818 CHAPTER 24. CRITICAL POINTS

Definition 24.1.12 A functional I satisfies the Palais Smale conditions if {I (uk)} is abounded sequence and I′ (uk)→ 0, then {uk} is precompact. That is, it has a subsequencewhich converges.

Here is a picture which illustrates the main conclusion of the following theorem. Theidea is that you modify the functional on some set making it smaller and leaving it un-changed off that set.

c−δ

c− ε

c

c+δ

c+ ε

I(η(1,u))

I(u)

I (η (1,u))≤ c−δ if I (u)≤ c+δ .

Theorem 24.1.13 Let I be C1, I is non constant, satisfy the Palais Smale condition, and I′

is bounded on bounded sets. Also suppose that c ∈ R is such that either

I−1 ([c−δ ,c+δ ]) = /0

for some δ > 0 or I−1 ([c−δ ,c+δ ]) ̸= /0 for all δ > 0 and IF I (u) = c, then I′ (u) ̸= 0.Then for each sufficiently small ε > 0, there is a constant δ ∈ (0,ε) and a function η :[0,1]×X → X such that

1. η (0,u) = u

2. η (1,u) = u on I−1 (X \ (c− ε,c+ ε))

3. I (η (t,u))≤ I (u)

4. η(1, I−1(−∞,c+δ ]

)⊆ I−1(−∞,c−δ ], so I (η (1,u))≤ c−δ if I (u)≤ c+δ .

The main part of this conclusion is the statement about u→ η (1,u) contained in parts2. and 4. The other two parts are there to facilitate these two although they are certainlyinteresting for their own sake.

Proof: Suppose I−1([

c− δ̂ ,c+ δ̂

])= /0 for some δ̂ > 0. Then

I−1

((−∞,c+

δ̂

2]

)⊆ I−1

((−∞,c− δ̂

2]

)

and you could take ε = δ and let η (t,u) = u. The conclusion holds with δ = δ̂/2.