We develop a variational framework to detect high energy solutions of the planar Schrödinger–Poisson system
with a positive function and . In particular, we deal with the periodic setting where the corresponding functional is invariant under -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 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 in and as 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. 169–197DOI 10.1016/J.ANIHPC.2014.09.008