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
  • Received: 2013-11-29.
  • Revised: 2014-02-03.
  • Accepted: 2014-02-04.
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Cite this article as:
Najib Tsouli, Omar Darhouche, Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator, Opuscula Math. 34, no. 3 (2014), 621-638, http://dx.doi.org/10.7494/OpMath.2014.34.3.621

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