Group invariant solutions for the planar Schrödinger-Poisson equations

This paper is concerned with the following planar Schrödinger-Poisson equations

$$-\mathrm{\Delta }u+V(x)u+\left(\ln |\cdot |\ast |u{|}^p\right)|u{|}^{p-2}u=f(x,u),x\in {\mathbb{R}}^2,$$

where $p\ge 2$ is a constant, and $V(x)$ and $f(x,u)$ are continuous, mirror symmetric or rotationally periodic functions. The nonlinear term $f(x,u)$ satisfies a certain monotonicity condition and has critical exponential growth in the Trudinger-Moser sense. We adopted a version of mountain pass theorem by constructing a Cerami sequence, which in turn leads to a ground state solution. Our method has two new insights. First, we observed that the integral ${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\ln (|x-y|)|u(x){|}^p|u(y){|}^{p}dxdy$ is always negative if $u$ belongs to a suitable space. Second, we built a new Moser type function to ensure the boundedness of the Cerami sequence, which further guarantees its compactness. In particular, by replacing the monotonicity condition with the Ambrosetti–Rabinowitz condition, our approach works also for the subcritical growth case.