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

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.