Isotopic linking and critical points of functionals

Nonlinear

Analysis,

Pergamon

Theory, Methods&Applications, Vol. 30, No. 7, pp. 41454149, 1997 Proc. 2nd World Congress of Nonlinear Analysts 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

PII; SO362-546X(97)00161-2

ISOTOPIC

LINKING

AND CRITICAL KYRIL

Uppsala

Key words flow,

and

University,

phruses: Linking,

Ljusternik-Schnirelman

OF FUNCTIONALS

TINTAREV

P.O.B.

minimax

POINTS

480, Uppsala

theorems,

multiple

751 06, Sweden’ critical

points,

pseudogradient

category.

Let E be a Banach space. We shall consider a classZ of isotopies on E defined as the class of functions h E C([O, l] x E; E), bounded on every bounded set, such that h1(0,x) = x, for each t E 10,11,h(k -1is a homeomorphism on E, and the function h given by h(t, .) = h-‘(t, .) is bounded on bounded sets. The latter implies that h E Z as well. We shall define topological linking as an axiomatic property. DEFINITION 1. Let B be a nonempty subset of E. We shall say that a collection Leg of bounded subsets of E is a linking classof B if the following relations hold:

non-empty

AcLCg+AnB=&

(1)

3cB > 0,VA E &, d(A, B) < CB.

(3)

We can now outline the content of the present note. We state below a version of a standard minimax theorem that uses the axiomatic linking; we give two equivalent definitions of a linking class which is the maximal classsatisfying Definition 1; we recall a symmetry property for linking and deal with a modification of the Ljusternik-Schnirelman category. 2. Let f E C’(E) and assumethat f’ is uniformly continuous on all bounded sets. Let B c E and A E LCgbe nonempty bounded sets, sich that THEOREM

sup f(A) < inf f(B).

(4) Then the number

is

finite and there is a sequence 21;satisfying f’(Zk)

(6) ‘Research

supported

by an NFR

grant

F-MA

+

0, f(zk)

10442-302

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+

c.

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PROOF. First note that c 5 sup f(A) < 00, since the condition on f’ implies that f is bounded on bounded sets. By (3), and since B is bounded, there exist R > 0 and y E S such that for every A’ E LB, A’ II BR(Y) # 0. Therefore, inf A'E.CB~EA'

sup f(z)

1

inf

f(z)

> -co.

zEBR(Y)

Existence of a sequence satisfying (6) is then derived from the Palais’ proof of the crtitical point theorem (cf. [2] or [3]), i f one notes that the integral flow of the Palais pseudogradient is a deformation of the class Z, if it is uniformly bounded on bounded sets. Indeed, since the gradient of f is uniformly continuous on all bounded sets, the pseudogradient in the Palais’ construction may be selected uniformly bounded on the same sets, which yields the same property for the flow. DEFINTION 3. We shall say that a map h E C([O, l] x E; E) is of a class Z’ if for every c > 0 the map hL(t, .) = h((1 - c)t, .) is of class Z and h(t, .) converges to a constant as t --t 1, uniformly on bounded sets. We will say that ‘p E 2s if Q : E -, E is a homeomorphism, and both Q and 'p-l are bounded on bounded sets. THEOREM 4. Assume that B c E is a closed bounded nonempty sets and that E \ B is connected. If A c E \ B is a nonempty closed bounded set with the connected complement, then the following statements are equivalent: (i) h E 2’ =+ h([O, l] x A) II B # 8; (ii) there is no Q E Zc and an open ball V such that B C Q(V), and A C E \ Q(V). When the latter is true, we will say that A E lg. Moreover, (7)

A E .C; =a B E L;,

.Ci satisfies Definition 1, and any class Cg satisfying Definition bounded sets A with connected E \ A is contained in .$.

1 and consisting

of closed

PROOF. Let us say that A E Pi (resp. .Cg) if A is a closed bounded set, E \ A is connected and A satisfies (i) (resp.(ii)). L et us fix an arbitrary class leg satisfying Definition 1 and consisting of closed bounded sets A with connected E \ A. 1. IfA~~~,leth~2’beamapsuchthat~h([O,1]~A)~B=0andh(l,A)={y}. Since E \ B is path-connected, we can select a 7 E C([O,oo)) such that y(0) = y and ]]7(s)]] -+ 00 when s --+ 00. Since B is bounded, d(7([0, oo)), B) > 0. Let E > 0 be so small that for every B). Let N > 0 and define h&t,z) = h(t,z) for 5 E A, ]]h(l - E,Z) - y]] < $(y([O,co)), t E [O,l -c] and hN(t,z) = h(l -6,~) -y+$N(t+~1)) for t E [l -c,l]. If A E LB, then B) + co when for every N, h~(l,A) E Le. However, this cannot be true, since d(h~(l,A), N ---f co, in contradiction to (3). This implies (8)

.cB c L&

2. If A 4 f$, there is a Q E Zc and an open ball V centered at some y E E, such that B C Q(V), and A C E \ Q(V). Let g(t,z) = (1 - t)z + ty. Let h(t).) = cp o g(t, .) o 'p-l. Clearly, h E Z’, h( [0, l] x B) II A = 0 and therefore B 4 CA. Then A $$ & by Proposition 1.1, [4]. Therefore, (9)

Jzg cc;;.

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3. Note that Pi satisfies the axioms of Definiton 1. Indeed, (1) and (2) are obvious. To verify (3), assume that. there is a sequence A,, E &g such that d(A,, B) > n. Let V be an open ball containing B. Then there exists an n such that A, n V = 0. This implies that A, 4 ~g, a contradiction. 4. Since the choice of LB was arbitrary, we can substitute rCg = Pg in (8), which, together with (9), yields J$ C J!& C pg. We can now use the notation Li for Pi = J$$. From (8) now follows that LB C J$,, and we have shown f$, satisfies the axioms of Definition 1. Finally, (7) follows from the corresponding identity for Li verified in [4].

REMARK 5. In general, A E CB does not imply B E t A: this can be observed by taking first the maximal class Ls, a set, A E ,C% and a maximal class JZ~, then choosing a set B’ E Li that is not 2-homotopic to B in E \ A. Ch oosing f> as an 2-homotopic (in E \ A) class of B’ we conclude that B 4 & while A E COB. REMARK 6. Isotopies in Definition 1 and Theorem 4 cannot be replaced by homotopies: Theorem 4 yields linking of an infinte-dimensional sphere S1 to a finite dimensional sphere Sz if Sz is linked to &. This is not true in the sense of Pi if any contraction is allowed. The following example shows that two infinite dimensional spheres in the similar geometric position do not, link even in the sense of Definition 1. PROPOSITION 7. Let H be a separable Hilbert space, V c H, W c H, and w. E H be such that V@W@Rwo = H. Let Pv, Pw and PO denote orthogonal projectors to the corresponding subspaces. Let A = {z E H : 11~11 = 1, Pwz = 0) and B = (3 E H : /[cc- wo[I = 1, PVX = 0). If dim V = dim w = 00 and LB is any class satisfying Definition 1, then A $ LB. PROOF. Let {uk, k E N,w 0, and Wj, j E N} be an orthonormal basis in H, subordinated to the orthogonal sum V @ Rwo $ W = H, and let ck, q and dj denote corresponding coordinates of an element x E H. Let us define the following linear isotopies: c&X

=

qWo

+

c

(ck

dk

COS t +

Sin

t)?&

+

c

&X

=

t

(-ck

Sin

(ck

COS t +

dk COS t)wk,

+

&:N

&N QWo

+

c

&k--1W2k-1

+

c

sin t)Dk +

d2k

&N

&N +

c

(-Ck

Sin

d2k COS t)WZk,

t +

LEN stx

It is immediate shows that

that ~~Qtz~~ = llR+ll

= ~~x~~f or a 11 real

(10)

A f~ QSB = 0 for s f (-7~/2,7r/2),

(11)

&An

t.

Q+B = 0 for t E [0, r/2],

and (12)

5’&12A

n

Qn@ = 0 for t E [0, oo).

=

x + tw1.

An elementary

computation

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Let now Tt = Q;’ for t E [0,7r/4], Tt = Qi/.,Rt-*,d for t E [7r/4,37r/4], and Tt = Q$4St--3a/4Rnp for t E [3s/4,oo). Let N > P and TN(t,-) = TN~, for t E [O,l]. Then TN E 5, TN([O, l] x A) n B = 8 and d(TN(l, A), B) -+ co as N -+ 00. This contradicts (3). We will now define Z of deformations.

a modification

of Ljusternik-Schnirelman

category

based on the class

DEFINITION 8. Let B C E, B # E, be a nonempty closed set with a connected complement and let the class .CB be fixed. Let A c E \ B b e a closed bounded set. We shall say that catBA is the smallest integer m such that there exist a covering of A by open sets Ok c E \ B and mappings hk E 20, k = 1, . .. . m such that for every t, hk(Ok) is an open ball. It is easy to see that this category to these properties when homotopies

is an index (cf .[3]) and that are restricted to 2.

it is maximal

with

respect

PROPOSITION 9. The category defined above satisfies the following properties: (i) catsA 2 O,cat~A = 0 e A = 0, (ii) A c A’ =+-&B(A) 5 catgA’, (iii) catB(A U A’) 5 catgA + cutgA’, (iv) h E 2,h([O, l] x A) C E \ B + cuts(h(l,A)) = catBA, _ (v) A is compact 3 catBA < 0;) and there is an open neighborhood N of A such that N C E \ B and catBA = CatBN, (vi) A is a finite set + catnA = 1. Moreover, category.

any integer-valued

mapping

satisfying

(i-vi)

has values

less or equal

than

the

The proof is repetitive of the correspondent proof for the standard Ljusternik-Schnirelman category and is omitted. Since the standard Ljusternik-Schnirelman category also satisfies (i-vi), it has smaller or equal values than the category of Definition 8. Since an infinitedimensional sphere can be contracted in itself, but cannot be contracted in the complement of its center by a contraction of the class T , the standard Ljusternik-Schnirelman category of the spehre in the complement of its center, cutE\{,$$, equals 1, while the corresponding category in the sense of Definition 8 equals 2. Since conjugations with integral flows bounded on bounded sets preserve the classes Z and Z’, the number of critical points of a functional (with derivatives, uniformly continuous on all bounded sets) can be estimated from below by the modified category, and might assume larger values. We formulate a conjecture concerning the modified category.

CONJECTURE 10. Let A, B complements. (13)

c

E be two non-empty

closed bounded

disjoint

set with

connected

Then

catBA = catAB,

where category is understood in the sense of Definiton 8. we can prove this conjecture for two following particular cases. First, if CutBA = 1, then B 6 Li which is equivalent, by Th eorem 4, to A 6 Li and so catAB = 1. Second, if E is a Hilbert space, B is a finite-dimensional sphere and A is any set satisfying the assumptions

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of Conjecture 10, then CatBA = 2 H catAB = 2. In fact, it suffices to verify that catBA 5 2 for catAB 5 2 and catBA = 1 M catAB = 1. Without loss of generality, assume that B is a unit sphere centered at the origin in a finite dimensional subspace V, and let W be an orthogonal complement of V. Let R > 2 be sufficiently large so that A is contained in a ball of radius R centered at the origin. Let now M = {z E E : IIxII < R,d(z, B) 2 E} with 0 < c < R/4 sufficiently small so that M n B = 0. Then catBA 5 catEM. Let us construct a covering for M. We set 01 = {z E E : ~~z~~< R + 43, d(z, B) > c/3, z1 > -e/3} and 02 = {z E E : ~~z~~< R + ~/3,d(z, B) > t/3, 51 < e/3}, where 21 is any coordinate in W. We conclude, therefore, that given the class Z of deformations, some progress in obtaining critical points is possible. Further restriction of Z may yield higher values for category and respectively, higher number of critical points. However, the class of deformations has to be closed under conjugations with a pseudogradient flow for a given functional. In special cases when the pseudogradient vector field may be compact or locally finite dimensional, there are many critical point results that are built on finite-dimensional approximations, cf. e.g. [l] Other restrictions of deformation classes seem to desrve a further study. REFERENCES 1. BENCI V., perquadratic 2. PALAIS

CAPOZZ! Potential

S.R.,

3. SCHECHTER to semilinear 4. STRUWE

Critical

A., FORTUNATO Annali Mat.Pura point

theory

M., TINTAREV elliptic equations, M.,

Variational

D., Periodic Solutions Applicata 4, l-46 (1986)

and the minimax

K., Pairs of saddle Bull.Soc.Math.Belg,

Methods,

Springer

1990

principle,

of Hamiltonian Proc.Symp.Pure

points produced by linking ser. B 44, 249-261 (1992)

Systems Math.

subsets

with

15 (1970) with

application

Su-