Infinitely many solutions for a Schrödinger-Poisson system withconcave and convex nonlinearities

Abstract

In this paper, we obtain theexistence of infinitely many solutions for the followingSchrödinger-Poisson system\begin{equation*}\begin{cases}-\Delta u+a(x)u+ \phi u=k(x)|u|^{q-2}u- h(x)|u|^{p-2}u,\quad&x\in \mathbb{R}^3,\\-\Delta \phi=u^2,\ \lim_{|x|\to +\infty}\phi(x)=0, &x\in\mathbb{R}^3,\end{cases}\end{equation*} where $1 < q < 2 < p < +\infty$, $a(x)$, $k(x)$ and $h(x)$ are measurable functions satisfying suitable assumptions.