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

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Abstract: In this paper we study the problem of prescribing fractional

Q

-curvature of order

2 s i g m a

for a conformal metric on the standard sphere

S n

with

s i g m a i n (0, n / 2)

and

n g e q 2

. Compactness and existence results are obtained in terms of the flatness order

of the prescribed curvature function

K

. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when

for all

s i g m a i n (0, n / 2)

. This work generalizes the corresponding results of Jin-Li-Xiong [Math. Ann. 369: 109--151, 2017] for

.

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