Gradient Estimates For $\Delta u + a(x)u\log u + b(x)u = 0$ and its Parabolic Counterpart Under Integral Ricci Curvature Bounds

TL;DR

This paper derives new local and global gradient estimates for solutions to a nonlinear elliptic equation and its parabolic version on Riemannian manifolds with integral Ricci curvature bounds, extending classical results.

Contribution

It introduces novel gradient estimates under integral Ricci curvature bounds for nonlinear equations, extending classical results and providing new corollaries and applications.

Findings

Derived local gradient estimates using Moser's iteration.

Extended classical results to manifolds with integral Ricci curvature bounds.

Established global gradient estimates under geometric conditions.

Abstract

In this paper, we consider a class of important nonlinear elliptic equations $$\Delta u + a(x)u\log u + b(x)u = 0$$ on a collapsed complete Riemannian manifold and its parabolic counterpart under integral curvature conditions, where $a(x)$ and $b(x)$ are two $C^1$-smooth real functions. Some new local gradient estimates for positive solutions to these equations are derived by Moser's iteration provided that the integral Ricci curvature is small enough. Especially, some classical results are extended by our estimates and a few interesting corollaries are given. Furthermore, some global gradient estimates are also established under certain geometric conditions. Some estimates obtained in this paper play an important role in a recent paper by Y. Ma and B. Wang [17], which extended and improved the main results due to B. Wang [29] to the case of integral Ricci curvature bounds.