The Keller-Osserman problem for the k-Hessian operator

TL;DR

This paper investigates the existence of boundary blow-up solutions for a class of elliptic equations involving the k-Hessian operator, establishing necessary and sufficient conditions on the nonlinearity g.

Contribution

It provides a complete characterization of when solutions exist for the boundary blow-up problem with the k-Hessian operator, using comparison principles and sub-supersolution methods.

Findings

Derived necessary and sufficient conditions on g for solution existence.

Established comparison principles for the k-Hessian boundary blow-up problem.

Proved existence of positive blow-up solutions under these conditions.

Abstract

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.