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$$\left\{\begin{array}{ll}- \Delta_{\vec{p}(x)} u+ \lambda(x)|u|^{p_{0}(x)-2}u = \alpha f(x,u)+ \beta g(x,u) \quad \mbox{in} \quad \Omega, \\\displaystyle\sum_{i=1}^{N}\Big| \frac{\partial u}{\partial x_{i}}\Big|^{p_{i}(x)-2}\frac{\partial u}{\partial x_{i}}\gamma_{i} =0 \quad \mbox{on} \quad \partial\Omega.\end{array}\right.$$We prove the existence of three weak solutions in the framework of anisotropic Sobolev spaces with variable exponent $W^{1,\vec{p}(\cdot)}(\Omega)$ under some hypotheses. The approach is based on a recent three critical points theorem for differentiable functionals.