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.
Categories: math.AP