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

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