  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.
