Necessary and sufficient conditions on solvability for Hessian inequalities

Abstract

In this paper, we discuss the solvability of the Hessian inequality $\sigma ^{\frac {1}{k}}_{k}(\lambda (D^{2}u)) \ge f(u)$ on the entire space $\mathbb {R}^{n}$ and provide a necessary and sufficient condition, which can be regarded as a generalized Keller-Osserman condition.