Mathematics

CLC number: O175.29

Existence of Ground State Solutions to Nonlinear Schrödinger-Poisson System with Doping Profile and Coercive Potential

含掺杂分布与强制势的非线性薛定谔-泊松系统基态解的存在性

School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, Gansu, China (西北民族大学　数学与计算机科学学院，甘肃　兰州　730030)

^† Corresponding author. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 31 December 2025

Abstract

This paper investigates the existence of ground state solutions to the following nonlinear Schrödinger-Poisson system by variational methods $\{\begin{array}{l}-\mathrm{\Delta }u+\omega u+V\left(x\right)u+e\varphi u={\left|u\right|}^{p-\mathrm{2}}u,x\in {\mathbb{R}}^{\mathrm{3}},\\ -\mathrm{\Delta }\varphi =\frac{e}{\mathrm{2}}\left(u^{\mathrm{2}}-\rho \left(x\right)\right),x\in {\mathbb{R}}^{\mathrm{3}},\end{array}$ provided that $\rho \left(x\right)\ge \mathrm{0}$ and $e^{\mathrm{2}}{\Vert \rho \Vert }_{\frac{\mathrm{6}}{\mathrm{5}}}\le {\rho }_{\mathrm{0}}$ with ${\rho }_{\mathrm{0}}>\mathrm{0}$ small enough, where $\mathrm{4}<p<\mathrm{6}$, $e>\mathrm{0}$ denotes a coupling constant, $\omega >\mathrm{0}$ is a Lagrange multiplier, the potential $V\left(x\right)$ is coercive and the doping profile $\rho \left(x\right)$ satisfies appropriate decay conditions. The introduction of $\rho \left(x\right)$ compromises the coercivity of the energy functional. Therefore, we establish the existence of ground state solutions by considering the minimization problem on Nehari manifold.

摘要

本文利用变分方法研究了当满足条件$\rho \left(x\right)\ge \mathrm{0}$且$e^{\mathrm{2}}{\Vert \rho \Vert }_{\frac{\mathrm{6}}{\mathrm{5}}}\le {\rho }_{\mathrm{0}}$(${\rho }_{\mathrm{0}}>\mathrm{0}$为足够小的常数)时下述非线性薛定谔-泊松系统基态解的存在性：$\{\begin{array}{l}-\mathrm{\Delta }u+\omega u+V\left(x\right)u+e\varphi u={\left|u\right|}^{p-\mathrm{2}}u,x\in {\mathbb{R}}^{\mathrm{3}},\\ -\mathrm{\Delta }\varphi =\frac{e}{\mathrm{2}}\left(u^{\mathrm{2}}-\rho \left(x\right)\right),x\in {\mathbb{R}}^{\mathrm{3}},\end{array}$其中$\mathrm{4}<p<\mathrm{6}$，$e>\mathrm{0}$为耦合常数，$\omega >\mathrm{0}$为拉格朗日乘子，势函数$V\left(x\right)$具有强制性，掺杂分布$\rho \left(x\right)$满足适当的衰减条件。$\rho \left(x\right)$的引入破坏了能量泛函的强制性。因此，本文通过考虑奈哈里流形上的极小化问题来研究基态解的存在性。