The planar Schrödinger-Poisson system with a positive potential

Abstract

In this paper we consider the problem \begin{equation*} \left \{ \begin{array}{l} -Δu \pm ϕu + W'(x,u) = 0\hbox{ in } \mathbb{R}^2,\newline Δϕ= 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.

13 pages

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