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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.19248 |
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| _version_ | 1866918242690269184 |
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| author | Dai, Wei Hu, Yichen Peng, Shaolong |
| author_facet | Dai, Wei Hu, Yichen Peng, Shaolong |
| contents | Assume $n\geq3$ and $u\in \dot{H}^1(\mathbb{R}^n)$. Recently, Piccione, Yang and Zhao \cite{Piccione-Yang-Zhao} established a nonlocal version of Struwe's decomposition in \cite{Struwe-1984}, i.e., if $Γ(u):=\left\|Δu+D_{n,α}\int_{\mathbb{R}^{n}}\frac{|u|^{p_α}(y) }{|x-y|^α}\mathrm{d}y |u|^{p_α-2} u\right\|_{\dot{H}^{-1}} \rightarrow 0$ and $u\geq 0$, then $dist(u,\mathcal{T})\to 0$, where $dist(u,\mathcal{T})$ denotes the $\dot{H}^1(\mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. In this paper, we establish the nonlocal version of the quantitative estimates of Struwe's decomposition in Ciraolo, Figalli and Maggi \cite{CFM} for one bubble and $n\geq3$, Figalli and Glaudo \cite{Figalli-Glaudo2020} for $3\leq n\leq5$ and Deng, Sun and Wei \cite{DSW} for $n\geq6$ and two or more bubbles. We prove that for $n\geq 3$ and $0<α<n$, \[dist (u,\mathcal{T})\leq C\begin{cases} Γ(u)\left|\log Γ(u)\right|^{\frac{1}{2}}\quad&\text{if } \,\, n\geq 6, \,\, ν\geq2 \,\, \text{and} \,\, α=\frac{n+2}{2}, \\ Γ(u) \quad&\text{for any other cases,}\end{cases}\] where $ν$ denotes the number of bubbles. Furthermore, we show that this inequality is sharp for $n\geq 6$ and $α=\frac{n+2}{2}$. It should be emphasized that, in our paper, we have developed new techniques to deal with the strong singular case $4<α<n$, which can not be handled by reduction methods in previous works. We believe that our method can also be applied to other problems related to the physically interesting Hartree equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the sharp quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality in $\mathbb{R}^{n}$ with $n\geq3$ Dai, Wei Hu, Yichen Peng, Shaolong Analysis of PDEs Assume $n\geq3$ and $u\in \dot{H}^1(\mathbb{R}^n)$. Recently, Piccione, Yang and Zhao \cite{Piccione-Yang-Zhao} established a nonlocal version of Struwe's decomposition in \cite{Struwe-1984}, i.e., if $Γ(u):=\left\|Δu+D_{n,α}\int_{\mathbb{R}^{n}}\frac{|u|^{p_α}(y) }{|x-y|^α}\mathrm{d}y |u|^{p_α-2} u\right\|_{\dot{H}^{-1}} \rightarrow 0$ and $u\geq 0$, then $dist(u,\mathcal{T})\to 0$, where $dist(u,\mathcal{T})$ denotes the $\dot{H}^1(\mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. In this paper, we establish the nonlocal version of the quantitative estimates of Struwe's decomposition in Ciraolo, Figalli and Maggi \cite{CFM} for one bubble and $n\geq3$, Figalli and Glaudo \cite{Figalli-Glaudo2020} for $3\leq n\leq5$ and Deng, Sun and Wei \cite{DSW} for $n\geq6$ and two or more bubbles. We prove that for $n\geq 3$ and $0<α<n$, \[dist (u,\mathcal{T})\leq C\begin{cases} Γ(u)\left|\log Γ(u)\right|^{\frac{1}{2}}\quad&\text{if } \,\, n\geq 6, \,\, ν\geq2 \,\, \text{and} \,\, α=\frac{n+2}{2}, \\ Γ(u) \quad&\text{for any other cases,}\end{cases}\] where $ν$ denotes the number of bubbles. Furthermore, we show that this inequality is sharp for $n\geq 6$ and $α=\frac{n+2}{2}$. It should be emphasized that, in our paper, we have developed new techniques to deal with the strong singular case $4<α<n$, which can not be handled by reduction methods in previous works. We believe that our method can also be applied to other problems related to the physically interesting Hartree equation. |
| title | On the sharp quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality in $\mathbb{R}^{n}$ with $n\geq3$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.19248 |