On the planar Schrödinger–Poisson system

Abstract

We develop a variational framework to detect high energy solutions of the planar Schrödinger–Poisson system{−Δu+a(x)u+γwu=0,Δw=u2inR2 with a positive function a∈L∞(R2) and γ>0. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z2-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 R2 and w(x)→−∞ as |x|→∞ 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.