for harmonic and minimal maps of surfaces into manifolds, Ann. Scuola NOfTn. Sup. PUB 9 (1982), 91-103. [Lil] Li, S., Multiple critical points of periodic functional and some applications, lCTP IC-86-191. [LiLI] Li, S. and Liu, J.Q., Morse theory and IIS)'JIlptotica1ly linear Hamiltonian systems, JDE 78 (1989), 53-73. [Liul] Liu, J.Q., A Morse index of a saddle point, preprint. [Liu2] Liu, J.Q., A generalized saddle point theorem. J. DiU. Eq. (1989). [Liu3] Liu, J.Q., Doctoral thesis, Academy of Science, Beijing, 1983. [Liu4] Liu, J.Q., A Hamiltonian system of second order, ICTP preprint. [Llol] Lloyd, N.G., Degree Thoory, Cambridge Univ. Press, Cambridge, 1977. [Lonl] Long, Y., Maslov index, degenerate critical points and asymptotically linear Hamilton systems, preprint. [Lon2] Long, Y., The structure of the singular symplectic matrix set, preprint. [Lon3] Long, Y., Multiple periodic solutions of perturbed superquadratic second order Hamiltonian systems, TMAS 311 (1989), 749-780. [LoZI] Long, Y. and Zehnder, E., Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, preprint. [LuSI] Lupo, D. and Solimini, S., A note on a resonance problem, Proc. Royal Soc. 0/ Edinburgh 102A (1986), 1-7. [Marl) Martin, R.H., Nonlinear Operntors and DiJTerentiBI Equa!ions in BBnach Spaces, John Wiley" Sons, 1976. [MaPl] Marino, A. and Prodi, G., La teoria di Morae per spazi di Hilbert, Rend. Sem. Mal. Univ. Padova 41 (1968), 43-68. [MaP2) Marino, A. and Prodi, G., Metodi perturbativi nella teoria di Morse, BoIL Un. Math. ltlll. Suppl. Fasc. 3 (1975), 1-32. [Mawl) Mawhin, J., Probl.mes de Dirichlet, variationnels nonlineaires, Univ. de Montreal, 104 (1987). [Maw2] Mawhin, J., Forced second order conservation systems with periodic nonlinearity, Anlllyse nonlineai..., Suppl. au vol. 6 (1989), 415-434. [MaWl] Mawhin, J. and Willem, M., Multiple solutions of the periodic BVP for some forced pendulum-type equations, J. DiJT. Equa. 52
306
Re/e<mcU
(1984), 264-287. [MaW2) Mawbin, J. and WiIlem, M., Critical point theory and Hamiltonian systems, Appl. Math. Sci 74 Springer-Verlag, 1989. [Mey1) Meyer, W., Kritische Mannigflatigkeiten in Hitbertmannigflatigkeiten, Math. Ann. 170 (1967), 45-66. [Mill) Milnor, J., Mon. Theorv, Princeton Univ. Press, Princeton, 1963. [Mil2) Milnor, J., Topology from the Differential Viewpoint, Univ. Press of Virginia, Charlottesville, 1969. (Morl) Morse, M., Relations between the critical points of a real function of n independent variahles, TAMS 27 (1925), 345-396. [Mor2) Morse, M., The calculus of variations in the large, Amer. Math. Soc. CoIl. Pub. No. 18, Providence, 1934. [MaCl) Morse, M. and Cairns, S.S., Critico/ Point Theo.,." in Global Anal",is and Diff....ntial Topology, Academic Press, New York, 1969. (MoTl) Morse, M. and Tompkins, C., The existence of minimal surfaces of general critical types, Ann. 01 Math. 40 (1939), 443-472. [MoT2) Morse, M. and Tompkins, C., Unstable minimal surfaces of higher structure, Duke Math. J. 8 (1941), 350--375. (Nil) Ni, W.M., Some minimax principles and their applications in nonlin"!"" elliptic equations, J. d'Analy.. Math. 37 (1980), 248-275. (Nikl) Nikol'ski, S.M., Approximation oll'Unctiona 01 SetJeral Variables and Imbedding Theorema, Springer-Verlag, 1975. (Nir I) Nirenberg, L., Topic.< in nonlinear functional analysis, Courant Institute Lecture Notes, New York, 1974. [Nir2) Nirenberg, L., Variational and topological methods in nonlinear problems, Bull. AMS 3 (1981), 267-302. [Nir3) Nirenberg, L., Comments on nonlinear problems, Le Matema!ische 16 (1981). (Nir4) Nirenberg, L., Variational Methods in Nonlinear Problems, LN in Math. 1365, 1989, Springer-Verlag, 100-119. [Ossll Osserman, R., A Survey 01 Minimal Surfaces, Van Nostrand, 1969. [Pall) Palais, R.S., Morse theory on Hilbert manifolds, Topology 2 (1963), 299-340. (PaI2] Palais, R.S., Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1-16. (Pal3] Palais, R.S., Ljusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. (Pal4] Palais, R.S., Critical point theory and the minimax principle, Global Anal",is, Proc. Symp. Pure Math. 15 (ed. Chern, S.S.), AMS, Providence (1970), 185-202. [PaS1) Palais, R.S. and Smale, S., A generalized Morse theory, BAMS 70 (1964), 165-171. (Pitl] Pitcher, E., Inequalities of critical point theory, BAMS 64 (1958), 1-30.
(PuSl) Pucci. P. and Serrin. J .• The structure of the critical set in the mountain pass theorem. TAMS 91 (1987). 115-132. (PuS2) Pucci. P. and Serrin. J .• Extensions of the mountain pass theorem. J. hnc!. Anal. 59 (1984). 185-210. (Qil) Qi. G.J .• Extension of mountain pass lemma, Kexue Tongbao 32 (1987). (Rabl) Rabinowitz. P.R .• Variational methods for nonlinear eigenvalue problems. Eigenvalues of Nonlinear Problems. Ed. Cremon.... Rmna (1974). 141-195. (Rab2) Rabinowitz. P.R .• A bifurcation theorem for potential operaton. J. Funct. Anal. 25 (1977). 412-424. (Rab3) Rabinowitz. P.R.. Periodic 8OIutioll8 of Hamiltonian systems. Comm. Pure Appl. Math. 31 (1978). 157-184. (Rab4) Rabinowitz. P.R.• Multiple critical points of perturbed symmetric functionals. TAMS 272 (1982). 753-770. (RaM) Rabinowitz. P.R .• Minimax methods in critical point theory with applications to differential equatioll8. CBMS Reg. Conf. Ser. in Math. 65 AMS (1986).
(Rab6) On a class of functionals invariant under a zn actioo. CMS Report #88-1. Univ. of Wisconsin. Madison (1987). (Reel] Reeken. M.• Stability of critical points under small perturbations. Part 2. analytic theory. Manwcripta Math. 8 (1973). 6~92. (Rotl] Rotbe. E .• Morse theory in Hilhert space. Rocky Mountain J. Math. 3 (1973). 251-274. (Rot2) Rothe. E.. Critical point theory in Hilbert space under regular boundary conditions. J. Math. Anal. Appl. 36 (1971). 377-431. (Rot3) Rotbe. E.. On the connection between critical point tbeory and Leray Scbauder degree. J. Math. Anal. A,ppl. 88 (1982). 265-269. [RoW) Rotbe. E.. Critical point theory in Hilhert space under general boundary conditions. J. Math. Anal. Appl. 2 (1965). 357-409. [Rybl) ~bakowski. K.P.. The Homotopy Indez and Partial Differential Equations, Springer-Verlag, 1987. [SaUl) Sacks. J. and UhJenheck. K.. The existence of minimal immenions of 2 spberes. Annales oJ Math. 113 (1981). 1-24. [SaU2) Sacks. J. and Ublenbeck. K.. Minimal immersions of cI06ed illemann surfaces. TAMS 211 (1982). 63~2. [SaZl) Salamon. D. and Zehnder. E .• Floer homology. the Morse index. and periodic orbits of Hamiltonian equations, preprint.
[ScJl) Schwartz. J.T .• Nonlinear Functional Analysis, Gordon and Breach. 1969. [SeRl) Schoen. R.. Conformal deformation of a illernanni8ll metric to con· st8llt scalar curvature. J. Diff. Geom. 20 (1984). 479-495. [Schl] Schoen. R. and Yau. S.T., Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Annale. oJ Math. 110 (1979). 127-142.
&/erencu
308
[Shil] Shiffman, M., The Plateau prohlem for minimal surfaces of arhitrary topological structure, Amer. J. Math. 61 (1939), 85:H!82. [Sikl] Sikarov, J.e., Points fixes d'uo symp1ectomorphisme homologue d'identit.e. J. Dil!. Geom. 22 (1985), 49-79. [Soli] Solimini, S., Existence of a third solution for a class of BVP with jumping nonlineBrities, Nonlinear AnalV'" TMA, (1983), 917-927. [SoI2] Solimini, S., Morse index estimates in Min-Max theorems, Manuscripta Math. 63 (1989), 421--454. [So13] Solimini, S., On the solvability of some elliptic PDE with linear part at resonance, JMAA 117 (1986), 138-152. [Spal] Spanier, E.H., A/gebrnic Topology, McGraw-Hill, 1966. [Strl] Struwe, M., Infinitely many critical points for functionals which are not even and applications to nonlinear BVP, Manuscripta Math. 32 (1982), 753-770. [Str2] Struwe, M., On the critical point theory for minimal surfaces spanning a wire in R", J. Rein. Angew. Math. 349 (1984), 1-23. [Str3] Struwe, M., A Morse theory for annulus type minimal surfaces, J. Reine Angew. Math. 368 (1986), 1-27. [Str4] Struwe, M., On the evolution of harmonic mappings, Commet. Math. Helvetici 60 (1985), 558-581. [StrS] Struwe, M., Plateau Problem and the Calculus of Variations, Princeton Univ. Press, 1988. [Szul] Szulkin, A., A relative category and applications to critical point theory for strongly indefinite functionais, preprint.
[Thul) Taubes, C.H., A framework for Morse theory for the Yang-Mills functional, preprint. [Thu2) Thubes, C.H., Minimax theory for the Yang-Mills-Higgs equations, Comm. Math. Phy>. 97 (1985), 473-540 [Tial] Tian, G., On the mountain pass theorem, Kexue Tongbao 29 (1984), 1150-1154. [Trol] Tromba, A., On Plateau prohlem for minimal surfaces of high genus in Rn I preprint.
[Tr02] Tromba, A., Degree theory on oriented infinite dimensional varieties and the Morse number of minimal surfaces spBIloing 8 curve in
R" I 1,
TAMS 290 (1985),385--413; 2, Manuscripta Math. 48 (1984), 139-161. [Tr03] Tromba, A., A general approach to Morse theory, J. Dil!. Gcom. 12 (1977), 4H15. [Uhll) Uhlenheck, K., Morse theory on Banach manifolds, J. FUnct. Anal. 10 (1972), 430--445. [UhI2] Uhlenbeck, K., Morse theory by perturbation methods with applications to barmonic maps, TAMS 267 (1981), 569-583. [Viti] Viterbo, C., Indice de Morse des points critiques obtenus par mini· max, Analyo. nonliniaire 5 (1988), 221-226. [WaTI] Wang, T.Q., Ljustemik-Schnirehnan category theory on closed subsets of Banach manifolds, preprint.
Re/erencu
309
[WaZl] Wang, Z.Q., Equivariant Morse theory for isolated critical orbits and its applications to nonlinear prohlems, LN in Math. 1306, Springer, 1988, 202-22l. (WaZ2] Wang, Z.Q., On a superlinear elliptic equation, Analyse nonlineaire 8 (1991), 43--58. [WaZ3] Wang, Z.Q., Multiple solutions for indefinite functionals and applications to asymptotically linear prohlems, Math. Sinica, New Serl.. 5 (1989), 101-113. (WaZ4] Wang, Z.Q., A note on the deformation theorem, Acta Math. Sinica 30 (1987), 106--110. [Warl] Ward, J.R., A boundary vaIue problem with periodic nonlinearity, Nonlinear Analy.ri&, TMA 10 (1986), 207-213. [Wasl] Wasserman, A.G., Equivariant differential topology, Topology 8 (1969), 127-150. [Weill Weinstein, A., Bifurcation and Hamilton's principle, MZ 159 (1978), 235--248. [Wei2] Weinstein, A., Critical point theory, symplectic geometry and Hamiltonian systems, Proc. 1983 Beijing Symp. on DD4, Gordon Breach, (1986), 261-288. [Witl] Witten, E., Supersymmetry and Morse theory, J. DijJ. Geom. 17 (1982), 661-692. [Wul] Wu, S.P., The nontrivial solution for .. class of quasilinear equations, Applied Math. A.J. of Chine.. Univ. 3 (1988), 33!M145. (YaSl] Yakubovich, V.A., Starzhinskii, V.M., Linear DijJerential Equations with Periodic Coefficients, Jobn Wiley &< Sons, 1975. [Yanl] Yang, X.F., The Morse critical groups of minimax theorem, preprint.
INDEX OF NOTATION
df
dilferential of /
19
f. K K.
level set of f, DOt above the level a
20
critical set critical set with critical value a
19 21
Paiais-Smale condition
20
(PS) exp(-)
exponential map
a
Laplacian operator
aM'"
Laplace-Beltrami operator
a
tension operator
"i1
gradient operator
mesO
measure
235
IAI
measure of A
(j)
direct sum
72 142 230 230
141,229
wT
transpose of the matrix W
175 ISO 182
A'A
loop space on A
204
IA
cardinal number of the set A
Fix(-)
fixed point set
A
exterior product
i..,
interior product
216 216 277 277
Id
identity operator
99
INDEX
Arnold oonjectwe on fixed point&, 216 on Lagrangian interaectlo"", 217
Euler characteristic, 6
Finsler manifold, 18
FInsIer structu..., 15 F\'edbolm operator, 47, 97 Banach mauifold, 14 Betti number, 3 bifurcation, 129, 161 blow up lIIl8IysIs, 232 Bott 79, 206
cap product, 9 catesorY, 105 relative catesorY, 109 oonformaI group, 360 convex set, 60 locally, 60 ant Lebesgue lemma, 268 critical group, 32 critical mauifold, 69 aitieal orbit, 61 critical point, 18 w.r.t. a locally convex closed.set, 62 critical . , 18 critical \'aIue, 18 cuplength, 9 cup product, 9
eo....
Deformation lemma, 21 Deformation retract, 20 strong, 21 Deformation theorem fim,29 equivariant first, 67 second, 23 equhariant~nd,68
degenerate critical point, 43
non,33;'41., 1
G-action, 66 G-cohomology, 75 G-critical group, 76 G-equharlant, 66 G-apace,66 GalerkiD approximation, 111 ~ boundary oonditlon, 55 genus, 96
cosenus,96 gradiem Dow, 19 Gromoll-Meyer pnir, 48 Gromoll-Meyer theory, 43
H8QJiItonian system, 179 handle hody theorem, 38 harmonic map, 229
bannoJiic ooclUation, 285 best Dow, 229 Hilbert Rielll8lUlian manifold, 19 Hilbert _ bundle, 70 Hodge theory, 274 homology group, 3 relative, 3 homotopy group, 12 relative, 12 Hurewicz isomorphism theorem, 13 hyperbolic operator, 41 invariant function, 111
isolated eritical manifold, 69
isolated critical orbit, 74 isolated eritical point, 43
312
Index
Jacobi operator, 251 jumping nonlinearity, 164
Poincare-Hop! theorem, 99 projective space
Kunneth rormula, 5,
real,6, 11 complex,6, III pseudo gradient vector field, 19
Landesman-Lazer coodltion, 153 Leray-Scbauder degree, 99 link homological, 84 homotoplcal, 83 Ljustemik-Schnire1man theorem, 105 1oeaI1y convex set, 60
regular set, 18
Marino-Prodi theorem, 53 G-equivariant, 80
Maslov index, 183 maximum principle, 143 strong, 143 minimal surface, 260 mini~8X principle, 87 Morse'decomposition, 250
Morae index, 33
regular point, 18 regular value, 18
saddle point reduction, 188 shifting theorem, 50 Sobolev embedding, 141 Sobo)evspace, 141, 231 splitting theorem, 44 strong resonance, 156 subordinate classes, 10
Bubsolution, 145 supersolutioD, 145 symplectic form, 215
symplectic matrix, 183
Morse inequality, 36, 79 Morse lemma, 33
Morse-.Tompkins-Shiffman theorem,
271
tangent bundle, 15 cotangent bundle, 15
Morae type number I 35 mountain pass point, 90 variational inequality, 65, 177 vector bundle, 15
Nemytcld operator, 141 Ronnal bundle, 70
Palais-Smale condition, 20 w.r.t. a convex Bet., 62
(PS)·,117 Palaia theorem, 14 pendulum, 209 periodic solution, 179 perturbation on critical manifold, 131
Uhlenbeck's method, 136
Plateau problem, 260
Witten complex, 282
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and Deportment of Mothematics Rutgen. University New Bnmswick, NJ 08903 U.s.A. ProS,.... in Nonlinear Diff'mllial Equa/_ and Th