Gradient Estimates For $\Delta u + au^{p+1}=0$ And Liouville Theorems

TL;DR

This paper develops a unified method to derive gradient estimates and Liouville theorems for positive solutions of a class of nonlinear elliptic equations on complete noncompact Riemannian manifolds, improving previous results.

Contribution

It introduces a unified approach to improve gradient estimates and Liouville theorems for $ riangle u + a u^{p+1}=0$ on manifolds, covering new cases and refining existing bounds.

Findings

Enhanced gradient estimates for $a>0$ cases, excluding $ ext{dim}=4$ and $ ext{dim} extless 3$.

Significant improvements for $a<0$, $p>0$ cases.

Liouville-type theorem established under nonnegative Ricci curvature.

Abstract

In this short note, we use a unified method to consider the gradient estimates of the positive solution to the following nonlinear elliptic equation $\Delta u + au^{p+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ where $a > 0$ and $ p <\frac{4}{n}$ or $a < 0$ and $p >0$ are two constants. For the case $a>0$, this improves considerably the previous known results except for the cases $\dim(M)=4$ and supplements the results for the case $\dim(M)\leq 2$. For the case $a<0$ and $p>0$, we also improve considerably the previous related results. When the Ricci curvature of $(M,g)$ is nonnegative, we also obtain a Liouville-type theorem for the above equation.