  In this paper, we deal with a class of planar Schr\"{o}dinger-Poisson
systems, namely, $-\Delta
u+V(x)u+\frac{\gamma}{2\pi}\bigl(\log(|\cdot|)\ast|u|^{2}\bigr)u=b|u|^{p-2}u\
\text{in}\ \mathbb{R}^{2}$, where $\gamma > 0$, $b \geq 0$, $p>2$ and $V \in
C(\mathbb{R}^2, \mathbb{R})$ is an unbounded potential function with
$\inf_{\mathbb{R}^2} V >0$. Suppose moreover that the potential $V$ satisfies
$\left|\{x \in \mathbb{R}^2:\: V(x)\leq M\}\right| < \infty$ for every $M>0$,
we establish the existence of ground state solutions for this system via
variational methods. Furthermore, we also explore the minimax characterization
of ground state solutions. Our main results can be viewed as a counterpart of
the result from Molle and Sardilli (Proc. Edinb. Math. Soc. 65:1133-1146,
2022), where the authors studied the existence of ground state solutions for
the above planar Schr\"{o}dinger-Poisson system in the case where $b>0$ and $p
>4$.
