Existence and Multiplicity of Solutions for a Robin Problem

Volume 10, Issue 3, pp 163-172 http://dx.doi.org/10.22436/jmcs.010.03.02

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Authors

Mostafa Allaoui - Department of Mathematics, University Mohamed I, Oujda, Morocco. Abdel Rachid El Amrouss - Department of Mathematics, University Mohamed I, Oujda, Morocco. Fouad Kissi - Department of Mathematics, University Mohamed I, Oujda, Morocco. Anass Ourraoui - Department of Mathematics, University Mohamed I, Oujda, Morocco.

Abstract

In this article we study the nonlinear Robin boundary-value problem \[ \begin{cases} -\Delta_{p(x)}u=\lambda f(x,u),\,\,\,\,\, \texttt{in}\quad \Omega,\\ |\nabla u|^{p(x)-2} \frac {\partial u}{\partial v} + \beta(x)|u|^{p(x)-2} u=0,\,\,\,\,\, \texttt{on}\quad \partial \Omega. \end{cases} \] Using the variational method, under appropriate assumptions on \(f\), we obtain a result on existence and multiplicity of solutions.

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Keywords

• \(p(x)\)-Laplace operator
• variable exponent Lebesgue space
• variable exponent Sobolev space
• Riccerifs variational principle.

MSC

•  34B09
•  47N20

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