# On the Schr\"{o}dinger-Poisson system with indefinite potential and $3$-sublinear nonlinearity

## TL;DR

This paper studies a Schr"{o}dinger-Poisson system with an indefinite potential and a nonlinearity that is not necessarily 3-superlinear, extending previous results to include the important Gross-Pitaevskii-Poisson case.

## Contribution

It extends the analysis of Schr"{o}dinger-Poisson systems to cases where the nonlinearity is 3-sublinear, covering the critical Gross-Pitaevskii-Poisson case previously unaddressed.

## Key findings

- Established existence of non-trivial solutions for the system.
- Included the critical case where the nonlinearity is |t|^2 t.
- Filled the gap in the theory for 3-sublinear nonlinearities.

## Abstract

We consider a class of stationary Schr\"{o}dinger-Poisson systems with a general nonlinearity $f(u)$ and coercive sign-changing potential $V$ so that the Schr\"{o}dinger 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.