Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8899796 | Journal of Mathematical Analysis and Applications | 2018 | 21 Pages |
Abstract
Let Nâ¥2, W1,N(RN) be the standard Sobolev space. For any real numbers Ï>0 and pâ¥N, we denoteλN,p,Ï=infuâW1,N(RN),uâ¢0â¡â«RN(|âu|N+Ï|u|N)dx(â«RN|u|pdx)N/p and define a norm in W1,N(RN) byâuâN,p,α,Ï=(â«RN(|âu|N+Ï|u|N)dxâα(â«RN|u|pdx)N/p)1/N for any fixed α, 0â¤Î±<λN,p,Ï. If Ï>0, 0<β<1, 0<α<λN,p,Ï, pâ¥Nâ¥2, and 0â¤Î³â¤Î±N, then using blow-up analysis, we prove that the supremumsupuâW1,N(RN),âuâN,p,α,Ïâ¤1â¡â«RN1|x|Nβ(eγ(1âβ)|u|NNâ1ââk=0Nâ2γk(1âβ)k|u|kNNâ1k!)dx can be attained by some function uâW1,N(RN) with âuâN,p,α,Ïâ¤1, where αN=NÏNâ11/(Nâ1) and ÏNâ1 is the area of the unit sphere in RN. This improves an inequality of Adimurthi and Yang and the result I obtained recently with Yang.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Xiaomeng Li,