Group invariant solutions for the planar Schr\"{o}dinger-Poisson system

Abstract

This paper is concerned with the following planar Schr\"{o}dinger-Poisson system \begin{equation*} \begin{cases} -\triangle{u}+V(x)u+\phi{(x)}|u|^{p-2}u=f(x,u),&\text{in $\mathbb{R}^{2}$}, \triangle{\phi}=|u|^{p},&\text{in $\mathbb{R}^{2}$}, \end{cases} \end{equation*} where $p\geq2$ is a constant, $V(x)$ and $f(x,t)$ are continuous, mirror symmetric or rotationally periodic functions. By assuming that the nonlinearity $f(x,t)$ has critical exponential growth, we obtain a nontrivial solution or a ground state solution of Nehari type to the above system. Our results extend previous works of Cao_Dai_Zhang and Chen-Tang. We handle more general nonlinearities $f$ with weaken constraint at infinity, and we assume only the (AR) type condition to take place of the monotonicity assumption. We considered all the cases $p\geq2$, and we show the existence of solutions with multiple types of symmetry. As in Chen_Tang, we adopt a version of mountain pass structure which provides a Cerami sequence, with two innovative points. First, we make a key observation for the sign of a crucial part of the energy functional corresponding to the nonlocal term $\phi|u|^{p-2}u$, and secondly we adopt a new Moser type functions to ensure the boundedness and compactness of the Cerami sequence. Moreover, our approach works also for the subcritical growth case, and generalizes recent works Liu_Radulescu_Tang_Zhang,Cao_Dai_Zhang,Chen_Tang.