On a superquadratic elliptic system with strongly indefinite structure

TL;DR

This paper proves the existence of infinitely many solutions for a superquadratic elliptic system with strongly indefinite structure using variational methods and a generalized fountain theorem.

Contribution

It introduces a new application of a generalized fountain theorem to establish multiple solutions for superquadratic elliptic systems with indefinite structure.

Findings

Proved existence of infinitely many solutions.

Improved upon previous results in the field.

Applied a generalized fountain theorem to strongly indefinite functionals.

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.