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

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 \(\beta \in (n-2\sigma ,n)\).