A complete study of the lack of compactness and existence results of a fractional Nirenberg equation via a flatness hypothesis, I

Abstract

We consider a nonlinear critical problem involving the fractional Laplacian operator arising in conformal geometry, namely the prescribed $\sigma$-curvature problem on the standard $n$-sphere, $n\ge 2$. Under the assumption that the prescribed function is flat near its critical points, we give precise estimates on the losses of the compactness and we provide existence results. In this first part, we will focus on the case $1<\beta \le n-2\sigma$, which is not covered by the method of Jin, Li, and Xiong (2014, 2015).