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In this paper we study the problem of prescribing fractional $Q$-curvature oforder $2\sigma$ for a conformal metric on the standard sphere $\Sn$ with$\sigma\in (0,n/2)$ and $n\geq2$. Compactness and existence results areobtained in terms of the flatness order $\beta$ of the prescribed curvaturefunction $K$. 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 thecorresponding results of Jin-Li-Xiong [Math. Ann. 369: 109--151, 2017] for$\beta\in (n-2\sigma,n)$.

Unified results of compactness and existence for prescribing fractional $Q$-curvatures problem