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Volume 24, Issue 3
On the Spectrum of a Class of Strongly Coupled p-Laplacian Systems

Abdelouahed El Khalil

DOI: 10.4208/jpde.v24.n3.1

J. Part. Diff. Eq., 24 (2011), pp. 195-206.

Published online: 2011-08

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  • 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.

  • Keywords

Coupled p-Laplacian systems eigencurves energy variational method Picones' identity simplicity

  • AMS Subject Headings

35J20 35J45 35J50 35J70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-24-195, author = {Abdelouahed El Khalil }, title = {On the Spectrum of a Class of Strongly Coupled p-Laplacian Systems}, journal = {Journal of Partial Differential Equations}, year = {2011}, volume = {24}, number = {3}, pages = {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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v24.n3.1}, url = {http://global-sci.org/intro/article_detail/jpde/5207.html} }
TY - JOUR T1 - On the Spectrum of a Class of Strongly Coupled p-Laplacian Systems AU - Abdelouahed El Khalil JO - Journal of Partial Differential Equations VL - 3 SP - 195 EP - 206 PY - 2011 DA - 2011/08 SN - 24 DO - http://doi.org/10.4208/jpde.v24.n3.1 UR - https://global-sci.org/intro/article_detail/jpde/5207.html KW - Coupled p-Laplacian systems KW - eigencurves KW - energy variational method KW - Picones' identity KW - simplicity AB -

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.

Abdelouahed El Khalil . (2011). On the Spectrum of a Class of Strongly Coupled p-Laplacian Systems. Journal of Partial Differential Equations. 24 (3). 195-206. doi:10.4208/jpde.v24.n3.1
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