Positive solutions for nonlinear Schr\"{o}dinger--Poisson Systems with general nonlinearity

TL;DR

This paper investigates the existence and properties of positive solutions for a class of nonlinear Schrödinger-Poisson systems with general nonlinearities, without relying on common growth conditions, and explores symmetry breaking phenomena.

Contribution

It introduces new estimates and variational methods to prove existence of ground state solutions without Ambrosetti-Rabinowitz conditions and analyzes symmetry breaking in solutions.

Findings

Existence of ground state solutions established.

Zero solutions when charge exceeds a positive threshold.

Multiple positive solutions and symmetry breaking when charge is radially symmetric.

Abstract

In this paper, we study a class of Schr\"{o}dinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the `charge' function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when `charge' function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions.