Semilinear problems with bounded nonlinear term
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We solve boundary value problems for elliptic semilinear equations in which no asymptotic behavior is prescribed for the nonlinear term. 1. Introduction Many authors (beginning with Landesman and Lazer [1]) have studied resonance problems for semilinear elliptic partial differential equations of the form −∆u − λ u = f (x,u)
in Ω,
u = 0 on ∂Ω,
(1.1)
where Ω is a smooth bounded domain in Rn , λ is an eigenvalue of the linear problem −∆u = λu
in Ω,
u = 0 on ∂Ω,
(1.2)
and f (x,t) is a bounded Carath´eodory function on Ω × R such that f (x,t) −→ f± (x)
a.e. as t −→ ±∞.
(1.3)
Sufficient conditions were given on the functions f± to guarantee the existence of a solution of (1.1). (Some of the references are listed in the bibliography. They mention other authors as well.) In the present paper, we consider the situation in which (1.3) does not hold. In fact, we do not require any knowledge of the asymptotic behavior of f (x,t) as |t | → ∞. As an example, we have the following. Theorem 1.1. Assume that
sup
v∈E(λ ) Ω
F(x,v)dx < ∞,
(1.4)
where E(λ ) is the eigenspace of λ and F(x,t) = Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:1 (2005) 1–8 DOI: 10.1155/BVP.2005.1
t 0
f (x,s)ds.
(1.5)
2
Semilinear problems with bounded nonlinear term
Assume also that if there is a sequence {uk } such that P uk −→ ∞,
2
Ω
I − P uk ≤ C,
F x,uk dx −→ b0 ,
f x,uk −→ f (x)
(1.6)
weakly in L2 (Ω),
where f (x) ⊥ E(λ ) and P is the projection onto E(λ ), then
b0 ≤ f ,u1 − B0 , where B0 = tion of
Ω W0 (x)dx,
(1.7)
W0 (x) = supt [(λ−1 − λ )t 2 − 2F(x,t)], and u1 is the unique solu−∆u − λ u = f ,
u ⊥ E λ .
(1.8)
Then (1.1) has at least one solution. In particular, the conclusion holds if there is no sequence satisfying (1.6). A similar result holds if (1.4) is replaced by
inf
v∈E(λ ) Ω
F(x,v)dx > −∞.
(1.9)
In proving these results we will make use of the following theorem [2]. Theorem 1.2. Let N be a closed subspace of a Hilbert space H and let M = N ⊥ . Assume that at least one of the subspaces M, N is finite dimensional. Let G be a C 1 -functional on H such that m1 := inf sup G(v + w) < ∞, w∈M v∈N
m0 := sup inf G(v + w) > −∞.
(1.10)
v∈N w∈M
Then there are a constant c ∈ R and a sequence {uk } ⊂ H such that m 0 ≤ c ≤ m1 ,
G uk −→ c,
G uk −→ 0.
(1.11)
2. The main theorem We now state our basic result. Let Ω be a domain in Rn , and let A be a selfadjoint operator on L2 (Ω) such that the following hold. (A) σe (A) ⊂ (0, ∞).
(2.1)
(B) There is a function V (x) > 0 in L2 (Ω) such that multiplication by V is a compact operator from D := D(|A|1/2 ) to L1 (Ω). (C) If u ∈ N(A) \ {0}, then u = 0 a.e. in Ω.
Martin Schechter 3 Let f (x,t) be a Carath´eodory function on Ω × R satisfying (D) f (x,t) ≤ V (x).
(2.2)
Let λ(λ) be the largest (smallest) negative (positive) point in σ(A), and define
W0 (x) := sup λt 2 − 2F(x,t) , t
(2.3)
W1 (x) := sup 2F(x,t) − λt 2 ,
(2.4)
t
where F(x,t)
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