Entire solutions of the generalized Hessian inequality

TL;DR

This paper investigates the generalized Hessian inequality, establishing necessary and sufficient conditions for the existence of entire solutions across various differential operators, thus extending classical results to a broader class.

Contribution

It introduces a unified framework for the global solvability of generalized Hessian inequalities, encompassing multiple well-known operators and deriving generalized Keller-Osserman conditions.

Findings

Derived necessary and sufficient conditions for solutions

Unified treatment of various differential operators

Extended classical Keller-Osserman conditions

Abstract

In this paper, we discuss the more general Hessian inequality $\sigma_{k}^{\frac{1}{k}}(\lambda (D_i (A\left(|Du|\right) D_j u)))\geq f(u)$ including the Laplacian, p-Laplacian, mean curvature, Hessian, k-mean curvature operators, and provide a necessary and sufficient condition on the global solvability, which can be regarded as generalized Keller-Osserman conditions.