Radial graphs of constant curvature and prescribed boundary

In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with constant curvature and that realizes the prescribed boundary. As an application we prove that if $\Omega\subset\mathbb{S}^n$ is a mean convex domain whose closure is contained in an open hemisphere of $\mathbb{S}^n$ then, for $0