Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation

摘要

We consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space (M, g, e^−f dv): Δfu + aulogu + bu = 0,  where a, b are two real constants. When the ∞-Bakry–Émery Ricci curvature is bounded from below, we obtain a global gradient estimate which is not dependent on |∇f|. In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions.