H+ (x) d x + H’ (x) d x ≥ 0

(8.107)

u>0 u<0

for all u ∈ D\{0} satisfying

Au = b+ (x)u + ’ b’ (x)u ’ ,

(8.108)

8.7. The proofs 81

where u ± (x) = max[±u(x), 0]. Assume that there is a q > 2 such that b1/q , W 1/q are

compact operators from D to L q ( ), where b(x) = max |b± (x)|, and that solutions of

(8.108) that are not identically zero a.e. are never zero a.e. Then (8.81) has a solution

u ∈ D\{0}.

Theorem 8.16. If we replace (8.103), (8.104) in Theorem 8.15 by

|H (x, t)| ¤ V (x)|t|γ + W (x), 0 < γ < 2,

(8.109)

and

H (x, t)/|t|γ ’ H± (x) as t ’ ±∞ a.e.,

(8.110)

the theorem will hold if

H± (x)|u|γ d x + H’ (x)|u|γ d x > 0

(8.111)

u>0 u<0

for all u ∈ D\{0} satisfying (8.108), and V 1/q is a compact operator from D to L q ( ).

In this case the requirement that solutions of (8.108) that are not identically zero a.e.

are never zero a.e. can be removed.

Theorem 8.17. Assume that there is a number γ such that 0 ¤ γ < 1 and

| f (x, t)| ¤ V (x)|t|γ + W (x),

(8.112)

with V 1/q , W 1/q compact operators from D to L q ( ), where q > 2. Then (8.81) has

a solution u ∈ D\{0}.

Theorem 8.18. Assume that

| f (x, t)| ¤ V (x)|t| + W (x)

(8.113)

and

f (x, t)/t ’ b± (x) as t ’ ±∞ a.e.,

(8.114)

with b1/q , V 1/q , and W 1/q compact operators from D to L q ( ), where b(x) =

max |b ± (x)|, and q > 2. Assume in addition that (8.108) has no nontrivial solutions.

Then (8.81) has a solution u ∈ D\{0}.

8.7 The proofs

In this section we give the proofs of Theorems 8.15 to 8.18. First, we give the

Proof of Theorem 8.15. We apply Theorem 8.14. We take r0 (t) = r1 (t) = 1,

ψ(t) = t ’1 . Inequality (8.70) is satis¬ed with q1 = q, V1 arbitrary, and W1 = W.

Since

‚(Ft ’2 )/‚t = ’2t ’3 H (x, t),

(8.115)

82 8. Semilinear Problems

we have

2 b+ (x)t + F0 (x, t), t > 0,

1 2

F(x, t) =

(8.116)

2 b’ (x)t + F0 (x, t), t < 0,

1 2

where

∞ ’3

H (x, s) ds, t > 0,

2t 2 ts

F0 (x, t) =

(8.117)

’2t 2 ’∞ s ’3 H (x, s) ds,

t

t < 0.

Thus,

∞

b+ (x)t + 4t ( t ) ’ 2H (x, t)/t, t > 0,

f (x, t) =

(8.118) t

b’ (x)t ’ 4t ( ’∞ ) ’ 2H (x, t)/t t < 0.

Consequently,

| f (x, t)| ¤ b(x)|t| + 4W |t|’1 , |t| ≥ 1.

(8.119)

This combined with (8.106) gives (8.69) with V0 = V = b1/q . Moreover, by (8.119),

f (x, st)/t ’ ±b± as s ’ a, t ’ ∞.

(8.120)

Thus, (8.79) becomes (8.108). Moreover, by (8.104), the left-hand side of (8.80) is

the left-hand side of (8.107). Hence, (8.107) assures us that (8.80) cannot hold for any

solution u ∈ D of (8.79). Thus, all of the hypotheses of Theorem 8.14 are satis¬ed.

Proof of Theorem 8.16. In this case we take r0 (t) = 1, r1 (t) = t γ . Since r1 = ∞,

the right-hand side of (8.80) vanishes. Thus, strict inequality in (8.111) is needed

to guarantee that no solutions of (8.78)“(8.80) exist. Moreover, we do not require

hypothesis V for this case.

Proof of Theorem 8.17. Here we take r0 (t) = t γ ’1 and r1 (t) = t γ +1 . Note that by

(8.112),

|F(x, t)| ¤ p’1 V (x)|t| p + W (x)|t|,

(8.121)

where p = 1 + γ . Thus, by (8.2),

1 1 3

|H (x, t)| ¤ + V (x)|t| p + W (x)|t|.

(8.122)

2 p 2

Since r0 = 0 and r1 = ∞, all of the hypotheses of Theorem 8.14 hold.

Proof of Theorem 8.18. Now we take r0 (t) = 1, r1 (t) = t 2 . By (8.113),

1

|F(x, t)| ¤ V (x)t 2 + W (x)|t|

2

8.8. Notes and remarks 83

and

1 3

|H (x, t)| ¤ V (x)t 2 + W (x)|t|,

(8.123)

2 2

Moreover, by (8.114),

2F(x, t)/t 2 ’ b± (x) a.e. as t ’ ±∞.

(8.124)

Consequently, by (8.2),

H (x, t)/t 2 ’ 0 a.e. as t ’ ±∞.

Thus r (x, a) ’ 0 a.e. and (8.80) is always satis¬ed. Thus, we must assume that (8.79)