Universal Gradient Estimates for the Trudinger Equation on Smooth Metric Measure Spaces

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 Δp,fu1p−1+buq+cur=0 on a complete smooth metric measure space with m-Bakry-Émery Ricci curvature bounded from below, where b,c∈R, p>1, and q≤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.