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 , are asymptotically symmetric in a neighborhood of the origin. As a consequence, we prove for a related problem of conformal deformation of metrics in that if a solution satisfies a Kazdan‐Warner‐type identity, then the conformal metric can be realized as a smooth metric on .