Solutions for Neumann boundary value problems involving $\big(p_{1}(x), p_{2}(x)\big)$-Laplace operators

In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=\lambda f(x,u) in a bounded smooth domain $\Omega\subset\mathbb{R}^{N}$ with Neumann boundary condition given by |\nabla u|^{p_{1}(x)-2}\frac{\partial u}{\partial\nu}+|\nabla u|^{p_{2}(x)-2}\frac{\partial u}{\partial\nu}=\mu g(x,u) on $\partial\Omega$. Under appropriate conditions on the source and boundary nonlinearities, we obtain a number of results on existence and multiplicity of solutions by variational methods in the framework of variable exponent Lebesgue and Sobolev spaces.