and

’2

2

(8.5) G(u) := u F(x, u) d x.

D

It is readily shown that G is a continuously differentiable functional on the whole of D

(cf., e.g., [122]). Since the embedding of D in L 2 ( ) is compact, the spectrum of A

consists of isolated eigenvalues of ¬nite multiplicity:

0 < »0 < »1 < · · · < » < · · · .

Let » , > 0, be one of these eigenvalues. We assume that the eigenfunctions of »

are in L ∞ ( ) and that the following hold:

2F(x, t) ¤ » t 2 + W1 (x), x∈ , t ∈R

(8.6)

for some W1 (x) ∈ L 1 (R),

» t 2 ¤ 2F(x, t), |t| ¤ δ,

(8.7)

for some δ > 0,

νt 2 ¤ 2F(x, t), x∈ , t ∈ R,

(8.8)

for some ν > » ’1 ,

H (x, t) := 2F(x, t) ’ t f (x, t) ¤ C(|t| + 1),

(8.9)

8.2. Bounded domains 65

and

σ (x) := lim sup H (x, t)/|t| < 0

(8.10) a.e.

|t |’∞

We wish to obtain a solution of

Au = f (x, u), u ∈ D.

(8.11)

By a solution of (8.11) we shall mean a function u ∈ D such that

(u, v) D = ( f (·, u), v), v ∈ D.

(8.12)

If f (x, u) is in L 2 ( ), then a solution of (8.12) is in D(A) and solves (8.11) in the

classical sense. Otherwise we call it a weak (or semi-strong) solution. We have

Theorem 8.1. Under the above hypotheses,

Au = f (x, u), u∈D

(8.13)

has at least one nontrivial solution.

Proof. Let N denote the subspace of L 2 ( ) spanned by the eigenfunctions of A cor-

responding to the eigenvalues »0 , . . . , » , and let M = N ⊥ © D. Thus, D = M • N.

This time we take

G(u) = 2 F(x, u) d x ’ u 2 , D

the negative of (8.5). We are therefore looking for solutions of G (u) = 0. Let N be

the set of those functions in N that are orthogonal to E(» ). It is spanned by those

eigenfunctions corresponding to »0 , . . . , » ’1 . Let v 0 be an eigenfunction of » with

norm 1. Let M1 = M • E(» ) {v 0 }. We can write

E = M1 • {v 0 } • N .

Consider the mapping

F(v + w + sv 0 ) = w + [s + ρ ’ ρ•( v /ρ 2 )]v 0 , v ∈ N, w ∈ M1 , s ∈ R,

2

where • satis¬es the hypotheses of Proposition 7.3 and ρ > 0 is to be chosen. We take

B = F ’1 (ρv 0 ).

A = M1 • {v 0 },

By Proposition 7.3, A, B form a sandwich pair.

For v ∈ N, we write v = v + y, where v ∈ N and y ∈ E(» ). Since E(» ) is

¬nite-dimensional and contained in L ∞ ( ), there is a ρ > 0 such that

¤ ρ implies y ¤ δ/2,

(8.14) y ∞

D

where δ is given by (8.7). Thus, if

v ¤ρ |v(x)| ≥ δ,

(8.15) and

D

66 8. Semilinear Problems

then

δ ¤ |v(x)| ¤ |v (x)| + |y(x)| ¤ |v (x)| + δ/2.

Hence,

|v(x)| ¤ 2|v (x)|

holds for all x ∈ ¯ satisfying (8.15). Thus by (8.7)

v 2d x ’ 2 {|V v|q + |V q v|W }d x ’ v 2

≥ »

G(v) D

|v|<δ |v|>δ

≥ »v ’» v 2d x ’ v ’C {|V v |q + δ 1’q |V v |q }d x

2 2

D

|v|>δ 2|v |>δ

≥ »v ’v ’C {|V v |q + δ 1’q |V v |q + δ 2’q |v |q }d x

2 2

D

2|v |>δ

q

≥ »v ’v ’C v

2 2

D D

» q’2 2

≥ ’1’C v v D.

» ’1 D

To see this, note that when (8.15) holds, we have |V v| ¤ 2|V v | and

|v|q’1

|V v| ¤ V |v| q’1 ¤ δ 1’q V q |2v |q .

q q

δ

Moreover,

v ¤C v ¤C v

q m,2 D

by the Sobolev inequality and the embedding of D in H m,2( ). From this we see that

there are positive constants , ρ such that

G(v) ≥ v D, v ¤ ρ, v ∈ N .

2

D

Moreover, this shows that for each positive ρ1 ¤ ρ,

G(v) ≥ 1, v = ρ1 , v ∈ N ,

(8.16) D