39.2. AN APPLICATION OF THE MOUNTAIN PASS THEOREM 1337

where C will be adusted as needed here and elsewhere. Thus, writing in terms of the innerproduct on H1

0 , (I′2 (u) ,v

)H1

0=(R−1 f (u) ,v

)H1

0

This is so if the 12 f ′ (û)v2 term is as it should be. We need to verify that∫

U

∣∣ 12 f ′ (û)v2

∣∣dx∥v∥H1

0

→ 0

However, we can use the estimate and write that this is no larger than∫U C(

1+ |v|p−1 + |u|p−1)∣∣v2

∣∣dx(∫U |v|

2n/(n−2) dx)(n−2)/2n

(39.2.11)

Then consider the term involving |u| .∫

U|u|p−1 |v|2 dx≤

(∫U|u|p+1

) p−1p+1(∫

U|v|p+1

)2/(p+1)

Now p+1≤ 2 nn−2 and so the first factor is finite. As to the second, it equals((∫

U|v|2n/(n−2)

) 12n (n−2)

)2

and so this term from 39.2.11 is o(v) on H10 (U) . The term involving |v|p−1 is obviously

o(v) . Consider the constant term.

∫U|v|2 dx≤

((∫U|v|2n/(n−2) dx

)(n−2)/2n)2

so it is also all right. Thus the derivative is as claimed. Is this derivative Lipschitz onbounded sets?

f (û)− f (u) =∫ 1

0f ′ (u+ t (û−u))(û−u)dt

Thus in H−1 and using the estimates,∣∣∣∣∫U( f (û)− f (u))vdx

∣∣∣∣≤ ∫U

∫ 1

0C(

1+ |u+ t (û−u)|p−1)|û−u|dtdx

=∫ 1

0

∫U

C(

1+ |u+ t (û−u)|p−1)|û−u|dxdt

≤ C(∫

U

(1+ |û|p−1 + |u|p−1

)2n/(n+2)) n+2

2n(∫

U|û−u|2n/(n−2)

)(n−2)/2n

≤ C(∫

U

(1+ |û|p−1 + |u|p−1

)2n/(n+2)) n+2

2n

∥û−u∥H10 (U)