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$$left{begin{array}{ll}- delta_{vec{p}(x)} u+ lambda(x)|u|^{p_{0}(x)-2}u = alpha f(x,u)+ beta g(x,u) quad mbox{in} quad omega, displaystylesum_{i=1}^{n}big| frac{partial u}{partial x_{i}}big|^{p_{i}(x)-2}frac{partial u}{partial x_{i}}gamma_{i} =0 quad mbox{on} quad partialomega.end{array}right.$$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.