Entire subsolutions of a kind of k-Hessian type equations with gradient terms

TL;DR

This paper investigates entire subsolutions of k-Hessian type equations with gradient terms, establishing conditions for their existence or nonexistence, extending classical results to a broader class of nonlinear PDEs.

Contribution

It provides a necessary and sufficient condition for the existence of entire admissible subsolutions of k-Hessian equations with gradient terms, generalizing the Keller-Osserman condition.

Findings

Derived conditions for existence and nonexistence of solutions

Extended classical results to fully nonlinear equations

Analyzed differences between semilinear and fully nonlinear cases

Abstract

In this paper, we consider a kind of $k$-Hessian type equations $S_k^{\frac{1}{k}}(D^2u+\mu|D u|I)= f(u)$ in $\mathbb{R}^n$, and provide a necessary and sufficient condition of $f$ on the existence and nonexistence of entire admissible subsolutions, which can be regarded as a generalized Keller-Osserman condition. The existence and nonexistence results are proved in different ranges of the parameter $\mu$ respectively, which embrace the standard Hessian equation case ($\mu=0$) by Ji and Bao (Proc Amer Math Soc 138: 175--188, 2010) as a typical example. The difference between the semilinear case ($k=1$) and the fully nonlinear case ($k\ge 2$) is also concerned.