Stability estimates for critical points of a nonlocal Sobolev-type inequality
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915139359342592 |
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| author | Yang, Minbo Zhao, Shunneng |
| author_facet | Yang, Minbo Zhao, Shunneng |
| contents | In this paper, we study the stability of the following nonlocal Soblev-type inequality
\begin{equation*}
C_{HLS}\big(\int_{\mathbb{R}^n}\big(|x|^{-μ} \ast u^{p}\big)u^{p} dx\big)^{\frac{1}{p}}\leq\int_{\mathbb{R}^n}|\nabla u|^2 dx , \quad \forall~u\in D^{1,2}(\mathbb{R}^n),
\end{equation*} which is induced by the classical Sobolev inequality and the Hardy-Littlewood-Sobolev inequality, where $p=\frac{2n-μ}{n-2}$, $n\geq3$ and $μ\in(0,n)$, is energy-critical exponent and $C_{HLS}$ is the best constant depending on $n$ and $μ$. Up to translation and scaling, the best constant of the nonlocal Soblev inequality can be achieved by a unique family of positive and radially symmetric extremal function $W(x)$ that satisfies, up to a suitable scaling, the classical critical Hartree equation
\begin{equation*}
Δu+(|x|^{-μ}\ast u^{p})u^{p-1}=0 \quad \mbox{in}\quad \mathbb{R}^n.
\end{equation*} Recently, Piccione, Yang and Zhao in \cite{p-y-z24} established a nonlocal version of Struwe's profile decomposition and they only proved the nonlocal version of the quantitative stability for the one bubble case without dimension restriction and the multiple bubbles case $κ\geq2$ if dimension $3\leq n<6-μ$ and $μ\in(0,n)$ with $μ\in(0,4]$ in Ciraolo-Figalli-Maggi \cite{CFM18} and Figalli-Glaudo \cite{FG20}. We establish the quantitative stability estimates for critical point of the nonlocal Soblev inequality for $n\geq6-μ$ and $μ\in(0,4)$, which is an extension of the recent works by Deng-Sun-Wei in \cite{DSW21} for the classical Sobolev inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01927 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability estimates for critical points of a nonlocal Sobolev-type inequality Yang, Minbo Zhao, Shunneng Analysis of PDEs In this paper, we study the stability of the following nonlocal Soblev-type inequality \begin{equation*} C_{HLS}\big(\int_{\mathbb{R}^n}\big(|x|^{-μ} \ast u^{p}\big)u^{p} dx\big)^{\frac{1}{p}}\leq\int_{\mathbb{R}^n}|\nabla u|^2 dx , \quad \forall~u\in D^{1,2}(\mathbb{R}^n), \end{equation*} which is induced by the classical Sobolev inequality and the Hardy-Littlewood-Sobolev inequality, where $p=\frac{2n-μ}{n-2}$, $n\geq3$ and $μ\in(0,n)$, is energy-critical exponent and $C_{HLS}$ is the best constant depending on $n$ and $μ$. Up to translation and scaling, the best constant of the nonlocal Soblev inequality can be achieved by a unique family of positive and radially symmetric extremal function $W(x)$ that satisfies, up to a suitable scaling, the classical critical Hartree equation \begin{equation*} Δu+(|x|^{-μ}\ast u^{p})u^{p-1}=0 \quad \mbox{in}\quad \mathbb{R}^n. \end{equation*} Recently, Piccione, Yang and Zhao in \cite{p-y-z24} established a nonlocal version of Struwe's profile decomposition and they only proved the nonlocal version of the quantitative stability for the one bubble case without dimension restriction and the multiple bubbles case $κ\geq2$ if dimension $3\leq n<6-μ$ and $μ\in(0,n)$ with $μ\in(0,4]$ in Ciraolo-Figalli-Maggi \cite{CFM18} and Figalli-Glaudo \cite{FG20}. We establish the quantitative stability estimates for critical point of the nonlocal Soblev inequality for $n\geq6-μ$ and $μ\in(0,4)$, which is an extension of the recent works by Deng-Sun-Wei in \cite{DSW21} for the classical Sobolev inequality. |
| title | Stability estimates for critical points of a nonlocal Sobolev-type inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.01927 |