数学物理学报 ›› 2024, Vol. 44 ›› Issue (1): 60-79.

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次临界 Choquard 方程的多解

温瑞江1(),刘范琴2(),徐子怡3,*()   

  1. 1.江西师范大学数学与统计学院 南昌 330022
    2.首都师范大学数学科学学院 北京 100048
    3.兰州大学数学与统计学院 兰州 730000
  • 收稿日期:2022-09-20 修回日期:2023-10-16 出版日期:2024-02-26 发布日期:2024-01-10
  • 通讯作者: 徐子怡, E-mail:XuZiyi0822@outlook.com
  • 作者简介:温瑞江, E-mail:ruijiangwen@126.com|刘范琴, E-mail:fanqliu@163.com

Multiplicity of Positive Solutions to Subcritical Choquard Equation

Wen Ruijiang1(),Liu Fanqin2(),Xu Ziyi3,*()   

  1. 1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022
    2. School of Mathematical Sciences, Capital Normal University, Beijing 100048
    3. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000
  • Received:2022-09-20 Revised:2023-10-16 Online:2024-02-26 Published:2024-01-10

摘要:

该文考虑次临界 Choquard 方程

{Δu+(λV(x)+1)u=(RN|u(y)|pε|xy|μdy)|u|pε2u,xRN,uH1(RN)(0.1)

多解的存在性, 其中N>3,λ是正实参数,pε=2με,ε>0,0<μ<N,2μ=2NμN2是 Hardy-Littlewood-Sobolev 不等式意义下的临界指数. 假定Ω:=intV1(0)RN中非空带光滑边界的有界区域, 利用 Lusternik-Schnirelman 定理,该文证明了当λ足够大及ε充分小时, 方程(0.1)至少有catΩ(Ω)个正解.

关键词: 次临界 Choquard 方程, Lusternik-Schnirelman 定理, 解的多重性

Abstract:

In this paper, we are concerned with the multiplicity of solutions for the following subcritical Choquard equation

{Δu+(λV(x)+1)u=(RN|u(y)|pε|xy|μdy)|u|pε2u,xRN,uH1(RN),

whereN>3,λis a real parameter,pε=2με,ε>0,μ(0,N)and2μ=2NμN2is the critical Hardy-Littlewood-Sobolev exponent. Suppose thatΩ:=intV1(0)is a nonempty bounded domain inRNwith smooth boundary, using Lusternik-Schnirelman theory, we prove the problem (0.1) has at leastcatΩ(Ω)positive solutions forλlarge andεsmall enough.

Key words: Subcritical Choquard equation, Lusternik-Schnirelman theory, Multiplicity of solutions

中图分类号: 

  • O175.25