On a Fractional Nirenberg problem involving the square root of the Laplacian on $\mathbb{S}^{3}$

TL;DR

This paper investigates the fractional Nirenberg problem on the 3-sphere involving the square root of the Laplacian, establishing new compactness, existence, and blow-up analysis results for solutions under specific curvature conditions.

Contribution

It extends Li's local results to nonlocal fractional cases, providing new compactness theorems, a degree-counting formula, and blow-up construction methods for solutions.

Findings

New optimal compactness results for fractional Nirenberg problem

A degree-counting formula for solutions

Construction of solutions blowing up at prescribed points

Abstract

In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $\sigma=1/2,$ when the prescribing $\sigma$-curvature function satisfies the $(n-2\sigma)$-flatness condition near its critical points. The compactness results are new and optimal. In addition, we obtain a degree-counting formula of all solutions. From our results, we can know where blow up occur. Moreover, for any finite distinct points, the sequence of solutions that blow up precisely at these points can be constructed. We extend the results of Li in \cite[CPAM, 1996]{LYY} from the local problem to nonlocal cases.