Stability of Hardy-Sobolev Inequality
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913260805029888 |
|---|---|
| author | Chakraborty, Souptik |
| author_facet | Chakraborty, Souptik |
| contents | Given $N\geq 3,$ we consider the critical Hardy-Sobolev equation $-Δu-\fracγ{|x|^2}u=\frac{|u|^{2^*(s)-2}u}{|x|^s}$ in $\mathbb{R}^N\setminus \{0\},$ where $0<γ<γ_{H}:=\left(\frac{N-2}{2}\right)^2,\,s\in (0,2)$ and $2^*(s)=\frac{2(N-s)}{(N-2)}.$ We prove a stability estimate for the corresponding Hardy-Sobolev inequality in the spirit of Bianchi-Egnell (1991). Also, we obtain a Struwe-type decomposition (1984) for the corresponding Euler-Lagrange equation. Finally, we prove a quantitative bound for one bubble, namely $\operatorname{dist}(u,\mathcal{M})\lesssim Γ(u)$ in the spirit of Ciraolo-Figalli-Maggi (2017). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06594 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of Hardy-Sobolev Inequality Chakraborty, Souptik Analysis of PDEs 35A23, 35B33, 35B35, 35J20, 35J61 (Primary) 26D10 (Secondary) Given $N\geq 3,$ we consider the critical Hardy-Sobolev equation $-Δu-\fracγ{|x|^2}u=\frac{|u|^{2^*(s)-2}u}{|x|^s}$ in $\mathbb{R}^N\setminus \{0\},$ where $0<γ<γ_{H}:=\left(\frac{N-2}{2}\right)^2,\,s\in (0,2)$ and $2^*(s)=\frac{2(N-s)}{(N-2)}.$ We prove a stability estimate for the corresponding Hardy-Sobolev inequality in the spirit of Bianchi-Egnell (1991). Also, we obtain a Struwe-type decomposition (1984) for the corresponding Euler-Lagrange equation. Finally, we prove a quantitative bound for one bubble, namely $\operatorname{dist}(u,\mathcal{M})\lesssim Γ(u)$ in the spirit of Ciraolo-Figalli-Maggi (2017). |
| title | Stability of Hardy-Sobolev Inequality |
| topic | Analysis of PDEs 35A23, 35B33, 35B35, 35J20, 35J61 (Primary) 26D10 (Secondary) |
| url | https://arxiv.org/abs/2403.06594 |