On a class of planar Schr\"{o}dinger-Poisson system with a bounded potential well

Abstract

In this paper, we deal with the planar Schr\"{o}dinger-Poisson system \begin{equation*}\begin{cases} -\Delta u + V(x) u + \phi u = b|u|^{p-2} u \ &\text{in}\ \mathbb{R}^{2},\\\Delta \phi= u^{2} &\text{in}\ \mathbb{R}^{2},\end{cases} \end{equation*} where $b \geq 0$, $p > 2 $ and $V \in C(\mathbb{R}^2, \mathbb{R})$ is a potential function with $\inf_{\mathbb{R}^2} V >0$. Suppose moreover that $V$ exhibits a bounded potential well in the sense that $\lim_{|x|\rightarrow \infty} V(x)$ exists and is equal to $\sup_{\mathbb{R}^2} V$. By using variational methods, we obtain the existence of ground state solutions for this system in the case where $p \geq 3$. Furthermore, we also present a minimax characterization of ground state solutions. The main feature of this work is that we do not assume any periodicity or symmetry condition on the external potential $V$, which is essential to establish the compactness condition of Cerami sequences.