  In this paper we establish existence results for the Fractional Nirenberg
Equation via the flatness hypothesis. We have been able to include the Morse
functions in our study. This extends the previous results obtained bY Yan Yan
Li and Coauthors. We also discuss the loss of compactness. Our method is
self-contained and hinges on the breakthrough results of Bahri andBahri and
Coron.
