Asymptotic analysis of singular solutions of the scalar and mean curvature equations - Semantic Scholar

We show that positive solutions of a semilinear elliptic problem in the Sobolev critical exponent with Newmann conditions, related to conformal deformation of metrics in R n + , are asymptotically symmetric in a neighborhood of the origin. As a consequence, we prove for a related problem of conformal deformation of metrics in R n + that if a solution satisfies a Kazdan-Warner-type identity, then the conformal metric can be realized as a smooth metric on S n + .