Multiplicity of weak solutions for a class of quasilinear elliptic Neumann problems using Variational methods

The existence of infinitely many weak solutions for the strongly nonlinear elliptic equation of the form $$\left\{\begin{array}{ll} -\mathrm{div}\Big( w_{1}(x)|\nabla u|^{p(x)-2}\nabla u\Big) + w_{0}(x){\mid u \mid}^{p(x)-2}u = f(x,u)+ g(x,u) \quad &\mbox{in} \quad \Omega, \\ \frac{\partial u}{\partial \gamma}=0\quad \textrm{on }\partial \Omega. \end{array}\right.$$ is proved by applying a critical point variational principle obtained by B. Ricceri in weighted variable exponent Sobolev space $W^{1,p(\cdot)}(\Omega,w_{0},w_{1})$.