The Neumann Problem for a Degenerate Elliptic System Near Resonance

Abstract

This paper studies the following system of degenerate equations -divpx∇u+qxu=αu+βv+g1x,v+h1x, x∈Ω, -div(p(x)∇v)+q(x)v=βu+αv+g2(x,u)+h2(x), x∈Ω, ∂u/∂ν=∂v/∂ν=0, x∈∂Ω. Here Ω⊂Rn is a bounded C2 domain, and ν is the exterior normal vector on ∂Ω. The coefficient function p may vanish in Ω¯, q∈Lr(Ω) with r>ns/(2s-n), s>n/2. We show that the eigenvalues of the operator -div(p(x)∇u)+q(x)u are discrete. Secondly, when the linear part is near resonance, we prove the existence of at least two different solutions for the above degenerate system, under suitable conditions on h1,h2,g1, and g2.