A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition

VicenŢiu D. RǍdulescu; Somayeh Saiedinezhad

doi:10.3934/cpaa.2018003

Article Contents

2018, Volume 17, Issue 1: 39-52. Doi: 10.3934/cpaa.2018003

This issue Previous Article Unilateral global interval bifurcation for Kirchhoff type problems and its applications Next Article Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential

A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition

VicenŢiu D. RǍdulescu^1, and
Somayeh Saiedinezhad^2, ,

1.
Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,014700 Bucharest, Romania, Department of Mathematics, University of Craiova, Street A.I. Cuza 13,200585 Craiova, Romania
2.
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

* Corresponding author: Somayeh Saiedinezhad
* Corresponding author: Somayeh Saiedinezhad

Received: July 31, 2016

Revised: June 30, 2017

Published: December 2017

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Abstract

We are concerned with the study of the following nonlinear eigenvalue problem with Robin boundary condition

$\begin{cases} -{\rm div}\,(a(x,\nabla u))=λ b(x,u)&\mbox{in} \ Ω\\\dfrac{\partial A}{\partial n}+β(x) c(x,u)=0&\mbox{on}\\partialΩ.\end{cases}$

The abstract setting involves Sobolev spaces with variable exponent. The main result of the present paper establishes a sufficient condition for the existence of an unbounded sequence of eigenvalues. Our arguments strongly rely on the Lusternik-Schnirelmann principle. Finally, we focus to the following particular case, which is a $p(x)$-Laplacian problem with several variable exponents:

$\begin{cases} -{\rm div}\,(a_0(x) |\nabla u|^{p(x)-2}\nabla u)=λ b_0(x)|u|^{q(x)-2}u&\mbox{in} \ Ω\\|\nabla u|^{p(x)-2}\dfrac{\partial u}{\partial n}+β(x)|u|^{r(x)-2} u=0&\mbox{on}\\partialΩ.\end{cases}$

Combining variational arguments, we establish several properties of the eigenvalues family of this nonhomogeneous Robin problem.

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Mathematics Subject Classification: Primary:34B09, 35P30;Secondary:35J20.

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