# On the planar Schrödinger–Poisson system

### Silvia Cingolani

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy### Tobias Weth

Institut für Mathematik, Goethe-Universität Frankfurt, D-60054 Frankfurt am Main, Germany

## Abstract

We develop a variational framework to detect high energy solutions of the planar Schrödinger–Poisson system

with a positive function $a \in L^{\infty }(\mathbb{R}^{2})$ and $\gamma > 0$. In particular, we deal with the periodic setting where the corresponding functional is invariant under $\mathbb{Z}^{2}$-translations and therefore fails to satisfy a global Palais–Smale condition. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. In the case where the external potential *a* is a positive constant, we also derive, as a special case of a more general result, the existence of nonradial solutions $(u,w)$ such that *u* has arbitrarily many nodal domains. Finally, in the case where *a* is constant, we also show that solutions of the above problem with $u > 0$ in $\mathbb{R}^{2}$ and $w(x)\rightarrow −\infty$ as $|x|\rightarrow \infty$ are radially symmetric up to translation. Our results are also valid for a variant of the above system containing a local nonlinear term in *u* in the first equation.

## Cite this article

Silvia Cingolani, Tobias Weth, On the planar Schrödinger–Poisson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 1, pp. 169–197

DOI 10.1016/J.ANIHPC.2014.09.008