The Keller-Osserman problem for the k-Hessian operator

Abstract

A delicate problem is to obtain existence of solutions to the boundary blow-up elliptic equation% \begin{equation*} σ_{k}^{1/k}\left( λ\left( D^{2}u\right) \right) =g\left( u\right) \text{ in }Ω\text{, }\underset{x\rightarrow x_{0}}{\lim }% u\left( x\right) =+\infty \text{ }\forall x_{0}\in \partial Ω\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.