Minimax Systems

2.1 Introduction

We have described how the variational approach to solving nonlinear problems

eventually leads to the search for critical points of related functionals. In the case

of semibounded functionals, one can look for extrema. Otherwise, one is forced to use

other methods. As we mentioned, linking provides a useful tool. There are several

approaches to linking. In this book we unify these approaches, providing one theory

that works for all of them.

The idea is as follows. For each A ‚ E, one wishes to ¬nd a collection K of sets

K with the following properties.

1. If G ∈ C 1 (E, R) satis¬es

a0 := sup G < a := inf sup G,

(2.1)

K ∈K K

A

then there is a sequence {u k } ‚ E such that (1.4) holds.

2. If

a0 ¤ sup G, K ∈ K,

(2.2)

K \A

then there is a sequence {u k } ‚ E such that (1.4) holds.

3. If A, B ‚ E satisfy

A © B = φ, B © K = φ, K ∈ K,

(2.3)

and

a0 := sup G < b0 := inf G,

(2.4)

B

A

then there is a sequence {u k } ‚ E such that (1.4) holds.

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_2,

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

8 2. Minimax Systems

4. If

g K = {v ∈ K \A : G(v) ≥ a0 } = φ, K ∈ K,

(2.5)

then there is a sequence {u k } ‚ E such that (1.4) holds.

In the next section we determine collections K that have these properties. More-

over, we shall show that all of these collections produce not only PS sequences (1.4),

but also Cerami sequences (1.5) as well.

Proofs of the theorems of this chapter will be given in Chapter 5.

2.2 De¬nitions and theorems

We begin by studying C 1 -functionals on a Banach space E.

De¬nition 2.1. We shall say that a map • : E ’ E is of class if it is a homeomor-

phism onto E, and both •, • ’1 are bounded on bounded sets.

De¬nition 2.2. For A ‚ E, we de¬ne

(A) = {• ∈ : •(u) = u, u ∈ A}.

De¬nition 2.3. For a nonempty set A ‚ E, we de¬ne a nonempty collection K = K(A)

of subsets K ‚ E to be a minimax system for A if it has the following property:

•(K ) ∈ K, •∈ (A), K ∈ K.

Every nonempty set has a minimax system. We have

Theorem 2.4. Let K be a minimax system for a nonempty subset A of E, and let G(u)

be a C 1 -functional on E. De¬ne

(2.6) a := inf sup G,

K ∈K K

and assume that a is ¬nite and satis¬es

a > a0 := sup G.

(2.7)

A

Let ψ(t) be a positive, nonincreasing, locally Lipschitz continuous function on [0, ∞)

such that

∞

ψ(r ) dr = ∞.

(2.8)

0

Then there is a sequence {u k } ‚ E such that

G(u k ) ’ a, G (u k )/ψ( u k ) ’ 0.

(2.9)

De¬nition 2.5. We shall call σ (t) ∈ C([0, T ] — E, E), T > 0, a ¬‚ow if σ (0) = I

and for each t ∈ [0, T ], σ (t) is a homeomorphism of E onto itself.

2.3. Linking subsets 9

Theorem 2.6. Let K be a minimax system for a nonempty set A and let G(u) be a C 1

functional on E such that

g K = {v ∈ K \A : G(v) ≥ a0 } = φ, K ∈ K,

(2.10)

and

a = inf sup G < ∞.

(2.11)

K K

Assume that for any b > a, K ∈ K and ¬‚ow σ (t) satisfying

G(σ (t)v) < b, v ∈ K , 0 < t ¤ T,

(2.12)

and

G(σ (t)u) < a, u ∈ A, 0 < t ¤ T,

(2.13)

˜

there is a K ∈ K such that

˜

K‚ σ (t)A ∪ σ (T )[E b ∪ K ],

(2.14)

t ∈[0,T ]

where

E ± = {u ∈ E : G(u) ¤ ±}.

(2.15)

Then the conclusions of Theorem 2.4 hold.

2.3 Linking subsets

We now show how linking can play a major role in ¬nding critical points.

De¬nition 2.7. We shall say that a set A in E links a set B ‚ E relative to a minimax

system K for A if

A©B =φ

(2.16)

and

B © K = φ, K ∈ K.

(2.17)

We shall say that A links B [mm] if there is a minimax system K for A such that A

links B relative to K.

Theorem 2.8. Let K be a minimax system for a nonempty set A, and assume that there

is a subset B of E such that A links B relative to K. Assume that G ∈ C 1 (E, R) satis¬es

a0 := sup G < b0 := inf G

(2.18)

B

A

and that the quantity a given by (2.6) is ¬nite. Then, for each positive, nonincreas-