Smoothness of radial solutions to Monge-Ampère equations

We prove that generalized convex radial solutions to the generalized Monge-Ampère equation $det{D}^{2}u=f(|x{|}^2/2,u,|\mathrm{\nabla }u{|}^2/2)$ with $f$ smooth are always smooth away from the origin. Moreover, we characterize the global smoothness of these solutions in terms of the order of vanishing of $f$ at the origin.