Abstract
In this paper, we investigate several Liouville-type theorems related to k k -Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S k [ D 2 u ] â„ h ( u , | D u | ) S_k[D^2u]\ge h(u,|Du|) in R n \mathbb {R}^n . The results depend on some qualified growth conditions of h h at infinity. A Liouville-type result to subsolutions of a prototype equation S k [ D 2 u ] = f ( u ) + g ( u ) Ï ( | D u | ) S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S k [ D 2 u ] = f ( u ) + g ( u ) | D u | q S_k[D^2u]=f(u)+g(u)|Du|^q for 0 †q > k + 1 0\le q>k+1 is also given.
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