0¤t¤t0

u∈A

We have

Theorem 3.1. Let G be a C 1 -functional on E, and let A be a subset of E. Assume that

a0 := sup G < a := inf sup G( (s)u) < ∞.

(3.2)

∈

A 0¤s¤1

u∈A

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

satisfying (2.8). Then there is a sequence {u k } ‚ E such that (2.9) holds. In particular,

there is a Cerami sequence satisfying (1.5).

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_3,

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

18 3. Examples of Minimax Systems

Proof. In this case, we let K be the collection of sets of the form

K = { (t)u : t ∈ [0, 1], u ∈ A, ∈ }.

(3.3)

To show that this is a minimax system, let • be a mapping in (A), and let (t) be in

. De¬ne the mapping

= • —¦ (t) —¦ • ’1 u,

1 (t)u u ∈ E.

1 (t) ∈ . Moreover,

Then

1 (t)u = • —¦ (t)u, u ∈ A.

Consequently,

1 (t)A = •({ (t)u : u ∈ A}),

showing that the collection K is a minimax system for A. Since (3.2) now becomes

(2.7), the result follows.

De¬nition 3.2. We say that A links B [hm] if A, B are subsets of E such that A© B = φ

and, for each (t) ∈ , there is a t ∈ (0, 1] such that (t)A © B = φ.

We have

Corollary 3.3. If A links B [hm], then it links B [mm].

We also have

Theorem 3.4. Let A, B be subsets of E such that A links B [hm], and let G be a

C 1 -functional on E, satisfying

a0 := sup G ¤ b0 := inf G.

(3.4)

B

A

Assume that

a := inf sup G( (s)u)

(3.5)

∈ 0¤s¤1

u∈A

is ¬nite. Let ψ(t) be a positive, nonincreasing, locally Lipschitz continuous function

on [0, ∞) satisfying (2.8). Then there is a sequence {u k } ‚ E such that (2.9) holds.

In particular, there is a Cerami sequence satisfying (1.5).

Proof. The theorem follows from Theorem 2.12. If

K = { (t)u : t ∈ [0, 1], u ∈ A}

(3.6)

∈ , and σ (t) is any ¬‚ow, let

for some

σ (2T s), 0 ¤ s ¤ 1,

˜ (s) = 2

(3.7)

σ (T ) (2s ’ 1), < s ¤ 1.

1

2

It is easily checked that ˜ ∈ . Moreover,

˜

K‚ σ (t)A ∪ σ (T )K .

(3.8)

t ∈[0,T ]

Hence, (2.14) is satis¬ed.

3.3. A method using metric spaces 19

3.3 A method using metric spaces

Another general procedure is described in Mawhin“Willem [96] and Brezis“Nirenberg

[28] as follows. One ¬nds a compact metric space and selects a closed subset — of

such that — = φ, — = . One then picks a map p— ∈ C( , E) and de¬nes

A ={ p ∈ C( , E) : p = p— on — },

a = inf max G( p(ξ )).

p∈A ξ ∈

They assume

—.

(A) For each p ∈ A, maxξ ∈ G( p(ξ )) is attained at a point in \

They then prove

Theorem 3.5. Under the above hypotheses, there is a sequence satisfying (1.4).

We shall prove

Theorem 3.6. Under the same hypotheses, let ψ(t) be a positive, nonincreasing,

locally Lipschitz continuous function on [0, ∞) satisfying (2.8). Then there is a

sequence {u k } ‚ E such that (2.9) holds. In particular, there is a Cerami sequence

satisfying (1.5).

Proof. If a > a0 , this follows from Theorem 2.4. In fact, we can take A = p— ( —)

and K as the collection of sets

K = { p(ξ ) : ξ ∈ , p ∈ A}.

If • ∈ (A), then

•( p(ξ )) = •( p— (ξ )) = p — (ξ ), —

ξ∈ .

Thus, •( p(ξ )) ∈ K. Hence, K is a minimax system for A, and we can apply Theo-

rem 2.4 to come to the desired conclusion.

Now assume that a0 = a. Since each K ∈ K is compact, the same is true of K ρ

given by (2.25) for each ρ > 0. Moreover, if σ (t) ∈ C(R+ — E, E) is such that

σ (0) = I, one has S(K ) ∈ K, where

S( p(ξ )) = σ (d( p(ξ ), A)) p(ξ ), ξ∈ .

The result follows from Theorem 2.14.

3.4 A method using homotopy-stable families

Another approach is found in Ghoussoub [74]. The author considers a closed subset A

of E, and a collection K of compact subsets of E having the property that if K ∈ K

and ψ ∈ C([0, 1] — E, E) satis¬es

ψ(0)u = u, u ∈ E,

20 3. Examples of Minimax Systems

and

ψ(t)u = u, t ∈ [0, 1], u ∈ A,

then ψ(1)K ∈ K. He calls such a collection a homotopy-stable family with extended

boundary A. He proves

Theorem 3.7. If K is a homotopy-stable family with extended closed boundary A, B is

a closed subset of E satisfying (1.1), and G is a C 1 -functional that satis¬es

sup G ¤ a ¤ inf G,

B

A

where the quantity a is given by (2.6), then there is a PS sequence satisfying (1.4).

We shall prove

Theorem 3.8. Under the same assumptions, for each function ψ(t) satisfying the

hypotheses of Theorem 2.4, there is a sequence satisfying (2.9). In particular, there

is a Cerami sequence satisfying (1.5).