广西师范大学学报（自然科学版） ›› 2018, Vol. 36 ›› Issue (4): 59-66.doi: 10.16088/j.issn.1001-6600.2018.04.008

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非线性薛定谔-麦克斯韦方程的基态解

姜影星, 黄文念*   

  1. 广西师范大学数学与统计学院,广西桂林541006
  • 收稿日期:2017-10-19 发布日期:2018-10-20
  • 通讯作者: 黄文念(1984—),男(壮族),广西河池人,广西师范大学副教授,博士。E-mail:csuhuangwn@163.com
  • 基金资助:
    国家自然科学基金(11771103);广西自然科学基金(2015GXNSFBA139018);广西师范大学科学研究基金(2014ZD001);广西研究生教育创新计划项目(XYCSZ2018060)

Ground State Solutions for the NonlinearSchrödinger-Maxwell Equations

JIANG Yingxing, HUANG Wennian*   

  1. College of Mathematics and Statistics, Guangxi Normal University, Guilin Guangxi 541006, China
  • Received:2017-10-19 Published:2018-10-20

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摘要： 在V、K和f的一些假设下,本文主要研究非线性薛定谔-麦克斯韦方程的基态解:
-Δu+V(x)u+K(x)φu=f(x,u), x∈R3,
-Δφ=K(x)u2, x∈R3。
首先利用山路定理得出薛定谔-麦克斯韦方程的非平凡解,然后证得泛函在Nehari流形上可达,最后证明薛定谔-麦克斯韦方程的基态解。本文弱化了已有文献中的某个条件,推广了已有文献中高能解的结论。

关键词: 薛定谔-麦克斯韦方程, 山路定理, 基态解, Nehari流形

Abstract: In this paper, under some assumptions of V, K and f, the following nonlinear Schrödinger-Maxwell equations are studied.
-Δu+V(x)u+K(x)φu=f(x,u), in R3,
-Δφ=K(x)u2, in R3.
First, the nontrivial solution of Schrödinger-Maxwell equations is obtained by using mountain pass theorem. Then, it is shown that the functional is achieved on Nehari manifold, and finally, ground state solutions of Schrödinger-Maxwell equations are obtained. Some conditions in previous literatures are weakened and the conclusions of high energy solutions in previous references are generalized in this paper.

Key words: Schrödinger-Maxwell equations, mountain pass theorem, ground state solution, Nehari manifold

中图分类号: 

  • O175.25
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