Remainder terms, profile decomposition and sharp quantitative stability in the fractional nonlocal Sobolev-type inequality with $n>2s$
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913727296569344 |
|---|---|
| author | Lu, Qikai Yang, Minbo Zhao, Shunneng |
| author_facet | Lu, Qikai Yang, Minbo Zhao, Shunneng |
| contents | In this paper, we study the following fractional nonlocal Sobolev-type inequality
\begin{equation*}
C_{HLS}\bigg(\int_{\mathbb{R}^n}\big(|x|^{-μ} \ast |u|^{p_s}\big)|u|^{p_s} dx\bigg)^{\frac{1}{p_s}}\leq\|u\|_{\dot{H}^s(\mathbb{R}^n)}^2\quad \mbox{for all}~~u\in \dot{H}^s(\mathbb{R}^n),
\end{equation*}
induced by the classical fractional Sobolev inequality and Hardy-Littlewood-Sobolev inequality for $s\in(0,\frac{n}{2})$, $μ\in(0,n)$ and where $p_{s}=\frac{2n-μ}{n-2s}\geq2$ is energy-critical exponent. The $C_{HLS}>0$ is a constant depending on the dimension $n$, parameters $s$ and $μ$, which can be achieved by $W(x)$, and up to translation and scaling, $W(x)$ is the unique positive and radially symmetric extremal function of the nonlocal Sobolev-type inequality. It is well-known that, up to a suitable scaling,
\begin{equation*}
(-Δ)^{s}u=(|x|^{-μ}\ast |u|^{p_s})|u|^{p_s-2}u\quad \mbox{for all}~~u\in\dot{H}^s(\mathbb{R}^n),
\end{equation*}
is the Euler-Lagrange equation corresponding to the associated minimization problem.
In this paper, we first prove the non-degeneracy of positive solutions to the critical Hartree equation for all $s\in(0,\frac{n}{2})$, $μ\in(0,n)$ with $0<μ\leq4s$. Furthermore, we show the existence of a gradient type remainder term and, as a corollary, derive the existence of a remainder term in the weak $L^{\frac{n}{n-2s}}$-norm for functions supported in domains of finite measure, under the condition $s\in(0,\frac{n}{2})$. Finally, we establish a Struwe-type profile decomposition and quantitative stability estimates for critical points of the above inequality in the parameter region $s\in(0,\frac{n}{2})$ with the number of bubbles $κ\geq1$, and for $μ\in(0,n)$ with $0<μ\leq4s$. In particular, we provide an example to illustrate the sharpness of our result for $n=6s$ and $μ=4s$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remainder terms, profile decomposition and sharp quantitative stability in the fractional nonlocal Sobolev-type inequality with $n>2s$ Lu, Qikai Yang, Minbo Zhao, Shunneng Analysis of PDEs Primary 35A23, 35R11, Secondly 35B35, 35J08 In this paper, we study the following fractional nonlocal Sobolev-type inequality \begin{equation*} C_{HLS}\bigg(\int_{\mathbb{R}^n}\big(|x|^{-μ} \ast |u|^{p_s}\big)|u|^{p_s} dx\bigg)^{\frac{1}{p_s}}\leq\|u\|_{\dot{H}^s(\mathbb{R}^n)}^2\quad \mbox{for all}~~u\in \dot{H}^s(\mathbb{R}^n), \end{equation*} induced by the classical fractional Sobolev inequality and Hardy-Littlewood-Sobolev inequality for $s\in(0,\frac{n}{2})$, $μ\in(0,n)$ and where $p_{s}=\frac{2n-μ}{n-2s}\geq2$ is energy-critical exponent. The $C_{HLS}>0$ is a constant depending on the dimension $n$, parameters $s$ and $μ$, which can be achieved by $W(x)$, and up to translation and scaling, $W(x)$ is the unique positive and radially symmetric extremal function of the nonlocal Sobolev-type inequality. It is well-known that, up to a suitable scaling, \begin{equation*} (-Δ)^{s}u=(|x|^{-μ}\ast |u|^{p_s})|u|^{p_s-2}u\quad \mbox{for all}~~u\in\dot{H}^s(\mathbb{R}^n), \end{equation*} is the Euler-Lagrange equation corresponding to the associated minimization problem. In this paper, we first prove the non-degeneracy of positive solutions to the critical Hartree equation for all $s\in(0,\frac{n}{2})$, $μ\in(0,n)$ with $0<μ\leq4s$. Furthermore, we show the existence of a gradient type remainder term and, as a corollary, derive the existence of a remainder term in the weak $L^{\frac{n}{n-2s}}$-norm for functions supported in domains of finite measure, under the condition $s\in(0,\frac{n}{2})$. Finally, we establish a Struwe-type profile decomposition and quantitative stability estimates for critical points of the above inequality in the parameter region $s\in(0,\frac{n}{2})$ with the number of bubbles $κ\geq1$, and for $μ\in(0,n)$ with $0<μ\leq4s$. In particular, we provide an example to illustrate the sharpness of our result for $n=6s$ and $μ=4s$. |
| title | Remainder terms, profile decomposition and sharp quantitative stability in the fractional nonlocal Sobolev-type inequality with $n>2s$ |
| topic | Analysis of PDEs Primary 35A23, 35R11, Secondly 35B35, 35J08 |
| url | https://arxiv.org/abs/2503.06636 |