  In this paper we prove non-existence and classification results for elliptic
fully nonlinear elliptic degenerate conformal equations on certain subdomains
of the sphere with prescribed constant mean curvature along its boundary. We
also consider non-degenerate equations. Such subdomains are the hemisphere (or
a geodesic ball in $\mathbb{S}^m$), punctured balls and annular domains.
  Our results extend those of Escobar in when $m\geq 3$, and Hang-Wang and
Jimenez in when $m=2$.
