含有临界或超临界非线性项的 Klein-Gordon-Maxwell 系统非平凡解的存在性

数学物理学报 ›› 2026, Vol. 46 ›› Issue (1): 190-199.

含有临界或超临界非线性项的 Klein-Gordon-Maxwell 系统非平凡解的存在性

孙歆¹, 段誉¹^,^*(), 柳鸠²

¹贵州工程应用技术学院理学院贵州毕节 551700
²湖南人文科技学院数学与金融学院湖南娄底 4117000

收稿日期:2025-04-02 修回日期:2025-07-19 出版日期:2026-02-26 发布日期:2026-01-19
通讯作者: 段誉 E-mail:duanyu3612@163.com
基金资助:
贵州省科技计划项目(ZK[2024]662);毕节市科学技术项目([2023]28);毕节市科学技术项目([2025]138)

Existence of Nontrivial Solution of Klein-Gordon-Maxwell Systems with Critical or Supercritical Nonlinearity

Xin Sun¹, Yu Duan¹^,^*(), Jiu Liu²

¹College of Science, Guizhou University of Engineering Science, Guizhou Bijie 551700
²College of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Hunan Loudi 417000

Received:2025-04-02 Revised:2025-07-19 Online:2026-02-26 Published:2026-01-19
Contact: Yu Duan E-mail:duanyu3612@163.com
Supported by:
Guizhou Scientific and Technological Program(ZK[2024]662);Bijie Scientific and Technological Program([2023]28);Bijie Scientific and Technological Program([2025]138)

摘要/Abstract

摘要：

研究如下 Klein-Gordon-Maxwell 系统

$$\begin{equation*} \begin{cases} -\Delta u+ V(x)u-(2\omega+\phi)\phi u=\lambda |u|^{s-2}u+ f(u), & x\in \mathbb{R}^{3},\\ \Delta \phi=(\omega+\phi)u^2, & x\in \mathbb{R}^{3}, \end{cases} \end{equation*}$$

其中 $\omega> 0$ 是一个常数, $\lambda> 0$ 是一个参数, $s\geq6$. 当 $V, f$ 满足适当条件且 $\lambda$ 比较小时, 利用变分法和 Moser 迭代证明了系统非平凡解的存在性. 该文结论完善了此系统解研究的已有结果.

关键词: Klein-Gordon-Maxwell 系统, 变分法, Moser 迭代, 非平凡解

Abstract:

This article concerns the following Klein-Gordon-Maxwell system

$$\begin{cases}-\Delta u+ V(x)u-(2\omega+\phi)\phi u=\lambda |u|^{s-2}u+ f(u), & x\in \mathbb{R}^{3},\\\Delta \phi=(\omega+\phi)u^2, & x\in \mathbb{R}^{3},\end{cases}$$

where $\omega> 0$ is a constant, $\lambda> 0$ is a real parameter, $s\geq6$. When $V, f$ satisfy suitable conditions and $\lambda$ is relatively small, existence of nontrivial solution can be proved via variational methods and Moser iteration. The result in this paper completes some recent works concerning research on solutions of this system.

Key words: Klein-Gordon-Maxwell system, variational methods, Moser iteration, nontrivial solutions

中图分类号:

O175.25

孙歆, 段誉, 柳鸠. 含有临界或超临界非线性项的 Klein-Gordon-Maxwell 系统非平凡解的存在性[J]. 数学物理学报, 2026, 46(1): 190-199.

Xin Sun, Yu Duan, Jiu Liu. Existence of Nontrivial Solution of Klein-Gordon-Maxwell Systems with Critical or Supercritical Nonlinearity[J]. Acta mathematica scientia,Series A, 2026, 46(1): 190-199.

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