On a planar Schrödinger-Poisson system involving a non-symmetric potential

Abstract

We prove the existence of a ground state positive solution of Schrödinger-Poisson systems in the plane of the form $$-\Delta u+V(x)u+\frac{\gamma }{2\pi }\left(\log |\cdot |\ast u^2\right)u=b|u{|}^{p-2}u\text{in}{\mathbb{R}}^2,$$ 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.