The planar Schr\"odinger-Poisson system with a positive potential

TL;DR

This paper investigates the existence of solutions to a planar Schr"odinger-Poisson system with a positive potential, providing new existence results for the case with a positive sign and examples for the negative case.

Contribution

It offers the first general existence theorem for the positive sign case in two dimensions and presents examples of solutions in the negative case.

Findings

Existence of solutions for the positive sign case in 2D.

Examples of solutions for the negative sign case.

Comparison with previous 3D results.

Abstract

In this paper we consider the problem \begin{equation*} \left \{ \begin{array}{l} -\Delta u \pm \phi u + W'(x,u) = 0\hbox{ in } \mathbb{R}^2,\newline \Delta \phi = u^2 \hbox{ in } \mathbb{R}^2, \end{array} \right. \end{equation*} where $W$ is assumed positive. In dimension three, the problem with the sign + (we call it $(\mathcal P_+)$) was considered and solved in \cite{M}, whereas in the same paper it was showed that no nontrivial solution exists if we consider the sign -- (say it $(\mathcal P_-)$). We provide a general existence result for $(\mathcal P_+)$ and two examples falling in the case $(\mathcal P_-)$ for which there exists at least a nontrivial solution.