Opuscula Math. 34, no. 3 (2014), 621-638
http://dx.doi.org/10.7494/OpMath.2014.34.3.621

Opuscula Mathematica

# Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator

Najib Tsouli
Omar Darhouche

Abstract. In this paper we study the following nonlinear boundary-value problem $-\Delta_{p(x)} u=\lambda f(x,u) \quad \text{ in } \Omega,$ $|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}+\beta(x)|u|^{p(x)-2}u=\mu g(x,u) \quad \text{ on } \partial\Omega,$ where $$\Omega\subset\mathbb{R}^N$$ is a bounded domain with smooth boundary $$\partial\Omega$$, $$\frac{\partial u}{\partial\nu}$$ is the outer unit normal derivative on $$\partial\Omega$$, $$\lambda, \mu$$ are two real numbers such that $$\lambda^{2}+\mu^{2}\neq0$$, $$p$$ is a continuous function on $$\overline{\Omega}$$ with $$\inf_{x\in \overline{\Omega}} p(x)\gt 1$$, $$\beta\in L^{\infty}(\partial\Omega)$$ with $$\beta^{-}:=\inf_{x\in \partial\Omega}\beta(x)\gt 0$$ and $$f : \Omega\times\mathbb{R}\rightarrow \mathbb{R}$$, $$g : \partial\Omega\times\mathbb{R}\rightarrow \mathbb{R}$$ are continuous functions. Under appropriate assumptions on $$f$$ and $$g$$, we obtain the existence and multiplicity of solutions using the variational method. The positive solution of the problem is also considered.

Keywords: critical points, variational method, $$p(x)$$-Laplacian, generalized Lebesgue-Sobolev spaces.

Mathematics Subject Classification: 35B38, 35D05, 35J20, 35J60, 35J66.

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• Najib Tsouli
• Department of Mathematics, University Mohamed I, Oujda, Morocco
• Omar Darhouche
• Department of Mathematics, University Mohamed I, Oujda, Morocco
• Revised: 2014-02-03.
• Accepted: 2014-02-04.