Ground states of planar Schrödinger–Poisson systems with an unbounded potential

In this paper, we deal with a class of planar Schrödinger–Poisson systems, namely, \(-\Delta u+V(x)u+\frac{\gamma }{2\pi }\bigl (\log (|\cdot |)*|u|^{2}\bigr )u=b|u|^{p-2}u\ \text {in}\ \mathbb {R}^{2}\) $-\mathrm{\Delta }u+V(x)u+\frac{\gamma }{2\pi }(log(|·{|)\ast |u|}^2)u=b{|u|}^{p-2}u\text{in}{\mathbb{R}}^2$ , where \(\gamma > 0\) $\gamma >0$ , \(b \ge 0\) $b\ge 0$ , \(p>2\) $p>2$ and \(V \in C(\mathbb {R}^2, \mathbb {R})\) $V\in C({\mathbb{R}}^2,\mathbb{R})$ is an unbounded potential function with \(\inf _{\mathbb {R}^2} V >0\) ${inf}_{{\mathbb{R}}^2}V>0$ . Suppose moreover that the potential V satisfies \(\left| \{x \in \mathbb {R}^2:\, V(x)\le M\}\right| < \infty \) $\left|\{,x,\in ,{\mathbb{R}}^2,:,,V,(x),\le ,M,\}\right|<\infty$ for every \(M>0\) $M>0$ , 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 results from Molle and Sardilli (Proc Edinb Soc 65:1133–1146, 2022), where the authors studied the existence of ground state solutions for the above planar Schrödinger–Poisson system in the case where \(b>0\) $b>0$ and \(p >4\) $p>4$ .