Ground states of planar Schr\"{o}dinger-Poisson systems with an unbounded potential

Abstract

In this paper, we deal with a class of planar Schr\"{o}dinger-Poisson systems, namely, $-\Delta u+V(x)u+\frac{\gamma}{2\pi}\bigl(\log(|\cdot|)\ast|u|^{2}\bigr)u=b|u|^{p-2}u\ \text{in}\ \mathbb{R}^{2}$, where $\gamma > 0$, $b \geq 0$, $p>2$ and $V \in C(\mathbb{R}^2, \mathbb{R})$ is an unbounded potential function with $\inf_{\mathbb{R}^2} V >0$. Suppose moreover that the potential $V$ satisfies $\left|\{x \in \mathbb{R}^2:\: V(x)\leq M\}\right| < \infty$ for every $M>0$, we establish the existence of ground state solutions for this system via variational methods. Furthermore, we also explore the minimax characterization of ground state solutions. Our main results can be viewed as a counterpart of the result from Molle and Sardilli (Proc. Edinb. Math. Soc. 65:1133-1146, 2022), where the authors studied the existence of ground state solutions for the above planar Schr\"{o}dinger-Poisson system in the case where $b>0$ and $p >4$.