while (8.23) implies

F(x, u k ) d x/ρk ¤ ν ˜

2

u 2 d x,

lim sup 2

showing that u ≡ 0. This contradiction proves the theorem as in the case of Theo-

˜

rem 8.1.

The proof of Theorem 8.4 implies

Corollary 8.5. If » is a simple eigenvalue, then hypothesis (8.21) in Theorem 8.4 can

be weakened to

2

, t ∈ R, for some W1 (x) ∈ L 1 (R).

» ¤ 2F(x, t) + W1 (x), x∈

(8.26) ’1 t

8.3 Some useful quantities

We now show how we can improve the results of the last section. For each ¬xed k, let

Nk denote the subspace of D := D(A1/2 ) spanned by the eigenfunctions corresponding

⊥

to »0 , . . . , »k , and let Mk = Nk © D. Then D = Mk • Nk . We de¬ne

±k := max{(Av, v) : v ∈ Nk , v ≥ 0, v = 1},

(8.27)

where v denotes the L 2 ( )-norm of v. We assume that A has an eigenfunction •0

of constant sign a.e. on corresponding to the eigenvalue »0 .

Next we de¬ne for a ∈ R

γk (a) := max{(Av, v) ’ a v ’ : v ∈ Nk , v + = 1}

2

(8.28)

and

:= inf{(Aw, w) ’ a w’ : w ∈ Mk , w+ = 1},

k (a)

2

(8.29)

where u ± = max{±u, 0}.

We take any integer ≥ 0 and let N denote the subspace of L 2 ( ) spanned

by the eigenspaces of A corresponding to the eigenvalues »0 , »1 , . . . , » . We take

M = N ⊥ © D, where D = D(A1/2 ). We assume that F(x, t) satis¬es

a1 (t ’ )2 + γ (a1 )(t + )2 ’ W1 (x) ¤ 2F(x, t)

(8.30)

¤ a2 (t ’ )2 + ν(t + )2 , x∈ , t ∈ R,

for numbers a1 , a2 satisfying ± < a1 ¤ a2 , where W1 is a function in L 1 ( ) and

ν < (a2 ). We also assume that

2F(x, t) ¤ » , |t| ¤ δ

2

(8.31) +1 t

70 8. Semilinear Problems

for some δ > 0,

| f (x, t)| ¤ C|t| + W (x), W ∈ L 2 ( ),

(8.32)

f (x, t)/t ’ ±± (x) a.e. as t ’ ±∞,

(8.33)

and the only solution of

Au = ±+ (x)u + ’ ±’ (x)u ’

(8.34)

is u ≡ 0. We have

Theorem 8.6. Under the above hypotheses, (8.13) has a nontrivial solution.

Proof. By (8.28),

¤ a1 v ’ + γ (a1 ) v + 2 ,

v v ∈ N,

2 2

(8.35) D

and by (8.29), we have

a2 w ’ (a2 ) w+

2 2 2

+ ¤w D, w ∈ M.

(8.36)

Hence,

G(v) ¤ B1 , v ∈ N.

(8.37)

Since ν < (a2 ), we see by continuity that there is an µ > 0 such that

a2

ν < (1 ’ µ) .

1’µ

Hence,

ν

a2

w+

2 2

G(w) ≥ µ w + (1 ’ µ) ’

(8.38) D

1’µ 1’µ

≥µ w D, w ∈ M,

2

by (8.30).

As in the proof of Theorem 8.1, we note that the following alternative holds:

Either

(a) there is an in¬nite number of eigenfunctions y ∈ E(» )\{0} such that

Ay = f (x, y) = » y,

(8.39)

or

(b) for each ρ > 0 suf¬ciently small, there is an µ > 0 such that

G(w) ≥ µ, w = ρ, w ∈ M .

(8.40) D

8.4. Unbounded domains 71

Since option (a) solves our problem, we may assume that option (b) holds. Let

v 0 ∈ E(» ), and let F be the mapping (7.10). Take A = N, B = F ’1 (δv 0 ). By (8.37),

(8.38), and (8.40) we see that (7.3) holds with b0 > 0 and a0 = B1 . By Proposition

7.3, we can conclude that there is a sequence {u k } ‚ D such that

G(u k ) ’ c, b 0 ¤ c ¤ B1 , G (u k ) ’ 0.

(8.41)

Thus,

2

G(u k ) = u k ’2 F(x, u k )d x ’ c

(8.42) D

and

(G (u k ), v) = 2(u k , v) D ’ 2( f (u k ), v) ’ 0, v ∈ D.

(8.43)

If ρk = u k D ’ ∞, let u k = u k /ρk . Then u k D = 1. Thus, there is a renamed

˜ ˜

subsequence such that u k ’ u weakly in D, strongly in L 2 ( ), and a.e. in . Hence,