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Gradient estimates for a nonlinear elliptic equation on complete Riemannian manifolds


Authors: Bingqing Ma, Guangyue Huang and Yong Luo
Journal: Proc. Amer. Math. Soc. 146 (2018), 4993-5002
MSC (2010): Primary 58J35; Secondary 35B45
DOI: https://doi.org/10.1090/proc/14106
Published electronically: August 7, 2018
MathSciNet review: 3856164
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Abstract: In this short paper, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold:

$\displaystyle \Delta u+cu^{\alpha }=0,$    

where $ c, \alpha $ are two real constants and $ c\neq 0$.


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Bingqing Ma
Affiliation: College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, People’s Republic of China; Department of Mathematics, Henan Normal University, Xinxiang 453007, People’s Republic of China
Email: bqma@henannu.edu.cn

Guangyue Huang
Affiliation: Department of Mathematics, Henan Normal University, Xinxiang 453007, People’s Republic of China
Email: hgy@henannu.edu.cn

Yong Luo
Affiliation: School of mathematics and statistics, Wuhan University, Wuhan 430072, People’s Republic of China
Email: yongluo@whu.edu.cn

DOI: https://doi.org/10.1090/proc/14106
Keywords: Gradient estimate, nonlinear elliptic equation, Liouville-type theorem.
Received by editor(s): October 26, 2017
Received by editor(s) in revised form: October 26, 2017, November 9, 2017, November 13, 2017, and January 22, 2018
Published electronically: August 7, 2018
Additional Notes: The research of the authors was supported by NSFC Nos. 11371018, 11401179, 11501421, 11671121
Communicated by: Guofang Wei
Article copyright: © Copyright 2018 American Mathematical Society