On a superquadratic elliptic system with strongly indefinite structure

Abstract: In this paper, we consider the elliptic system \begin{equation*} \left{\begin{array}{ll} -\Delta u=g(x,v)\,\, \textnormal{in}\Omega, & \hbox{} -\Delta v=f(x,u)\,\,\textnormal{in}\Omega, & \hbox{} u=v=0\textnormal{on}\partial\Omega, & \hbox{} \end{array} \right. \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, and $f$ and $g$ satisfy a general superquadratic condition. By using variational methods, we prove the existence of infinitely many solutions. Our argument relies on the application of a generalized variant fountain theorem for strongly indefinite functionals. Previous results in the topic are improved.