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

Abstract

Abstract We consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space (M,g,e−fdv) $(M, g,e^{-f}\,dv)$: Δfu+aulogu+bu=0, $$\Delta_{f} u+au\log u+bu=0, $$ 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| $|\nabla f|$. In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions.

Keywords

• Gradient estimate
• Nonlinear elliptic equation
• Liouville-type theorem