On the prescribed scalar curvature problem on Sn: Part 1, asymptotic estimates and existence results

Abstract

There have been many works on the problem of finding a conformal metric on the standard sphere Sn,n≥3, when the prescribed scalar curvature function is flat near its critical points with order of flatness β. All the existence results up to this research concern the case 1<β<n. The present paper deals with the case β≥n. We provide a complete analysis of the asymptotic expansion of the associated gradient vector field for any β>1. Moreover, we establish (in this first part) existence results concerning some cases of β≥n.