On the planar Schrödinger–Poisson system

We develop a variational framework to detect high energy solutions of the planar Schrödinger–Poisson system {cases} −Δ u + a(x)u + γ wu = 0, \\ Δ w = u² \\ {cases} in R² 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.