On a planar Schr\"odinger-Poisson system involving a non-symmetric potential

TL;DR

This paper establishes the existence of positive ground state solutions for a Schr"odinger-Poisson system in the plane with a non-symmetric, unbounded potential using variational methods.

Contribution

It proves the existence of solutions without symmetry or periodicity assumptions on the potential, extending previous results to more general potentials.

Findings

Existence of positive ground state solutions proven

Solutions exist for potentials unbounded at infinity

Variational methods successfully applied to non-symmetric potentials

Abstract

We prove the existence of a ground state positive solution of Schr\"odinger-Poisson systems in the plane of the form $$ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \qquad\text{in}\ \mathbb{R}^2, $$ where $p>4$, $\gamma,b>0$ and the potential $V$ is assumed to be positive and unbounded at infinity. On the potential we do not require any symmetry or periodicity assumption, and it is not supposed it has a limit at infinity. We approach the problem by variational methods, using a variant of the mountain pass theorem and the Cerami compactness condition.