带有非稳定位势的 Schrödinger-Poisson 方程解的存在性

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1492-1518.

带有非稳定位势的 Schrödinger-Poisson 方程解的存在性

刘泉^1,²(),叶江华^3,^*(),郑雄军¹()

¹江西师范大学数学与统计学院南昌 330022
²赣南师范大学数学与计算机科学学院江西赣州 341000
³华中师范大学数学与统计学学院武汉 430079

收稿日期:2024-03-27 修回日期:2025-05-12 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 叶江华, E-mail:jhye@mails.ccnu.edu.cn
作者简介:刘泉, E-mail:quan_liu@jxnu.edu.cn|郑雄军,E-mail:xjzh1985@126.com
基金资助:
国家自然科学基金(12361023);江西省自然科学基金重点项目(20242BAB26001);中国科协青年人才托举工程博士生专项计划

Existence of Solution for Schrödinger-Poisson Equation with Nonstabilizing Potential

Quan Liu^1,²(),Jianghua Ye^3,^*(),Xiongjun Zheng¹()

¹School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022
²School of Mathematics and Computer, Gannan Normal University, Jiangxi Ganzhou 341000
³School of Mathematics and Statistics, Central China Normal University, Wuhan 430079

Received:2024-03-27 Revised:2025-05-12 Online:2025-10-26 Published:2025-10-14
Supported by:
NSFC(12361023);Key Project of Jiangxi Provincial NSF(20242BAB26001);doctoral student special plan of the China Association for Science and Technology Youth Talent Lifting Project

摘要/Abstract

摘要：

该文研究了如下 Schrödinger-Poisson 系统基态解以及束缚态解的存在性

$\begin{align*} \left\{\begin{array}{ll} -\Delta u+V(x)u+\phi (x)u=|u|^{p-2}u, \quad x\in\mathbb{R}^{3},\\ -\Delta \phi (x)=u^{2}, \quad x\in\mathbb{R}^{3}, \end{array} \right. \end{align*}$

其中 $V$ 是非稳定的位势函数并且 $p\in(4,6)$. 当位势函数 $V$ 是 $\mathbb{R}^{3}$ 上的概周期函数时, 该文利用集中紧原理证明了上述方程的壳方程 (shell equation) 存在一个基态解. 并且, 当 $V(x)=V(x^{1}, x^{2}, x^{3})$ 关于 $x^{i} (i=2,3)$ 是 $T_{i}$-周期, 且对 $(x^{2},x^{3})\in [T_{2}]\times[T_{3}]$ 关于第一个变量 $x^{1}$ 是一致概周期时, 上述系统存在一个束缚态解.

关键词: Schrödinger-Poisson 系统, 非稳定位势, 概周期函数, 集中紧原理

Abstract:

In this paper, the existence of ground state and bound state solutions for the following Schrödinger-Poisson system

$\begin{align*} \left\{\begin{array}{ll} -\Delta u+V(x)u+\phi (x)u=|u|^{p-2}u \quad \text{in} \quad\mathbb{R}^{3},\\ -\Delta \phi (x)=u^{2} \quad \text{in} \quad\mathbb{R}^{3} \end{array} \right. \end{align*}$

is studied, where $V$ is a nonstabilizing continuous potential and $p\in(4,6)$. It is proved that the shell equation has a ground state solution by using the concentration-compactness principle when the potential function $V$ is almost periodic on $\mathbb{R}^{3}$. Moreover, a bound state solution is obtained when $V(x)=V(x^{1}, x^{2}, x^{3})$ is $T_{i}$-periodic in $x^{i}$ for $i=2,3$ and almost periodic in $x^{1}$ uniformly with respect to $(x^{2},x^{3})\in [0,T_{2}]\times[0,T_{3}]$.

Key words: Schrödinger-Poisson system, nonstabilizing potential, almost periodic function, concentration-compactness principle

中图分类号:

O175.25

刘泉, 叶江华, 郑雄军. 带有非稳定位势的 Schrödinger-Poisson 方程解的存在性[J]. 数学物理学报, 2025, 45(5): 1492-1518.

Quan Liu, Jianghua Ye, Xiongjun Zheng. Existence of Solution for Schrödinger-Poisson Equation with Nonstabilizing Potential[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1492-1518.

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