Liouville-type theorems for partial trace equations with nonlinear gradient terms

In this paper, we will study various Liouville-type theorems for partial trace equations with nonlinear gradient terms. Specifically, we will provide sufficient conditions for non-negative viscosity subsolutions of these equations in $R^n$ to vanish identically. For a prototype of such equations, we will give necessary and sufficient conditions for non-negative subsolutions in $R^n$ to be identically zero.