数学物理学报 ›› 2022, Vol. 42 ›› Issue (6): 1682-1704.

• 论文 • 上一篇    下一篇

分数阶临界Choquard方程的多解

陈琳(),刘范琴*()   

  1. 江西师范大学数学与统计学院, 南昌 330022
  • 收稿日期:2021-12-03 出版日期:2022-12-26 发布日期:2022-12-16
  • 通讯作者: 刘范琴 E-mail:chenlinshutong@jxnu.edu.cn;fanqliu@163.com
  • 作者简介:陈琳, E-mail: chenlinshutong@jxnu.edu.cn

Multiplicity of Solutions to Fractional Critical Choquard Equation

Lin Chen(),Fanqin Liu*()   

  1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022
  • Received:2021-12-03 Online:2022-12-26 Published:2022-12-16
  • Contact: Fanqin Liu E-mail:chenlinshutong@jxnu.edu.cn;fanqliu@163.com

摘要:

该文考虑分数阶临界Choquard方程 $\begin{equation} \left\{\begin{array}{ll} { } (-{\Delta})^s{u}=\lambda|u|^{q-2}{u}+ \bigg(\int_\Omega\frac{|u(y)|^{2^\ast_{\mu, s}}}{|x-y|^\mu}{\rm d}y\bigg)|u|^{2^\ast_{\mu, s}-2}u, & x\in\Omega, \\ {u=0}, &x\in{{{\Bbb R}} ^N}\setminus\Omega \end{array}\right. \end{equation}$ 多解的存在性, 其中$\Omega\subset\mathbb{R}^N$是具有光滑边界的有界开集, $N > 2s$, $s\in(0, 1)$, $0 < \mu < N$, $\lambda$是正实参数, $q\in[2, 2^\ast_s)$, $^\ast_{s}=\frac{2N}{N-2s}$是分数阶临界Sobolev指数, $^\ast_{\mu, s}=\frac{2N-\mu}{N-2s}$是Hardy-Littlewood-Sobolev不等式意义下的临界指数.利用Lusternik-Schnirelman定理, 证明了当$q=2$$N\geq4$$q\in(2, 2^\ast_s)$$N > \frac{2s(q+2)}{q}$时, 存在$\bar{\lambda} > 0$, 对$\lambda\in(0, \bar{\lambda})$, 方程至少有cat$_\Omega(\Omega)$个非平凡解.

关键词: Choquard方程, 临界指数, Lusternik-Schnirelman定理

Abstract:

In this paper, we are concerned with the multiplicity of solutions for the following fractional Laplacian problemwhere $\Omega\subset\mathbb{R} ^N$ is an open bounded set with continuous boundary, $N>2s$ with $s\in(0, 1)$, $\lambda$ is a real parameter, $\mu\in(0, N)$ and $q\in[2, 2^\ast_s)$, where $^\ast_{s}=\frac{2N}{N-2s}$, $^\ast_{\mu, s}=\frac{2N-\mu}{N-2s}$. Using Lusternik-Schnirelman theory, there exists $\bar{\lambda}>0$ such that for any $\lambda\in(0, \bar{\lambda})$, the problem has at least $cat_\Omega(\Omega)$ nontrivial solutions provided that $q=2$ and $N\geq4s$ or $q\in(2, 2^\ast_s)$ and $N>\frac{2s(q+2)}{q}$.

Key words: Choquard equation, Critical exponent, Lusternik-Schnirelman theory

中图分类号: 

  • O175.2