Divergent operator with degeneracy and related sharp inequalities

Abstract

In this paper we classify all nonnegative extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves a divergent operator with degeneracy on the boundary. As an application of the results, we derive a sharp Sobolev type inequality involving Baouendi-Grushin operator, and classify certain extremal functions for all τ>0 and m≠2 or n≠1.