Ground state solutions for Schrödinger-Poisson system with a doping profile

TL;DR

This work analyzes ground state solutions of the Schrödinger–Poisson system with a doping profile in $\mathbb{R}^3$, introducing the energy functional with a nonlocal term coupled to $\rho$ and establishing a variational framework. Ground states are obtained by minimizing on the Nehari–Pohozaev set, aided by a key energy inequality that enables a unique scaling and clarifies the relationship between action and energy GSS under an $L^2$ constraint. The paper provides existence results for $2<p<5$, a radial ground state for $1<p<2$, and a treatment of the case where $\rho$ is a characteristic function, all under smallness conditions on the doping profile; it also connects energy minimizers under mass constraint to action minimizers, deepening understanding of nonlocal coupling with inhomogeneous doping. These results advance the mathematical understanding of inhomogeneous Schrödinger–Poisson systems and have potential implications for semi-conductor models with impurities.

Abstract

This paper is devoted to the study of the nonlinear Schrödinger-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.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5