Saddle solutions for the planar Schrödinger–Poisson system with exponential growth

In this paper, we are interested in the following planar Schrödinger–Poisson system 0.1 \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+a(x)u+2\pi \phi u=|u|^{p-2}ue^{\alpha _0|u|^\gamma }, \ {} &{} x\in {\mathbb {R}}^2,\\ \Delta \phi =u^2,\ {} &{} x\in {\mathbb {R}}^2, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+a(x)u+2\pi \varphi u={|u|}^{p-2}u{e}^{{\alpha }_0{|u|}^{\gamma }},\hfill & x\in {\mathbb{R}}^2,\hfill \\ \mathrm{\Delta }\varphi =u^2,\hfill & x\in {\mathbb{R}}^2,\hfill \end{array}\right)\end{array}$$ where \(p>2\) $p>2$ , \(\alpha _0>0\) ${\alpha }_0>0$ and \(0<\gamma \le 2\) $0<\gamma \le 2$ , the potential \(a:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) $a:{\mathbb{R}}^2\to \mathbb{R}$ is invariant under the action of a closed subgroup of the orthogonal transformation group O(2). As a consequence, we obtain infinitely many saddle type nodal solutions for equation (0.1) with their nodal domains meeting at the origin if \(0<\gamma <2\) $0<\gamma <2$ and \(p>2\) $p>2$ . Furthermore, in the critical case \(\gamma =2\) $\gamma =2$ and \(p>4\) $p>4$ , we prove that equation (0.1) possesses a positive solution which is invariant under the same group action.