#### Vol. 306, No. 2, 2020

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A new local gradient estimate for a nonlinear equation under integral curvature condition on manifolds

### Liang Zhao and Shouwen Fang

Vol. 306 (2020), No. 2, 755–765
##### Abstract

In this paper, we consider a nonlinear elliptic equation

$\Delta u+aulogu+bu=0$

on complete Riemannian manifolds under integral curvature condition, where $a\ge 0,b$ are two real constants. A new local gradient estimate for positive solutions to this equation under integral curvature condition is derived, and as an application, we give a corresponding Harnack inequality.

##### Keywords
integral curvature, nonlinear equation, gradient estimate
Primary: 58J05
Secondary: 58J35
##### Milestones
Received: 22 May 2019
Revised: 26 November 2019
Accepted: 2 January 2020
Published: 13 July 2020
##### Authors
 Liang Zhao Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing China Shouwen Fang College of Mathematical Science Yangzhou University Yangzhou China