含周期位势的 Schrödinger-Poisson 方程的基态解

数学物理学报 ›› 2026, Vol. 46 ›› Issue (5): 1706-1720.

含周期位势的 Schrödinger-Poisson 方程的基态解

冯静雯, 王征平^*()

武汉理工大学数学与统计学院武汉 430070

收稿日期:2025-05-06 修回日期:2025-09-30 出版日期:2026-10-26 发布日期:2026-09-07
通讯作者: 王征平 E-mail:zpwang@whut.edu.cn
基金资助:
国家自然科学基金(12371118);中央高校基本科研业务费专项资金(104972025KFYjc0094)

Ground State Solutions for the Schrödinger-Poisson Equations with Periodic Potential

Jingwen Feng, Zhengping Wang^*()

School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070

Received:2025-05-06 Revised:2025-09-30 Online:2026-10-26 Published:2026-09-07
Contact: Zhengping Wang E-mail:zpwang@whut.edu.cn
Supported by:
NSFC(12371118);Fundamental Research Funds for the Central Universities(104972025KFYjc0094)

摘要/Abstract

摘要：

该文考虑如下一类非线性 Schrödinger-Poisson 方程

$\begin{equation*}\ -\varDelta u+\left( V\left( x \right) -\frac{\mu}{\left| x \right|} \right) u+\lambda \left( \frac{1}{\left| x \right|}*\left| u \right|^2 \right) u=\left| u \right|^{p-1}u,\ x\in \mathbb{R}^3, \ \lambda>0, \end{equation*}$

其中 $V(x)$ 是周期位势函数, $\mu,\ \lambda \in \mathbb{R}$ 是参数, 且 $p\in \left( 1,2 \right)$. 应用变分方法和 Cerami 序列的 Profile 分解, 该文在适当的参数假设下证明了上述方程基态解的存在性与非存在性.

关键词: 基态解, 变分法, Nehari 流形, 周期位势, Schrödinger-Poisson 方程

Abstract:

A class of nonlinear Schrödinger-Poisson equations is considered,

$ -\varDelta u+\left( V\left( x \right) -\frac{\mu}{\left| x \right|} \right) u+\lambda \left( \frac{1}{\left| x \right|}*\left| u \right|^2 \right) u=\left| u \right|^{p-1}u,\ x\in \mathbb{R}^3, \ \lambda>0, $

where $V(x)$ denotes a periodic potential function, $\mu,\ \lambda \in \mathbb{R}$ are parameters, and $p\in \left( 1,2 \right)$. By employing variational methods and the Cerami sequence profile decomposition, the existence and nonexistence of ground state solutions to the above equation are established under suitable assumptions on the parameters.

Key words: ground state, variational methods, Nehari manifold, periodic potential, Schrödinger-Poisson equation

中图分类号:

O175.2

冯静雯, 王征平. 含周期位势的 Schrödinger-Poisson 方程的基态解[J]. 数学物理学报, 2026, 46(5): 1706-1720.

Jingwen Feng, Zhengping Wang. Ground State Solutions for the Schrödinger-Poisson Equations with Periodic Potential[J]. Acta mathematica scientia,Series A, 2026, 46(5): 1706-1720.

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