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

TL;DR

This paper establishes unified compactness and existence results for prescribing fractional Q-curvature on spheres, extending previous work to a broader range of flatness orders using integral and perturbation methods.

Contribution

It provides a unified framework for compactness and existence of solutions for fractional Q-curvature problems across a wider flatness order range, generalizing prior results.

Findings

Unified approach for all b2 [n-2c, n)

Extended results to fractional orders c  (0, n/2)

Generalized Jin-Li-Xiong's results to broader flatness range

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 $\Sn$ with $\sigma\in (0,n/2)$ and $n\geq2$. 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 $\beta\in (n-2\sigma,n)$.