  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.
