Journal of Mathematical Physics, Analysis, Geometry 2022, vol. 18, No 1, pp. 3-32   https://doi.org/10.15407/mag18.01.003     ( to contents , go back )

### A Complete Study of the Lack of Compactness and Existence Results of a Fractional Nirenberg Equation via a Flatness Hypothesis. Part II

Azeb Alghanemi

Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
E-mail: aalghanemi@kau.edu.sa

Wael Abdelhedi

Sfax University, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia
E-mail: wael hed@yahoo.fr

Hichem Chtioui

Sfax University, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia
E-mail: Hichem.Chtioui@fss.rnu.tn

Received October 27, 2020, revised January 13, 2021.

Abstract

This is a sequel to [2] where the prescribed $\sigma$-curvature problem on the standard sphere was studied under the hypothesis that the flatness order at critical points of the prescribed function lies in (1, n - 2$\sigma$]. We provide a complete description of the lack of compactness of the problem when the flatness order varies in (1, n) and we establish an existence theorem based on an Euler-Hopf type formula. As a product, we extend the existence results of [2, 17, 18] and deliver a new one.

Mathematics Subject Classification 2010: 35A15, 35J60, 58E30
Key words: conformal geometry, fractional curvature, variational calculus, critical points at infinity