Existence of Ground State Solutions for the Schrödinger–Poisson System in $\mathbb{R}^{2}$

Abstract

This paper concerns the following planar Schrödinger–Poisson system \[ \begin{cases} -\Delta u + V(x) u + \phi u = |u|^{p-2} u &\textrm{in $\mathbb{R}^{2}$}, \\ \Delta \phi = u^{2}, \end{cases} \] where $p \geq 3$. By developing some new analytic techniques and variational methods, we establish a local compactness splitting lemma, and prove that this system possesses ground state solutions. We extend the case where $V(x)$ is a constant coefficient to the case where $V(x)$ is a variable coefficient. Some related results are improved.