TY  - JOUR
ID  - Tsouli2014
LB  - Tsouli2014
AU  - Tsouli, Najib
AU  - Darhouche, Omar
TI  - Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator
ST  - Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2014
VL  - 34
IS  - 3
SP  - 621
EP  - 638
DO  - http://dx.doi.org/10.7494/OpMath.2014.34.3.621
UR  - http://dx.doi.org/10.7494/OpMath.2014.34.3.621
AB  - 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.
KW  - critical points
KW  - variational method
KW  - \(p(x)\)-Laplacian
KW  - generalized Lebesgue-Sobolev spaces
SN  - 1232-9274
ER  -