Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8898841 | Journal of Differential Equations | 2018 | 43 Pages |
Abstract
In a previous work (Adimurthi and Yang, 2010 [2]), Adimurthi-Yang proved a singular Trudinger-Moser inequality in the entire Euclidean space RN(Nâ¥2). Precisely, if 0â¤Î²<1 and 0<γâ¤1âβ, then there holds for any Ï>0,supuâW1,N(RN),â«RN(|âu|N+Ï|u|N)dxâ¤1â¡â«RN1|x|Nβ(eαNγ|u|NNâ1ââk=0Nâ2αNkγk|u|kNNâ1k!)dx<â, where αN=NÏNâ11/(Nâ1) and ÏNâ1 is the area of the unit sphere in RN. The above inequality is sharp in the sense that if γ>1âβ, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case β=0 has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11]. We shall investigate the singular case 0<β<1 and prove that for all Ï>0, 0<β<1 and 0<γâ¤1âβ, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Xiaomeng Li, Yunyan Yang,