Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations in $\mathbb{R}^{N}$ with indefinite weight functions

Abstract

This paper considers the following sublinear Schrödinger-Maxwell system \begin{equation}\label{man} \begin{cases} -\Delta u+V(x)u+K(x) \phi u =a(x) |u |^{q-1}u, \ \ \ \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ -\Delta \phi = K(x)u^{2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation} where $N=3, $ $ 0 <q <1,$ and $ a,K,V\in{L^{\infty}} (\mathbb{R}^{3} )$. We suppose that $K$ is a positive function and $a, V$ both change sign in $\mathbb{R}^{3}$. There seems to be no results on the existence of infinitely many solutions to problem (0.1). The proof is based on the Symmetric Mountain Pass theorem.