Spectral problem for the Laplacian and a selfadjoint nonlinear elliptic boundary value problem

Abstract

In this paper, we present some connections between the spectral problem,

−Δu(x) = λ1u(x) in Ω,
u(x) = 0 on ∂Ω

and selfadjoint boundary value problem,

Δu(x) − λ1u(x) + g(x, u(x)) = h(x) in Ω,
u(x) = 0 on ∂Ω,

where λ₁ is the smallest eigenvalue of −∆, Ω ⊆ Rⁿ is a bounded domain, h ∈ L²(Ω) and the nonlinear function g is a Caratheodory function satisfying a growth condition. We initially investigate the existence of solutions for the spectral problem by considering the selfadjoint boundary value problem. The selfadjoint boundary value problem is then considered for both existence and estimation results. We use degree argument in order to show that the selfadjoint boundary value problem has a solution instead of the Landesman-Lazer condition or the monotonocity assumption on the second argument of the function g.

In this paper, we present some connections between the spectral problem,

and selfadjoint boundary value problem,

where λ₁ is the smallest eigenvalue of −∆, Ω ⊆ Rⁿ is a bounded domain, h ∈ L²(Ω) and the nonlinear function g is a Caratheodory function satisfying a growth condition. We initially investigate the existence of solutions for the spectral problem by considering the selfadjoint boundary value problem. The selfadjoint boundary value problem is then considered for both existence and estimation results. We use degree argument in order to show that the selfadjoint boundary value problem has a solution instead of the Landesman-Lazer condition or the monotonocity assumption on the second argument of the function g.