manifold. For instance, if E = M • N, where M, N are closed, in¬nite-dimensional

subspaces of E and B R is the ball centered at the origin of radius R in E, it is unknown

if the set A = M © ‚ B R links B = N. (If either M or N is ¬nite-dimensional, then

A does link B; cf. Example 2 of Section 3.4.) Very little is known for the in¬nite-

dimensional case. Unfortunately, this situation arises in some important applications,

including Hamiltonian systems, the wave equation, and elliptic systems, to name a few.

The purpose of the present chapter is to study linking when both M and N are

in¬nite-dimensional and G has some additional continuity property. The property we

have chosen is that of weak-to-weak continuity:

De¬nition 10.1. Let E be a Banach space. We shall call a functional G ∈ C 1 (E, R)

weak-to-weak continuously differentiable if, for each sequence

u k ’ u weakly in E,

(10.4)

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_10,

© Birkh¤user Boston, a part of Springer Science+Business Media, LLC 2009

108 10. Weak Linking

there exists a renamed subsequence such that

G (u k ) ’ G (u) weakly.

(10.5)

We restrict our attention to Hilbert space and prove the following.

Theorem 10.2. Let E be a separable Hilbert space, and let G be a weak-to-weak

continuously differentiable functional on E. Let N be a closed subspace of E, and

let Q be a bounded, convex, open subset of N containing the point p. Let F be a

continuous map of E onto N such that

(a) F| Q = I,

and

(b) for each ¬nite-dimensional subspace S of E containing p such that F S = {0},

there is a ¬nite-dimensional subspace S0 = {0} of N containing p such that

¯

v ∈ Q © S0 , w ∈ S ’ F(v + w) ∈ S0 .

(10.6)

(The restriction F S = {0} is made in case p = 0.) Set A = ‚ Q, B = F ’1 ( p). If

a1 = sup G < ∞

(10.7)

¯

Q

and (10.1) holds, then there is a sequence {u k } ‚ E such that (10.2), (10.3) hold and

c ¤ a1 .

¯

If F does not satisfy (a), but the restriction F0 of F to Q is a homeomorphism

¯ ‚ N, then this can replace (a).

of Q onto the closure of a convex, open subset

Moreover, most, if not all, sets A, B known to link when one of the subspaces M, N is

¬nite-dimensional will link now as well.

This leads to the following.

De¬nition 10.3. A subset A of a Banach space E links a subset B weakly if, for every

G ∈ C 1 (E, R) that is weak-to-weak continuously differentiable and satis¬es (10.1),

there are a sequence {u k } ‚ E and a constant c such that (10.2) and (10.3) hold.

Thus we have

Corollary 10.4. Under the hypotheses of Theorem 10.2, the set A = ‚ Q links the set

B = F ’1 ( p) weakly.

Theorem 10.2 will be proved in the next section.

10.2 Another norm

In this section we prove Theorem 10.2 by introducing a different norm that is equivalent

to the weak topology on bounded sets.

10.2. Another norm 109

Proof. Assume that there is no such sequence. Then there is a positive number δ > 0

such that

G (u) ≥ 2δ

(10.8)

whenever u belongs to the set

ˆ

E = {u ∈ E : b0 ’ 2δ ¤ G(u) ¤ a1 + 2δ}.

(10.9)

Since E is separable, we can norm it with a norm |u|w satisfying

|u|w ¤ u , u ∈ E,

(10.10)

and such that the topology induced by this norm is equivalent to the weak topology of

E on bounded subsets of E.

This can be done as follows. Let {ek } be an orthonormal basis for E. We then set

∞

|(u, ek )|2

|u|2 = .

w

k2

k=1

˜

We denote E equipped with this norm by E. Let

E = {u ∈ E : G (u) = 0}.

For u ∈ E , let h(u) = G (u)/ G (u) . Then, by (10.8),

ˆ

(G (u), h(u)) ≥ 2δ, u ∈ E.

(10.11)

Let

T = (a1 ’ b0 + 4δ)/δ,

B R = {u ∈ E : u < R},

(10.12) R = sup u + T,

Q

ˆ ¯ ˆ

B = B R © E.

ˆ ˜

For each u ∈ B, there is an E neighborhood W (u) of u such that

ˆ

(G (v), h(u)) > δ, v ∈ W (u) © B.

(10.13)

ˆ

Otherwise, there would be a sequence {v k } ‚ B such that

|v k ’ u|w ’ 0 (G (v k ), h(u)) ¤ δ.

(10.14) and

ˆ

Since B is bounded in E, v k ’ u weakly in E and (10.5) implies that

(G (v k ), h(u)) ’ (G (u), h(u)) ¤ δ

(10.15)

110 10. Weak Linking

˜ ˆ

in view of (10.14). This contradicts (10.11). Let B be the set B with the inherited

˜ ˜

topology of E. It is a metric space, and W (u) © B is an open set in this space. Thus,

˜ ˜ ˜

{W (u) © B}, u ∈ B, is an open covering of the paracompact space B (cf., e.g., [112]).

Consequently, there is a locally ¬nite re¬nement {W„ } of this cover. For each „, there

is an element u „ such that W„ ‚ W (u „ ). Let {ψ„ } be a partition of unity subordinate