This shows that u ≡ 0. Let

˜ on which u = 0. Then

˜

be the subset of

0

|u k (x)| = ρk |u k (x)| ’ ∞,

˜ x∈ 0.

(8.60)

= \

If 0, then we have

1

H (x, u k ) d x = + ≥ H (x, u k ) d x ’ W1 (x) d x ’ ∞.

(8.61)

0 1 0 1

This contradicts (8.57), and we see that ρk = u k D is bounded. Once we know

that the ρk are bounded, we can apply Theorem 3.4.1 of [122] to obtain the desired

conclusion.

Remark 8.8. It should be noted that the crucial element in the proof of Theorem 8.7

was (8.56). If we had been dealing with an ordinary Palais“Smale sequence, we could

only conclude that

u k 2 ’ ( f (·, u k ), u k ) = o(ρk ),

D

which would imply only that

H (x, u k )d x = o(ρk ).

This would not contradict (8.61), and the argument would not go through.

74 8. Semilinear Problems

We also have

Theorem 8.9. The conclusion of Theorem 8.7 holds if in place of (8.47), (8.48) we

assume that

H (x, t) ¤ W1 (x) ∈ L 1 ( ), x∈ , t ∈ R,

(8.62)

and

H (x, t) ’ ’∞ a.e. as |t| ’ ∞.

(8.63)

Proof. We use (8.62) and (8.63) to replace (8.61) with

H (x, u k ) d x = + ¤ H (x, u k ) d x + W1 (x) d x ’ ’∞.

(8.64)

0 1 0 1

We then proceed as before.

As a speci¬c example of an operator A satisfying the hypotheses of Theorems 8.7

and 8.9, let g(x) be a measurable function satisfying

g(x) ≥ c0 > 0, x ∈ Rn ,

for some positive constant c0 . We consider the problem

’ u(x) + g(x)2 u(x) = f (x, u(x)), x ∈ Rn .

(8.65)

We de¬ne the operator A on L 2 = L 2 (Rn ) by u ∈ D(A) and Au = f if u ∈ D =

H 1,2 = H 1,2(Rn ) and

(u, v) D = (∇u, ∇v) + (gu, gv) = ( f, v), v ∈ H 1,2.

We assume that g(x) ∈ L 2 and that multiplication by g ’1 is a compact operator from

loc

H 1,2 to L 2 . It follows that A is a self-adjoint operator on L 2 that is bijective. Moreover,

A’1 is a compact operator on L 2 . It follows that the spectrum of A consists of isolated

eigenvalues of ¬nite multiplicity satisfying (8.49). Thus, A satis¬es the hypotheses of

Theorems 8.7 and 8.9. Solutions of (8.11) satisfy (8.65). Hence, Theorems 8.7 and 8.9

produce weak solutions of (8.65). We summarize this as

Theorem 8.10. Let g(x) be a function satisfying the conditions described above. Then

there exists a sequence of eigenvalues for the equation

’ u(x) + g(x)2 u(x) = »u(x), x ∈ Rn ,

(8.66)

satisfying (8.49). Let f (x, t) be a Carath´ odory function satisfying hypothesis (A) for

e

=R n , and assume that (8.47), (8.48), and (8.50) hold for some > 0. Then (8.65)

has at least one solution.

8.5. Further applications 75

Remark 8.11. We could have assumed

a1 ¤ lim inf 2F(x, t)/t 2 ¤ lim sup 2F(x, t)/t 2 ¤ a2

t ’’∞ t ’’∞

and

γ (a1 ) ¤ lim inf 2F(x, t)/t 2 ¤ lim sup 2F(x, t)/t 2 ¤ (a2 )

t ’∞ t ’∞

in place of (8.50).

Remark 8.12. This theorem generalizes results of several authors, including [10],

[46], [71], [98], and [104], with various conditions on the function g(x) to ensure that

the spectrum of (8.66) is discrete. We guarantee it by assuming that multiplication by

g ’1 is a compact operator from H 1,2 to L 2 . A suf¬cient condition for this is given in

[106]. Since g ’1 is bounded, a simple suf¬cient condition is that for each constant

b > 0,

m{x ∈ Rn : |x ’ y| < 1, g(x) < b} ’ 0 as |y| ’ ∞.

(8.67)

Remark 8.13. If we choose a1 = » and a2 = » +1 , inequality (8.50) reduces to

» t 2 ’ W1 (x) ¤ 2F(x, t) ¤ » + W2 (x), x ∈ Rn , t ∈ R.

2

(8.68) +1 t

By choosing a1 , a2 to be different values, we allow a wider range of possibilities for

F(x, t).

8.5 Further applications

Now we look for solutions of (8.13) under different conditions. Let A be a self-adjoint

operator on L 2 ( ). We assume that A ≥ »0 > 0 and that

∞

C0 ( ) ‚ D := D(A1/2 ) ‚ H m,2( )

for some m > 0. Let q — be given by

q— = 2n/(n ’ 2m), 2m < n

= ∞, n ¤ 2m,

and let f (x, t) be a Carath´ odory function on — R. We assume the following.

e

I. There are positive functions V (x), V0 (x), V1 (x), W (x), W1 (x), r0 (t), r1 (t) and

constants q, q1 such that the following hold.

(a) 2 < q < q — , 1 < q1 < q — .

(b) Multiplication by V or W is a bounded operator from D to L 2 ( ), multiplica-

tion by V is compact from D to L q’1 (K ) for each compact subset K of , and W is in

L q ( ). Multiplication by V0 is compact from D to L q ( ), V0 is locally bounded, and

| f (x, st)| ¤ V0 (x)(|V (x)s|q’1 + W (x)q’1 )tr0 (t), s ∈ R, t ≥ 1.

(8.69)

76 8. Semilinear Problems

(c) Multiplication by V1 is a compact operator from D to L q1 ( ), W1 is in L 1 ( ),