  We study the Dirichlet problem for a class of curvature equations arising
from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where
$n \geq 3$. We prove there exists a unique solution using the continuity method
which is based on \emph{a priori} estimates for admissible solutions. In
deriving the estimates, a crucial step is to derive a lower bound for the
gradient on the boundary. This is overcome by constructing a cluster of
subsolutions.
