Year: 2011
Journal of Partial Differential Equations, Vol. 24 (2011), Iss. 3 : pp. 195–206
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v24.n3.1
Journal of Partial Differential Equations, Vol. 24 (2011), Iss. 3 : pp. 195–206
Published online: 2011-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 12
Keywords: Coupled p-Laplacian systems