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

In this paper, we are interested in the following planar Schrödinger–Poisson system

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+a(x)u+2\pi \phi u=|u|^{p-2}ue^{\alpha _0|u|^\gamma }, \ {} &{} x\in {\mathbb {R}}^2,\\ \Delta \phi =u^2,\ {} &{} x\in {\mathbb {R}}^2, \end{array} \right. \end{aligned}$$(0.1)

where \(p>2\), \(\alpha _0>0\) and \(0<\gamma \le 2\), the potential \(a:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) 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 \(0<\gamma <2\) and \(p>2\). Furthermore, in the critical case \(\gamma =2\) and \(p>4\), we prove that equation (0.1) possesses a positive solution which is invariant under the same group action.