On the Spectrum of a Class of Strongly Coupled p-Laplacian Systems

Abstract

Consider the nonlinear coupled elliptic system  -Δ_pu-λV(x)|u|^α|v|^βv=μ|u|^α|v|^βv, in Ω,  -Δ_qv-λV(x)|u|^α|v|^βu=μ|u|^α|v|^βu, in Ω,  u=v=0, in Ω,  where Δ_{ρ}ζ=∇⋅(|∇ζ|^{ρ-2}∇ζ),ρ > 1, Ω is a bounded domain and V(x) is a potential weight function. We prove that for any real parameter λ, there is at least a sequence of eigencurves (μ_k(λ))_k by using an energy variational method. We prove also via an homogeneity type condition that the eigenvector corresponding to the principal frequency μ_1(λ) is unique modulo scaling, bounded, regular and positive, without any condition on regularity of the domain. We end this work by giving a new proof technique to prove the simplicity of μ_1(λ) via a new version of Picones' identity.