In this paper, we consider the following $ f $-Laplacian equation with a gradient term and a nonlinear term $ \begin{equation*} \Delta_{f} u+W\left(u\right)\left|\nabla u\right|^2+F\left(u\right) = 0 \end{equation*} $ on smooth metric measure spaces with nonnegative $ m $-Bakry-Émery Ricci curvature, where $ W(t) $ is a continuous function for $ t > 0 $ and $ F(t) $ is a differentiable function in $ (0, \infty) $. Under certain assumptions on $ W $ and $ F $, by using the Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method, we derive some Liouville-type theorems for positive solutions to the above equation. Our results extend those of Lu (2025) and McCoy (2007).

In this paper, we employ the Nash–Moser iteration technique and the Saloff–Coste’s Sobolev inequality to study the local and global properties of positive solutions to the Trudinger equation ${\Delta }_{p,f}{u}^{{\displaystyle \frac{1}{p-1}}}+b{u}^q+c{u}^r=0$ on a complete smooth metric measure space with m-Bakry-Émery Ricci curvature bounded from below, where $b,c\in \mathbb{R}$, $p>1$, and $q\le r$ are real constants. We first give universal gradient estimates for the above equation under certain assumptions on b, c, p, q, and r. As their natural corollary, Harnack inequalities and Liouville-type theorems for positive solutions are obtained. Later, we consider the explicit global gradient estimates for such entire solutions through the global gradient estimates obtained.