  In this paper we prove classification results to elliptic fully nonlinear
conformal equations on certain subdomains of the sphere with prescribed
constant mean curvature on its boundary. Such subdomains are the hemisphere (or
a geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribed
constant mean curvature on its boundary, and annular domains with minimal
boundary. Our results extend the classifications of Escobar in \cite{E0} when
$n\geq 3$, and Hang-Wang in \cite{HaWa} and Jimenez in \cite{J} when $n=2$.
