A planar Schrödinger-Poisson system with vanishing potentials and exponential critical growth

In this paper we look for ground state solutions of the elliptic system $$ \begin{cases} -\Delta u+V(x)u+\gamma\phi K(x)u = Q(x)f(u), &x\in\mathbb{R}^{2}, \\ \Delta \phi =K(x) u^{2}, &x\in\mathbb{R}^{2}, \end{cases} $$% where $\gamma> 0$ and the continuous potentials $V$, $K$, $Q$ satisfy some mild growth conditions and the nonlinearity $f$ has exponential critical growth. The key point of our approach is a new version of the Trudinger-Moser inequality for weighted Sobolev space.