Existence of three weak solutions for an anisotropic quasi-linear elliptic problem

We consider in this paper a Neumann \vec{p}(x)-elliptic problems of the type\left\{\begin{array}{ll}- \Delta_{\vec{p}(x)} u+ \lambda(x)|u|^{p_{۰}(x)-۲}u = \alpha f(x,u)+ \beta g(x,u) \quad &\mbox{in} \quad \Omega, \\\displaystyle\sum_{i=۱}^{N}\Big| \frac{\partial u}{\partial x_{i}}\Big|^{p_{i}(x)-۲}\frac{\partial u}{\partial x_{i}}\gamma_{i} =۰ \quad &\mbox{on} \quad \partial\Omega.\end{array}\right.We prove the existence of three weak solutions in the framework of anisotropic Sobolev spaces with variable exponent W^{۱,\vec{p}(\cdot)}(\Omega) under some hypotheses. The approach is based on a recent three critical points theorem for differentiable functionals.

Keywords:

Neumann elliptic problem , Weak solutions , Variational principle , Anisotropic variable exponent Sobolev spaces