39.2. AN APPLICATION OF THE MOUNTAIN PASS THEOREM 1339

| f (u)v| ≤C (1+ |u|p) |v| ≤C(

1+ |u|p+1)p/(p+1)

|v|

∣∣∣∣∫Uf (u)vdx

∣∣∣∣ ≤ C(∫

U

(1+ |u|p+1

))p/(p+1)(∫U|v|p+1

)1/(p+1)

≤ C(∫

U

(1+ |u|p+1

))p/(p+1)

∥v∥H10 (U)

and so

∥ f (u)∥H−1(U) ≤ ∥ f (u)∥Lp+1(U) ≤C(∫

U

(1+ |u|p+1

))p/(p+1)

It follows thatf (uk)→ f (u) pointwise

and also| f (uk)− f (u)|p+1 ≤Cp

(| f (uk)|p+1 + | f (u)|p+1

)where

limk→∞

∫U

(| f (uk)|p+1 + | f (u)|p+1

)dx =

∫U

2 | f (u)|p+1 dx

then by the dominated convergence theorem or more precisely Corollary 11.4.10,

limk→∞

(∫U| f (uk)− f (u)|p+1

)1/(p+1)

= 0

It follows that f (uk)→ f (u) in H−1 (U). Hence R−1 f (uk)→ R−1 f (u) in H10 (U) and so

from 39.2.13, uk → u strongly in H10 (U) also. Thus {uk} is precompact. This verifies the

Palais Smale conditions.

mountain pass conditions

It is clear that I (0) = 0. It remains to verify that for some r > 0, I (u)≥ a > 0 whenever∥u∥H1

0 (U) = r and for some v with ∥v∥> r, I (v) = 0. Now consider ru where ∥u∥= 1.

I (ru) =12

r2−∫

UF (ru)dx

From the assumed estimates and Sobolev embedding,

I (ru) ≥ 12

r2−∫

UA |u|p+1 rp+1dx≥ 1

2r2−CArp+1 ∥u∥p+1

H10 (U)

=12

r2−CArp+1

Now this is independent of u such that ∥u∥= 1. Then the derivative of the right side is

r− (p+1)CArp