Mathematische Nachrichten
Volume 297, Issue 5 pp. 1652-1667
ORIGINAL ARTICLE

Remainder terms of a nonlocal Sobolev inequality

Shengbing Deng, 

Shengbing Deng

School of Mathematics and Statistics, Southwest University, Chongqing, China

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Xingliang Tian, 

Xingliang Tian

School of Mathematics and Statistics, Southwest University, Chongqing, China

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Minbo Yang, 

Corresponding Author

Minbo Yang

School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, China

Correspondence

Minbo Yang, School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, Zhejiang, China.

Email: [email protected]

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Shunneng Zhao, 

Shunneng Zhao

School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, China

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First published: 06 December 2023
https://doi.org/10.1002/mana.202300172

Abstract

In this note, we study a nonlocal version of the Sobolev inequality

∫ R N | ∇ u | 2 d x ≥ S HLS ∫ R N | x | − α * u 2 α * u 2 α * d x 1 2 α * , ∀ u ∈ D 1 , 2 ( R N ) , $$\begin{equation*} \int _{\mathbb {R}^N}|\nabla u|^2 dx \ge S_{\text{HLS}}{\left(\int _{\mathbb {R}^N}{\left(|x|^{-\alpha} \ast u^{2_\alpha ^{\ast}}\right)}u^{2_\alpha ^{\ast}} dx\right)}^{\frac{1}{2_\alpha ^{\ast}}}, \quad \forall u\in \mathcal {D}^{1,2}(\mathbb {R}^N), \end{equation*}$$
where S HLS $S_{\text{HLS}}$ is the best constant, * $\ast$ denotes the standard convolution and D 1 , 2 ( R N ) $\mathcal {D}^{1,2}(\mathbb {R}^N)$ denotes the classical Sobolev space with respect to the norm ∥ u ∥ D 1 , 2 ( R N ) = ∥ ∇ u ∥ L 2 ( R N ) $\Vert u\Vert _{\mathcal {D}^{1,2}(\mathbb {R}^N)}=\Vert \nabla u\Vert _{L^2(\mathbb {R}^N)}$ . By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak L N N − 2 $L^{\frac{N}{N-2}}$ -norm of above inequality for all 0 < α < N $0&lt;\alpha &lt;N$ with 0 < α ≤ 4 $0&lt;\alpha \le 4$ .

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