Maryam Mirzapour - Department of Mathematics, Faculty of Mathematical Sciences, Farhangian University, Tehran, Iran

In this article, we study the nonlocal p(x)-Laplacian problem of the following form\left\{\begin{array}{ll}M\Big (\int_{\Omega}\frac{۱}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\Big)\Big(-\mathrm{div}(|\nabla u|^{p(x)-۲}\nabla u+|u|^{p(x)-۲}u\Big) =\lambda f(x,u) &\text{ in } \Omega,\\M\Big (\int_{\Omega}\frac{۱}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\Big)|\nabla u|^{p(x)-۲}\nabla \frac{\partial u}{\partial \nu}=\mu g(x,u) & \textrm{ on } \partial\Omega,\end{array}\right.By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence and multiplicity of solutions for the problem.

Generalized Lebesgue-Sobolev spaces, Nonlocal condition, Mountain Pass Theorem, Fountain theorem, Dual fountain theorem