considered by these authors for simple eigenvalues.

Theorems 14.2 and 14.3 were given in [116].

Chapter 15

Weak Sandwich Pairs

15.1 Introduction

In Chapter 7 we discussed the situation in which one cannot ¬nd linking sets that

separate the functional, i.e., satisfy (7.1). Are there sets such that the opposite of (7.1)

will imply (3.31)? More precisely, are there sets A, B such that (7.3) implies that there

is a sequence satisfying (7.4)? This was answered in the af¬rmative in Chapter 7. Such

pairs exist. This has led to

De¬nition 15.1. We say that a pair of subsets A, B of a Banach space E forms a

sandwich, if, for any G ∈ C 1 (E, R), inequality (7.3) implies the existence of a PS

sequence (7.4).

It follows from Theorem 3.17 that M, N form a sandwich pair if one of them is

¬nite-dimensional. (Note that m 0 ¤ m 1 .) This is a severe drawback in many applica-

tions.

The purpose of the present chapter is to ¬nd a counterpart of sandwich pairs that

deals with the case when both sets in the pair are in¬nite-dimensional. In order to do

this, we required weak-to-weak continuous differentiability of the functional as we did

in Theorem 15.2. We call such pairs weak sandwiches.

The purpose of this chapter is to solve systems of equations of the form

Av = f (x, v, w)

(15.1)

Bw = g(x, v, w),

(15.2)

where A, B are linear partial differential operators. The variational approach to solving

such a system is to study a functional G(u) chosen so that the system is equivalent to

G (u) = 0.

(15.3)

(In very many cases, such a functional can be found.) The sandwich theorem, Theo-

rem 3.17, is very useful in dealing with equations or systems for which the correspond-

ing functional is semibounded in one of the directions only on a subspace of ¬nite

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_15,

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

178 15. Weak Sandwich Pairs

dimension. However, there are many systems for which this is not the case. On the

other hand, the theorem is probably not true if both subspaces are in¬nite-dimensional.

In the present chapter we shall show that the theorem is indeed true if we require more

than mere continuous differentiability of the functional. The requirement we have cho-

sen is present in many applications. It is the weak-to-weak continuous differentiability

de¬ned in Chapter 10 (cf. De¬nition 10.1). For such functionals, we have

Theorem 15.2. Let N be a closed subspace of a Hilbert space E and let M = N ⊥ .

Let G be a weak-to-weak continuously differentiable functional on E such that

m 0 := inf G(w) = ’∞

(15.4)

w∈M

and

m 1 := sup G(v) = ∞.

(15.5)

v∈N

Then there are a constant c ∈ R and a sequence {u k } ‚ E such that

G(u k ) ’ c, m 0 ¤ c ¤ m 1, quG (u k ) ’ 0.

(15.6)

We shall prove Theorem 15.2 in the next section, where we introduce weak sand-

wich pairs. Applications will be given in Section 15.3.

15.2 Weak sandwich pairs

We now introduce the corresponding de¬nition for the case when both sets A, B are

in¬nite-dimensional.

De¬nition 15.3. We shall say that a pair of subsets A, B of a Banach space E forms a

weak sandwich pair if, for any weak-to-weak continuously differentiable G ∈

C 1 (E, R), the inequality

’∞ < b0 := inf G ¤ a0 := sup G < ∞

(15.7)

B A

implies that there is a sequence {u k } satisfying

G(u k ) ’ c, b0 ¤ c ¤ a0 , G (u k ) ’ 0.

(15.8)

We have

Theorem 15.4. Let E be a separable Hilbert space, let N be a closed subspace of E,

and let p be any point of N. Let F be a Lipschitz continuous map of E onto N such

that F| N = I,

F(g) ’ F(h) ¤ K g ’ h , g, h ∈ E,

(15.9)

and, 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 ∈ S0 , w ∈ S ’ F(v + w) ∈ S0 .

(15.10)

Then A = N, B = F ’1 ( p) form a weak sandwich pair.

15.2. Weak sandwich pairs 179

Proof. Assume that the theorem is false. Let G be a weak-to-weak continuously dif-

ferentiable functional on E satisfying (15.7), where A, B are the subsets of E speci¬ed

in the theorem, such that there is no sequence satisfying (15.8). Then there is a positive

number δ > 0 such that

G (u) ≥ 2δ

(15.11)

whenever u belongs to the set

ˆ

E = {u ∈ E : b0 ’ 2δ ¤ G(u) ¤ a0 + 2δ}.

(15.12)

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

|u|w ¤ u , u ∈ E,

(15.13)

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

˜ ˆ

We denote E equipped with this norm by E. For u ∈ E, let h(u) = G (u)/ G (u) .

Then, by (15.11),

ˆ

(G (u), h(u)) ≥ 2δ, u ∈ E.

(15.14)

Let

T = (a0 ’ b0 + 4δ)/δ,

B R = {u ∈ E : u < R},

(15.15) R = sup u + T,

ˆ ¯ ˆ

B = B R © E,

where is a bounded, open subset of N containing the point p such that

ρ(‚ , p) > K T + δ,

(15.16)

ˆ ˜

and ρ is the distance in E. 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.