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
DOI: 10.2140/pjm.2020.306.755
Abstract

In this paper, we consider a nonlinear elliptic equation

Δu + aulogu + bu = 0

on complete Riemannian manifolds under integral curvature condition, where a ≥ 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.

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Keywords
integral curvature, nonlinear equation, gradient estimate
Mathematical Subject Classification 2010
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