  In this paper we prove the existence of complete, noncompact convex
hypersurfaces whose $p$-curvature function is prescribed on a domain in the
unit sphere. This problem is related to the solvability of Monge-Amp\`ere type
equations subject to certain boundary conditions depending on the value of $p$.
The special case of $p=1$ was previously studied by Pogorelov and Chou-Wang.
Here, we give some sufficient conditions for the solvability for general
$p\neq1$.
