8.8 Notes and remarks

The material in this chapter is from [120], [110], [107], and [132].

Chapter 9

Superlinear Problems

9.1 Introduction

Consider the problem

’ u = f (x, u), x ∈ ; u = 0 on ‚ ,

(9.1)

where ‚ Rn is a bounded domain whose boundary is a smooth manifold, and f (x, t)

is a continuous function on ¯ — R. This semilinear Dirichlet problem has been studied

by many authors. It is called sublinear if there is a constant C such that

| f (x, t)| ¤ C(|t| + 1), x∈ , t ∈ R.

Otherwise, it is called superlinear. Beginning in [7], almost all researchers studying

the superlinear problem assumed

(a1 ) There are constants c1 , c2 ≥ 0 such that

| f (x, t)| ¤ c1 + c2 |t|s ,

where 0 ¤ s < (n + 2)/(n ’ 2) if n > 2.

(a2 ) f (x, t) = o(|t|) as t ’ 0.

(a3 ) There are constants μ > 2, r ≥ 0 such that

0 < μF(x, t) ¤ t f (x, t), |t| ≥ r,

(9.2)

where

t

F(x, t) = f (x, s) ds.

0

They proved

Theorem 9.1. Under hypotheses (a1 )“(a3 ), problem (9.1) has a nontrivial weak

solution.

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_9,

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

86 9. Superlinear Problems

The condition (a3 ) is convenient, but it is very restrictive. In particular, it implies

that there exist positive constants c3 , c4 such that

F(x, t) ≥ c3 |t|μ ’ c4 , x∈ , t ∈ R.

(9.3)

Although this condition is weaker, it still eliminates many superlinear problems.

A much weaker condition that implies superlinearity is

(a3 ) Either

F(x, t)/t 2 ’ ∞ as t ’ ∞

or

F(x, t)/t 2 ’ ∞ as t ’ ’∞.

The purpose of the present chapter is to explore what happens when (a3 ) is replaced

with (a3 ). Surprisingly, we ¬nd the following to be true.

Theorem 9.2. Under hypotheses (a1 ), (a2 ), (a3 ), the boundary-value problem

’ u = β f (x, u), x ∈ ; u = 0 on ‚ ,

(9.4)

has a nontrivial solution for almost every positive β.

We generalize this theorem and present some variations below.

9.2 The main theorems

We consider the boundary-value problems described in Sections 8.2 and 8.4 under

assumption (A) stipulated there. We allow the domain to be unbounded. We also

assume

(B) The point »0 is an isolated simple eigenvalue with a bounded eigenfunction

•0 (x) = 0 a.e. in .

(C) There is a δ > 0 such that

2F(x, t) ¤ »0 t 2 , |t| ¤ δ, x ∈ ,

where

t

f (x, s)ds.

(9.5) F(x, t) :=

0

There is a function W (x) ∈ L 1 ( ) such that either

(D)

W (x) ¤ F(x, t)/t 2 ’ ∞ as t ’ ∞, x∈

or

W (x) ¤ F(x, t)/t 2 ’ ∞ as t ’ ’∞, x∈ .

(The function W (x) need not be positive.)

(E) There are constants μ > 2, C ≥ 0 such that

[μF(x, t) ’ t f (x, t)]/(t 2 + 1) ¤ C, t ∈ R, x ∈ .

9.2. The main theorems 87

We shall prove

Theorem 9.3. Under the above hypotheses, the problem

Au = f (x, u), u∈D

(9.6)

has at least one nontrivial solution.

We also have

Theorem 9.4. If we replace hypothesis (E) with

(E ) The function

H (x, t) := t f (x, t) ’ 2F(x, t)

(9.7)

is convex in t.

then problem (9.6) has at least one nontrivial solution.

As we noted, problem (9.6) is called sublinear if f (x, t) satis¬es

| f (x, t)| ¤ C(|t| + 1), x∈ , t ∈ R.

(9.8)

Otherwise, it is called superlinear. Hypothesis (D) requires (9.6) to be superlinear.

If we drop hypothesis (E) completely, then we are able to prove the following

theorems.

Theorem 9.5. If we replace hypotheses (C), (D) with

˜

(C ) There are a δ > 0 and a » > »0 such that

˜

2F(x, t) ≥ »t 2 , |t| ¤ δ, x ∈

and

there is a function W (x) ∈ L 1 ( ) such that

(D )

W (x) ≥ P(x, t) ’ ’∞ as |t| ’ ∞, x∈ ,

where

1

P(x, t) := F(x, t) ’ »0 t 2 .

(9.9)

2

and drop hypothesis (E), then problem (9.6) has at least one nontrivial solution.

We also have

Theorem 9.6. Assume that (A)“(D) hold. Then, for almost every β ∈ (0, 1), the

equation

Au = β f (x, u)

(9.10)

has a nontrivial solution. In particular, the eigenvalue problem (9.10) has in¬nitely

many solutions.

88 9. Superlinear Problems

Theorem 9.7. If we replace hypothesis (C) in Theorem 9.6 with

˜

(C ) There are a δ > 0 and a » ¤ »0 such that

˜

2F(x, t) ¤ »t 2 , |t| ¤ δ, x ∈ .

and (D) with