A complete study of the lack of compactness and existence results of a   Fractional Nirenberg Equation via a flatness hypothesis: Part I
  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. 
W. Abdelhedi, H. Chtioui, H. Hajaiej
math.AP