The Neumann Problem for a Degenerate Elliptic System Near Resonance

Abstract EN

This paper studies the following system of degenerate equations - d i v p x ∇ u + q x u = α u + β v + g 1 x , v + h 1 x , x ∈ Ω , - d i v ( p ( x ) ∇ v ) + q ( x ) v = β u + α v + g 2 ( x , u ) + h 2 ( x ) , x ∈ Ω , ∂ u / ∂ ν = ∂ v / ∂ ν = 0 , x ∈ ∂ Ω .

Here Ω ⊂ R n is a bounded C 2 domain, and ν is the exterior normal vector on ∂ Ω .

The coefficient function p may vanish in Ω ¯ , q ∈ L r ( Ω ) with r > n s / ( 2 s - n ) , s > n / 2 .

We show that the eigenvalues of the operator - d i v ( 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 h 1 , h 2 , g 1 , and g 2 .