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Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential
Cao, Daomin1,2; Yan, Shusen3
2010-07-01
Source PublicationCALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
ISSN0944-2669
Volume38Issue:3-4Pages:471-501
AbstractIn this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth and a Hardy potential: -Delta u - mu/vertical bar x vertical bar(2) u = vertical bar u vertical bar(2*-2)u + au in Omega, u = 0 on partial derivative Omega, (*) under the assumptions that N >= 7, mu is an element of [0, (N-2)(2)/4 - 4) and a > 0, where 2* = 2N/N-2, and Omega is an open bounded domain in RN which contains the origin. To achieve this goal, we consider the following perturbed problem of (*), which is of subcritical growth, -Delta u - mu/vertical bar x vertical bar(2) u = vertical bar u vertical bar(2*-2-epsilon n)u + au in Omega, u = 0 on partial derivative Omega, (**)(n) where epsilon(n) > 0 is small and epsilon(n) -> 0 as n -> +infinity. By the critical point theory for the even functionals, for each fixed epsilon(n) > 0 small, (**)(n) has a sequence of solutions u(k,epsilon n) is an element of H-0(1)(Omega). We obtain the existence of infinitely many solutions for (*) by showing that as n -> infinity, u(k,epsilon n) converges strongly in H-0(1)(Omega) to u(k), which must be a solution of (*). Such a convergence is obtained by applying a local Pohozaev identity to exclude the possibility of the concentration of {u(k,epsilon n)}.
DOI10.1007/s00526-009-0295-5
Language英语
Funding ProjectNSFC[10631030] ; NSFC[10721101] ; ARC in Australia
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000277541500008
PublisherSPRINGER
Citation statistics
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/10435
Collection应用数学研究所
Corresponding AuthorCao, Daomin
Affiliation1.Chinese Acad Sci, Inst Appl Math, AMSS, Beijing 100190, Peoples R China
2.Chinese Acad Sci, Key Lab Random Complex Struct & Data Sci, Beijing 100190, Peoples R China
3.Univ New England, Dept Math, Armidale, NSW 2351, Australia
Recommended Citation
GB/T 7714
Cao, Daomin,Yan, Shusen. Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential[J]. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS,2010,38(3-4):471-501.
APA Cao, Daomin,&Yan, Shusen.(2010).Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential.CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS,38(3-4),471-501.
MLA Cao, Daomin,et al."Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential".CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 38.3-4(2010):471-501.
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