数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (2): 527-536.doi: 10.1016/S0252-9602(16)30018-2

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INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRÉZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL

张靖1, 马世旺2   

  1. 1. Mathematics Science College, Inner Mongolia Normal University, Hohhot 010022, China;
    2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
  • 收稿日期:2014-10-30 修回日期:2015-10-22 出版日期:2016-04-25 发布日期:2016-04-25
  • 作者简介:Jing ZHANG,E-mail:jinshizhangjing@eyou.com;Shiwang MA,E-mail:shiwangm@163.net
  • 基金资助:

    Research supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China.

INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRÉZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL

Jing ZHANG1, Shiwang MA2   

  1. 1. Mathematics Science College, Inner Mongolia Normal University, Hohhot 010022, China;
    2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
  • Received:2014-10-30 Revised:2015-10-22 Online:2016-04-25 Published:2016-04-25
  • Supported by:

    Research supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China.

摘要:

In this article, we give a new proof on the existence of infinitely many sign-changing solutions for the following Brézis-Nirenberg problem with critical exponent and a Hardy potential -Δu-μu/|x|2u+|u|2*-2u in Ω, u=0 on Ω, where Ω is a smooth open bounded domain of RN which contains the origin, 2*=2N/N-2 is the critical Sobolev exponent. More precisely, under the assumptions that N≥7, μ∈[0, μ-4), and μ=(N-2)2/4, we show that the problem admits infinitely many sign-changing solutions for each fixed λ>0. Our proof is based on a combination of invariant sets method and Ljusternik-Schnirelman theory.

关键词: Critical exponent, sign-changing solutions, minimax method, hardy potential

Abstract:

In this article, we give a new proof on the existence of infinitely many sign-changing solutions for the following Brézis-Nirenberg problem with critical exponent and a Hardy potential -Δu-μu/|x|2u+|u|2*-2u in Ω, u=0 on Ω, where Ω is a smooth open bounded domain of RN which contains the origin, 2*=2N/N-2 is the critical Sobolev exponent. More precisely, under the assumptions that N≥7, μ∈[0, μ-4), and μ=(N-2)2/4, we show that the problem admits infinitely many sign-changing solutions for each fixed λ>0. Our proof is based on a combination of invariant sets method and Ljusternik-Schnirelman theory.

Key words: Critical exponent, sign-changing solutions, minimax method, hardy potential

中图分类号: 

  • 35J60