Ground states and high energy solutions of the planar Schr\"odinger-Poisson system

TL;DR

This paper investigates the existence and properties of ground states and high energy solutions for the planar Schrödinger-Poisson system, extending analysis to physically relevant exponents in the case where the spatial dimension is two.

Contribution

It introduces a new variational approach to analyze the Schrödinger-Poisson system in two dimensions for exponents 2<p<4, overcoming previous restrictions to p≥4.

Findings

Established existence of ground states in 2D for 2<p<4

Analyzed the functional geometry of the system in the planar case

Extended the range of physically relevant exponents studied

Abstract

In this paper, we are concerned with the Schr\"{o}dinger-Poisson system \begin{equation} (0.1)\qquad -\Delta u + u +\phi u = |u|^{p-2}u \quad \text{in}\ \mathbb{R}^{d},\qquad \Delta \phi= u^{2} \quad \text{in}\ \mathbb{R}^{d}. \end{equation} Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case $d \ge 3$. In contrast, much less information is available in the planar case $d=2$ which is the focus of the present paper. It has been observed by Cingolani and the second author \cite{Cingolani-Weth-2016} that the variational structure of $(0.1)$ differs substantially in the case $d=2$ and leads to a richer structure of the set of solutions. However, the variational approach of \cite{Cingolani-Weth-2016} is restricted to the case $p \ge 4$ which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case $2<p<4$ with a different variational approach.