On the planar Schrodinger-Poisson system

We develop a variational framework to detect high energy solutions of the planar Schrodinger-Poisson system {-Delta u + a(x)u + gamma wu = 0, {Delta w = u(2) in R-2 with a positive function a is an element of L-infinity(R-2) and gamma > 0. In particular, we deal with the periodic setting where the corresponding functional is invariant under 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 R-2 and w(x) -> -infinity as vertical bar x vertical bar -> infinity 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.