数学物理学报(英文版)

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SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN RN

王征平; 周焕松   

  1. 中国科学院武汉物理与数学研究所, 武汉 430071
  • 收稿日期:2004-10-30 修回日期:2005-08-25 出版日期:2006-07-20 发布日期:2006-07-20
  • 通讯作者: 周焕松
  • 基金资助:

    This work was supported by NNSFC (10571174)

SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN RN

Wang Zhengping; Zhou Huansong   

  1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences P.O.Box 71010, Wuhan 430071, China
  • Received:2004-10-30 Revised:2005-08-25 Online:2006-07-20 Published:2006-07-20
  • Contact: Zhou Huansong

摘要:

For the following elliptic problem
-△ u - μu/|x|2={|u|2*(s)-2u}/|x|s+h(x), on RN
u ∈ D1,2(RN), N≥3, 0≤μ<\bar\mu=(N-2)2 /4, 0≤ s<2,
where 2*(s)=2(N-s)/(N-2) is the critical Sobolev-Hardy exponent, h(x)∈ D1,2(RN))* , the dual space of (D1,2(RN)), with h(x)≥(≠) 0. By Ekeland's variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if
||h||*N,s As (n-s)/(4-2s)(1-μ/{\bar\mu})1/2,

CN,s=(4-2s)/(N-2) ((N-2)/(N+2-2s)(N+2-2s)/(4-2s)
and As=inf u∈ D1,2(RN)\{0} {∫RN(|▽u|2-μu2/|x|2) dx}/{(∫RN|u|2*(s)/|x|s dx)2/2*(s)}.

关键词: Critical Sobolev-Hardy exponent, elliptic equation, Mountain Pass theorem, subsuper solutions, nonhomogeneous

Abstract:

For the following elliptic problem
-△ u - μu/|x|2={|u|2*(s)-2u}/|x|s+h(x), on RN
u ∈ D1,2(RN), N≥3, 0≤μ<\bar\mu=(N-2)2 /4, 0≤ s<2,
where 2*(s)=2(N-s)/(N-2) is the critical Sobolev-Hardy exponent, h(x)∈ D1,2(RN))* , the dual space of (D1,2(RN)), with h(x)≥(≠) 0. By Ekeland's variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if
||h||*N,s As (n-s)/(4-2s)(1-μ/{\bar\mu})1/2,

CN,s=(4-2s)/(N-2) ((N-2)/(N+2-2s)(N+2-2s)/(4-2s)
and As=inf u∈ D1,2(RN)\{0} {∫RN(|▽u|2-μu2/|x|2) dx}/{(∫RN|u|2*(s)/|x|s dx)2/2*(s)}.

Key words: Critical Sobolev-Hardy exponent, elliptic equation, Mountain Pass theorem, subsuper solutions, nonhomogeneous

中图分类号: 

  • 35J60