Unified results for existence and compactness in the prescribed fractional Q-curvature problem

Yan Li, Zhongwei Tang, Heming Wang, Ning Zhou
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Abstract

In this paper we study the problem of prescribing fractional Q-curvature of order \(2\sigma \) for a conformal metric on the standard sphere \(\mathbb {S}^n\) with \(\sigma \in (0,n/2)\) and \(n\ge 3\). Compactness and existence results are obtained in terms of the flatness order \(\beta \) of the prescribed curvature function K. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when \(\beta \in [n-2\sigma ,n)\) for all \(\sigma \in (0,n/2)\). This work generalizes the corresponding results of Jin-Li-Xiong (Math Ann 369:109–151, 2017) for \(\beta \in (n-2\sigma ,n)\).

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规定分数 Q曲率问题中存在性和紧凑性的统一结果
在本文中,我们研究了在标准球面 \(\mathbb {S}^n\) 上为共形度量规定阶为 \(2\sigma \) 的分数 Q 曲率问题,该度量具有 \(\sigma \in (0,n/2)\) 和 \(n\ge 3\) 。利用积分表征和扰动结果,我们开发了一种统一的方法来获得这些结果,即当\(\beta\in [n-2\sigma ,n)\)对于所有\(\sigma\in (0,n/2)\)时。这项工作概括了熊金力(Math Ann 369:109-151,2017)对于(n-2 sigma ,n)的相应结果。
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