带有部分调和位势以及临界增长的 Schrödinger-Poisson 系统基态解的存在性——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1529-1547.

带有部分调和位势以及临界增长的 Schrödinger-Poisson 系统基态解的存在性——献给邓引斌教授 70 寿辰

单沥莹(), 帅伟^*(), 杨平(), 叶江华()

华中师范大学数学与统计学学院武汉 430079

收稿日期:2026-01-04 修回日期:2026-05-09 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 帅伟 E-mail:shanliying@mails.ccnu.edu.cn;wshuai@ccnu.edu.cn;yangping0427@163.com;jhye@mails.ccnu.edu.cn
作者简介:单沥莹,E-mail: shanliying@mails.ccnu.edu.cn;
杨平,E-mail: yangping0427@163.com;
叶江华,E-mail: jhye@mails.ccnu.edu.cn
基金资助:
国家自然科学基金(124711070)

Existence of Ground State Solution for Schrödinger-Poisson System with Partial Confinement and Critical Growth

Liying Shan(), Wei Shuai^*(), Ping Yang(), Jianghua Ye()

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079

Received:2026-01-04 Revised:2026-05-09 Online:2026-08-26 Published:2026-06-10
Contact: Wei Shuai E-mail:shanliying@mails.ccnu.edu.cn;wshuai@ccnu.edu.cn;yangping0427@163.com;jhye@mails.ccnu.edu.cn
Supported by:
NSFC(12471107)

摘要/Abstract

摘要：

该文研究如下带有部分调和位势以及临界增长的 Schrödinger-Poisson 系统

$\begin{eqnarray*} \begin{cases} -\Delta u+(x_1^2+x_2^2)u+\phi u=\vert u\vert^{p-2}u+\vert u\vert^{4}u, & x\in \mathbb{R}^3,\\ -\Delta \phi =u^{2}, & x\in \mathbb{R}^3, \end{cases} \end{eqnarray*}$

其中 $p\in (4,6)$. 利用变分方法, 该文得到了此系统正基态解的存在性, 还通过比较能量的方法得到了极小能量变号解的非存在性.

关键词: Schr?dinger-Poisson 系统, 部分调和位势, 临界增长, 基态解

Abstract:

In this paper, the following Schrödinger-Poisson system with partial confinement and critical growth

$\begin{eqnarray*} \begin{cases} -\Delta u+(x_1^2+x_2^2)u+\phi u=\vert u\vert^{p-2}u+\vert u\vert^{4}u, & x\in \mathbb{R}^3,\\ -\Delta \phi =u^{2}, & x\in \mathbb{R}^3, \end{cases} \end{eqnarray*}$

is studied, where $p\in (4,6)$. By using variational method, we prove the existence of a positive ground state solution. Moreover, via energy comparison, we also prove the nonexistence of least-energy sign-changing solution.

Key words: Schr?dinger-Poisson system, partial confinement, critical growth, ground state solution

中图分类号:

O175.25

单沥莹, 帅伟, 杨平, 叶江华. 带有部分调和位势以及临界增长的 Schrödinger-Poisson 系统基态解的存在性——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1529-1547.

Liying Shan, Wei Shuai, Ping Yang, Jianghua Ye. Existence of Ground State Solution for Schrödinger-Poisson System with Partial Confinement and Critical Growth[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1529-1547.

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