数学物理学报 ›› 2024, Vol. 44 ›› Issue (5): 1205-1215.

• • 上一篇    下一篇

次线性 Klein-Gordon-Maxwell 系统解的多重性

孙歆,段誉*()   

  1. 贵州工程应用技术学院理学院 贵州毕节 551700
  • 收稿日期:2023-10-22 修回日期:2024-02-21 出版日期:2024-10-26 发布日期:2024-10-16
  • 通讯作者: *段誉, E-mail: duanyu3612@163.com
  • 基金资助:
    国家自然科学基金(11661021);毕节市科学技术项目([2023]28);毕节市科学技术项目([2023]52)

Multiplicity of Solutions for Sublinear Klein-Gordon-Maxwell Systems

Sun Xin,Duan Yu*()   

  1. College of Science, Guizhou University of Engineering Science, Guizhou Bijie 551700
  • Received:2023-10-22 Revised:2024-02-21 Online:2024-10-26 Published:2024-10-16
  • Supported by:
    NSFC(11661021);Bijie Scientific and Technological Program([2023]28);Bijie Scientific and Technological Program([2023]52)

摘要:

该文研究如下 Klein-Gordon-Maxwell 系统 $\begin{equation*} \begin{cases} -\Delta u+u-(2\omega+\phi)\phi u=\lambda Q(x)f(u), & x\in \mathbb{R}^{3},\\ \Delta \phi=(\omega+\phi)u^2, &x\in \mathbb{R}^{3}, \end{cases} \end{equation*}$ 其中 $\omega> 0$ 是一个常数, $\lambda> 0$ 是一个参数, $Q$ 是一个正的函数. 当非线性项 $f$ 在无穷远处是次线性增长时, 利用变分方法及三临界点定理获得此系统至少存在两个非平凡解. 另外, 当 $f$ 仅在原点附近满足次线性增长时, 利用变分方法及临界点定理获得此系统解的存在性及多重性. 完善了此系统解的多重性的已有结果.

关键词: Klein-Gordon-Maxwell 系统, 变分法, 次线性, 临界点定理, 多重性

Abstract:

This article concerns the following Klein-Gordon-Maxwell system $\begin{equation*} \begin{cases} -\Delta u+u-(2\omega+\phi)\phi u=\lambda Q(x)f(u), & x\in \mathbb{R}^{3},\\ \Delta \phi=(\omega+\phi)u^2, &x\in \mathbb{R}^{3}, \end{cases} \end{equation*}$ where $\omega> 0$ is a constant, $\lambda> 0$ is a parameter, $Q$ is a positive function. When the nonlinear term $f$ is sublinear at infinity, two nontrivial solutions for the system are established via variational methods and three critical points theorem. Furtermore, when $f$ is sublinear only in a neighbourhood of the origin, existence and multiplicity of non-trivial solutions are obtained via variational methods and critical point theorem. Our result completes some recent works concerning the multiplicity of solutions of this system.

Key words: Klein-Gordon-Maxwell system, Variational methods, Sublinearity, Critical point theorem, Multiplicity

中图分类号: 

  • O175.25