Gradient Estimates and Liouville-Type Theorems for a Nonlinear Elliptic Equation

Abstract

In this paper, we consider the following nonlinear elliptic equation △_f^u+hu^α=0 on the complete smooth metric space (Rⁿ, g₀, e −f dv_g0), where g₀ is the Euclidean metric on Rⁿ and f = |x|²/4. We prove gradient estimates and Liouville-Type theorems for positive solutions of the above equation.