数学物理学报 ›› 2022, Vol. 42 ›› Issue (2): 470-490.

• 论文 • 上一篇    下一篇

分数阶Choquard方程正解的存在性、多重性和集中现象

张伟强*(),赵培浩()   

  1. 兰州大学数学与统计学院 兰州 730000
  • 收稿日期:2021-04-22 出版日期:2022-04-26 发布日期:2022-04-18
  • 通讯作者: 张伟强 E-mail:zhangwq19@lzu.edu.cn;zhaoph@lzu.edu.cn
  • 作者简介:赵培浩, E-mail: zhaoph@lzu.edu.cn
  • 基金资助:
    国家自然科学基金(11471147)

Existence, Multiplicity and Concentration of Positive Solutions for a Fractional Choquard Equation

Weiqiang Zhang*(),Peihao Zhao()   

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000
  • Received:2021-04-22 Online:2022-04-26 Published:2022-04-18
  • Contact: Weiqiang Zhang E-mail:zhangwq19@lzu.edu.cn;zhaoph@lzu.edu.cn
  • Supported by:
    the NSFC(11471147)

摘要:

该文考虑了下面的次临界的分数阶Choquard方程 正解的存在性、多重性和集中现象, 这里$\varepsilon>0$是一个常数, $s\in (0, 1)$, $(-\Delta)^{s}$是分数阶Laplace算子, 位势$V:\mathbb{R} ^{N}\rightarrow\mathbb{R} $是正的且有全局极小, $0<\mu<\min\{4s, N\}$, 非线性项$f\in C^{1}(\mathbb{R} , \mathbb{R} )$是次临界增长的, $F$是$f$的原函数.该文的主要研究方法是变分法和Ljusternik-Schnirelmann理论.

关键词: 分数阶Choquard方程, 变分法, Ljusternik-Schnirelmann理论, 正解, 集中现象

Abstract:

We are concerned with the existence, multiplicity and concentration of positive solutions for the following fractional Choquard equation with subcritical nonlinearity where $\varepsilon>0$ is a parameter, $s\in(0, 1)$, $(-\Delta)^{s}$ is the fractional Laplace operator, $V:\mathbb{R} ^{N}\rightarrow\mathbb{R} $ is a positive potential having global minimum, $0<\mu<\min\{4s, N\}$, and $F$ is the primitive of $f\in C^{1}(\mathbb{R} , \mathbb{R} )$ which is subcritical growth. The main research methods of this article are variational method and the Ljusternik-Schnirelmann theory.

Key words: Fractional Choquard equation, Variational method, Ljusternik-Schnirelmann theory, Positive solution, Concentrating phenomenon.

中图分类号: 

  • O175.2