Gradient estimates and Liouville type theorems for a nonlinear elliptic equation

TL;DR

This paper establishes gradient estimates and Liouville theorems for positive solutions to a nonlinear elliptic equation on Riemannian manifolds, generalizing classical results by analyzing solutions under curvature conditions.

Contribution

It provides new gradient estimates and Liouville theorems for a specific nonlinear elliptic equation, extending Yau's classical results to broader settings.

Findings

Bounded positive solutions are constant under certain conditions for a<0.

Generalization of Yau's classical Liouville theorem to a nonlinear elliptic equation.

Conditions relating Ricci curvature bounds to solution behavior.

Abstract

Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where $a$ is a nonzero constant. In particular, for $a<0$, we prove that any bounded positive solution of the above equation with a suitable condition for $a$ with respect to the lower bound of Ricci curvature must be $u\equiv 1$. This generalizes a classical result of Yau.