Gradient Estimates And Liouville Theorems For A Class of Nonlinear Elliptic Equations

TL;DR

This paper derives gradient estimates and Liouville theorems for a class of nonlinear elliptic equations on Riemannian manifolds, extending previous results to more general equations and conditions.

Contribution

It introduces new gradient estimates for nonlinear elliptic equations on manifolds that are independent of solution bounds and Laplacian of the distance, and extends Liouville theorems to broader classes.

Findings

Gradient estimates for positive solutions without solution bounds

Liouville-type theorems under nonnegative Ricci curvature

Harnack inequalities as a consequence of the estimates

Abstract

In this paper, first we study carefully the positive solutions to $\Delta u+\lambda_{1}u\ln u +\lambda_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as Lichnerowicz-type equations, and obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on $(M, g)$. Then, we extend our techniques to a class of more general semilinear elliptic equations $\Delta u(x)+uh(\ln u)=0$ and obtain some similar results under some suitable analysis conditions on these equations. Moreover, we also obtain some Liouville-type theorems for these equations when $Ric(g)\geq 0$ and establish some Harnack inequalities as consequences.