On the Schrödinger-Poisson system with indefinite potential and $3$-sublinear nonlinearity

Abstract

We consider a class of stationary Schrödinger-Poisson systems with a general nonlinearity $f(u)$ and coercive sign-changing potential $V$ so that the Schrödinger operator $-\Delta +V$ is indefinite. Previous results in this framework required $f$ to be strictly $3$-superlinear, thus missing the paramount case of the Gross-Pitaevskii-Poisson system, where $f(t)=|t{|}^{2}t$; in this paper we fill this gap, obtaining non-trivial solutions when $f$ is not necessarily $3$-superlinear.

23 pages, comments welcome!