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

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.