  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.
