Existence of infinitely many solutions to the Neumann problem for quasilinear elliptic systems involving the $\vec{p}(x)$ and $\vec{q}(x)$-Laplacian

Abstract

In this paper, we consider the following quasilinear Neumann boundary-value problem of the type
\begin{equation*}\left\{\begin{array}{ccc} && -\Delta_{\vec{p}(x)}+a(x)|u|^{p_0(x)-2}u=\alpha(x)f(u,v)\quad \textrm{in }\Omega\\ && -\Delta_{\vec{q}(x)}+b(x)|v|^{q_0(x)-2}v=\alpha(x)g(u,v)\quad \textrm{in }\Omega\\
&&\frac{\partial u}{\partial \gamma}=0\quad \textrm{on }\partial \Omega\\
&&\frac{\partial v}{\partial \gamma}=0\quad \textrm{on }\partial \Omega
\end{array}\right. \end{equation*} We prove the existence of infinitely many solutions of the problem under weaker hypotheses by applying a variational principle due to B. Ricceri and the theory of the Anisotropic variable exponent Sobolev spaces. Our results are an improvement and generalization of the relative results \cite{Danila}.