·(u) = 1, u ∈ Q 1 ; ·(u) = 0, u ∈ Q 2 ; 0 < ·(u) < 1, otherwise.

Consider the differential equation

σ (t) = W (σ (t)), t ∈ R, σ (0) = u ∈ N,

(7.14)

where

W (u) = ’·(u)Y (u).

60 7. Sandwich Pairs

The mapping W is locally Lipschitz continuous on the whole of E and is bounded in

norm by 1. Hence, by Theorem 4.5, (7.13) has a unique solution for all t ∈ R. Let us

denote the solution of (7.13) by σ (t)u. The mapping σ (t) is in C(E — R, E) and is

called the ¬‚ow generated by W (u). Note that

d G(σ (t)u)/dt = (G (σ (t)u), σ (t)u)

(7.15)

= ’·(σ (t)u)(G (σ (t)u), Y (σ (t)u))

¤ ’2δ·(σ (t)u).

Let

E ± = {u ∈ E : G(u) ¤ ±}.

I claim that there is a T > 0 such that

σ (T )E a0 +δ ‚ E b0 ’δ .

(7.16)

In fact, we can take T > (a0 ’ b0 + δ)/2δ. Let u be any element in E a0 +δ . If there is

a t1 ∈ [0, T ] such that σ (t1 )u ∈ Q 1 , then

/

G(σ (T )u) ¤ G(σ (t1 )u) < b0 ’ δ

by (7.15). Hence, σ (T )u ∈ E b0 ’δ . On the other hand, if σ (t)u ∈ Q 1 for all t ∈ [0, T ],

then ·(σ (t)u) = 1 for all t, and (7.15) yields

G(σ (T )u) ¤ G(u) ’ 2δT ¤ a0 ’ 2δT ¤ b0 ’ δ.

Hence, (7.16) holds. Let be a bounded open subset of N containing the point p such

that

ρ(‚ , p) > K T + δ,

(7.17)

where ρ is the distance in E and K is the constant in (7.11). If v ∈ ‚ , then

v ’ p ¤ v ’ Fσ (t)v + Fσ (t)v ’ p .

Then

Fσ (t)v ’ p > K T + δ ’ t K > 0, v ∈ ‚ , 0 ¤ t ¤ T,

(7.18)

since

t

Fσ (t)v ’ v ¤ K σ (s)v ds ¤ K t.

0

Let

H (t) = Fσ (t).

Then H (t) is a continuous map of into N for 0 ¤ t ¤ T. Moreover, H (t)v = p for

v ∈ ‚ by (7.18). Hence, the Brouwer degree d(H (t), , p) is de¬ned. Consequently,

d(H (T ), , p) = d(H (0), , p) = d(I, , p) = 1.

7.3. Notes and remarks 61

This means that there is a v ∈ such that

Fσ (T )v = p.

But then

σ (T )v ∈ F ’1 ( p) = B.

This is not consistent with (7.16). Hence, A, B form a sandwich pair.

7.3 Notes and remarks

The material of this chapter comes from [135].

Chapter 8

Semilinear Problems

8.1 Introduction

In the present chapter we study several nonlinear boundary value problems that arise

frequently in applications and illustrate the techniques described in the book.

8.2 Bounded domains

We assume that is a bounded domain in Rn with boundary ‚ suf¬ciently regular so

that the Sobolev inequalities hold and the embedding of H m,2( ) in L 2 ( ) is compact

(cf., e.g., [1]). 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, where C0 ( ) denotes the set of test functions in (i.e., in¬nitely

differentiable functions with compact supports in ) and H m,2( ) denotes the Sobolev

space. If m is an integer, the norm in H m,2 ( ) is given by

⎛ ⎞1/2

:= ⎝ Dμ u 2 ⎠ .

(8.1) u m,2

| μ|¤m

Here D μ represents the generic derivative of order |μ| and the norm on the right-hand

side of (8.1) is that of L 2 ( ). We shall not assume that m is an integer. Let q be any

number satisfying

¤ q ¤ 2n/(n ’ 2m), 2m < n

(8.2) 2

¤ q < ∞, n ¤ 2m

2

and let f (x, t) be a Carath´ odory function on — R. This means that f (x, t) is

e

continuous in t for a.e. x ∈ and measurable in x for every t ∈ R. We make the

following assumption.

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_8,

© Birkh¤user Boston, a part of Springer Science+Business Media, LLC 2009

64 8. Semilinear Problems

(A) The function f (x, t) satis¬es

| f (x, t)| ¤ V (x)q (|t|q’1 + W (x))

and

f (x, t)/V (x)q = o(|t|q’1 ) as |t| ’ ∞,

where V (x) > 0 is a function in L q ( ) such that

¤C u D, u ∈ D,

(8.3) Vu q

and W is a function in L ∞ ( ). Here

1/q

|u(x)| d x q

u :=

q

and

:= A1/2 u .

(8.4) u D

With the norm (8.4), D becomes a Hilbert space. De¬ne

t

f (x, s) ds

F(x, t) :=