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

TL;DR

This paper establishes the existence of symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth, using variational methods and novel techniques to handle nonlocal terms and general nonlinearities.

Contribution

It extends previous results by considering more general nonlinearities, weaker conditions, and multiple symmetry types, with innovative methods for energy functional analysis.

Findings

Existence of nontrivial solutions with various symmetries.

Handling of critical exponential growth nonlinearities.

Introduction of new Moser type functions for compactness.

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.