On the planar Schrödinger–Poisson system

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 $w(x)\to -\infty$ as $|x|\to \infty$ 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.