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.