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 2σ for a conformal metric on the standard sphere 𝕊^n with σ∈ (0,n/2) and n≥ 3 . Compactness and existence results are obtained in terms of the flatness order β of the prescribed curvature function K. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when β∈ [n-2σ ,n) for all σ∈ (0,n/2) . This work generalizes the corresponding results of Jin-Li-Xiong (Math Ann 369:109–151, 2017) for β∈ (n-2σ ,n) .