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

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