  A delicate problem is to obtain existence of solutions to the boundary
blow-up elliptic equation% \begin{equation*} \sigma _{k}^{1/k}\left( \lambda
\left( D^{2}u\right) \right) =g\left( u\right) \text{ in }\Omega \text{,
}\underset{x\rightarrow x_{0}}{\lim }% u\left( x\right) =+\infty \text{
}\forall x_{0}\in \partial \Omega \text{,} \end{equation*}% where $\sigma
_{k}^{1/k}\left( \lambda \left( D^{2}u\right) \right) $ is the $k$-Hessian
operator and $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain. Our
goal is to provide a necessary and sufficient condition on $g$ to ensure
existence of at least one positive blow-up solution. The main tools for proving
existence are the comparison principle and the method of sub and
supersolutions.
