Compactness and existence results of the prescribing fractional $Q$-curvatures problem on $\mathbb{S}^n$

TL;DR

This paper establishes new compactness and existence results for solutions to the prescribing fractional Q-curvature problem on spheres, including blow-up analysis and solutions with multiple harmonic components.

Contribution

It provides the first compactness results that are both novel and optimal, along with a degree-counting formula for existence and blow-up analysis.

Findings

Compactness results are novel and optimal.

A degree-counting formula for solutions is established.

Solutions with prescribed blow-up points and multiple harmonic cases are constructed.

Abstract

This paper is devoted to establishing the compactness and existence results of the solutions to the prescribing fractional $Q$-curvatures problem of order $2\sigma$ on $n$-dimensional standard sphere when $ n-2\sigma=2$, $\sigma=1+m/2,$ $m\in \mathbb{N}_{+}.$ The compactness results are novel and optimal. In addition, we prove a degree-counting formula of all solutions to achieve the existence. From our results, we can know where blow up occur. Furthermore, the sequence of solutions that blow up precisely at any finite distinct location can be constructed. It is worth noting that our results include the case of multiple harmonic.