On a class of planar Schrödinger-Poisson system with a bounded potential well

Abstract

In this paper, we deal with the planar Schrödinger-Poisson system \begin{equation*}\begin{cases} -Δu + V(x) u + ϕu = b|u|^{p-2} u \ &\text{in}\ \mathbb{R}^{2},\\Δϕ= u^{2} &\text{in}\ \mathbb{R}^{2},\end{cases} \end{equation*} where $b\ge 0$, $p>2$ 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 ${\lim }_{|x|\to \infty }V(x)$ 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 $p\ge 3$. 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.