带有渐近线性项和库伦位势的薛定谔-泊松系统

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 650-660.

带有渐近线性项和库伦位势的薛定谔-泊松系统

常金华,魏娜^*()

中南财经政法大学统计与数学学院武汉 430073

收稿日期:2023-04-27 修回日期:2023-10-14 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *魏娜, Email:weina@zuel.edu.cn
基金资助:
国家自然科学基金(12071482)

On the Schrödinger-Poisson System with Asymptotically Linear Term and Coulomb Potential

Chang Jinhua,Wei Na^*()

hongnan University of Economics and Law, Wuhan 430073

Received:2023-04-27 Revised:2023-10-14 Online:2024-06-26 Published:2024-05-17
Supported by:
NSFC(12071482)

摘要/Abstract

摘要：

该文研究下列薛定谔-泊松系统

(P)$\begin{equation} \left\{\begin{array}{ll} -\Delta u + \Big(\omega-\sum\limits_{i=1}^m\frac{1}{|x-x_i|}\Big)u+\lambda\phi (x)u =f(u)\,\,\,& x\in \mathbb{R}^3, \\ -\Delta\phi = |u|^{2},\, \ &u\in H^1(\mathbb{R}^3), \end{array}\right. \end{equation}$

其中 $\omega>0$, $\lambda>0$, $x_i\in\mathbb{R}^3$, $m\in\mathbb{N}$, $f(u)\sim lu\ (u\rightarrow+\infty)$ 是渐近线性项. 该文应用变分方法研究参数 $\omega$, $\lambda$ 及渐近系数 $l$ 的取值范围对系统 (P) 的基态解的存在性及解的多重性的影响等.

关键词: 椭圆型方程, 渐近线性, 变分方法

Abstract:

The purpose of this paper is to study the following Schrödinger-Poisson system

(P)$\begin{equation} \left\{\begin{array}{ll} -\Delta u + \Big(\omega-\sum\limits_{i=1}^m\frac{1}{|x-x_i|}\Big)u+\lambda\phi (x)u =f(u)\,\,\, &x\in \mathbb{R}^3, \\ -\Delta\phi = |u|^{2},\, \ &u\in H^1(\mathbb{R}^3), \end{array}\right. \end{equation}$

where $\omega>0$, $\lambda>0$, $x_i\in\mathbb{R}^3$, $ m\in\mathbb{N}$, $f(u)\sim lu$ (as $u\rightarrow+\infty$) is the asymptotically linear term. We study the effect of values of parameters $\omega$, $\lambda$ and asymptotic coefficient $l$ on the existence of ground state, multiple solutions to system (P), by using the variational method.

Key words: Elliptic equation, Asymptotically linear, Variational method

中图分类号:

O175.25

常金华, 魏娜. 带有渐近线性项和库伦位势的薛定谔-泊松系统[J]. 数学物理学报, 2024, 44(3): 650-660.

Chang Jinhua, Wei Na. On the Schrödinger-Poisson System with Asymptotically Linear Term and Coulomb Potential[J]. Acta mathematica scientia,Series A, 2024, 44(3): 650-660.

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