On the spectrum of Robin boundary p-Laplacian problem

Abdelouahed El Khalil

doi:10.2478/mjpaa-2019-0020

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On the spectrum of Robin boundary p-Laplacian problem

Moroccan Journal of Pure and Applied Analysis

Volume 5 (2019): Issue 2 (December 2019)

By: Abdelouahed El Khalil

Open Access

|Jan 2020

Abstract

We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition

${\begin{matrix} - Δ_{p} u = λ {| u |}^{p - 2} u i n Ω, \\ {| \nabla u |}^{p - 2} \nabla u . v + {| u |}^{p - 2} u = 0 o n Γ . \end{matrix}$ \left\{ {\matrix{ { - {\Delta _p}u = \lambda {{\left| u \right|}^{p - 2}}u\,\,\,in\,\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u.v + {{\left| u \right|}^{p - 2}}u = 0\,\,\,on\,\,\Gamma .} \hfill \cr } } \right.

We successfully investigate the existence at least of one nondecreasing sequence of positive eigenvalues λ_n↗^∞. To this end we endow W^1,^p(Ω) with a norm invoking the trace and use the duality mapping on W^1,^p (Ω) to apply mini-max arguments on C¹-manifold.

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