Ground state solutions for Schr\"odinger-Poisson system with a doping profile

TL;DR

This paper investigates the existence of ground state solutions for the nonlinear Schrödinger-Poisson system with a doping profile, addressing challenges posed by the doping profile and establishing key relations and conditions for solutions.

Contribution

It introduces a new approach to prove existence of ground states considering doping profiles and relates solutions to geometric properties of the domain.

Findings

Existence of ground state solutions under certain conditions.

Relation between ground states and $L^2$-constraint minimizers.

Geometric quantities influence solution existence.

Abstract

This paper is devoted to the study of the nonlinear Schr\"odinger-Poisson system with a doping profile. We are interested in the existence of ground state solutions by considering the minimization problem on a Nehari-Pohozaev set. The presence of a doping profile causes several difficulties, especially in the proof of the uniqueness of a maximum point of a fibering map. A key ingredient is to establish the energy inequality. We also establish the relation between ground state solutions and $L^2$-constraint minimizers. When the doping profile is a characteristic function supported on a bounded smooth domain, some geometric quantities related to the domain, such as the mean curvature,are responsible for the existence of ground state solutions.