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Gradient estimates for the $ p$-Laplacian Lichnerowicz equation on smooth metric measure spaces


Authors: Liang Zhao and Dengyun Yang
Journal: Proc. Amer. Math. Soc. 146 (2018), 5451-5461
MSC (2010): Primary 58J05; Secondary 58J35
DOI: https://doi.org/10.1090/proc/13997
Published electronically: September 17, 2018
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Abstract: In this paper, we consider the weighted $ p$-Laplacian Lichnerowicz equation

$\displaystyle \triangle _{p,f} u+cu^{\sigma }=0$    

on smooth metric measure spaces, where $ c\geq 0, p>1,$ and $ \sigma \leq p-1 $ are real constants. A local gradient estimate for positive solutions to this equation is derived, and as applications, we give a corresponding Liouville property and Harnack inequality.

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Additional Information

Liang Zhao
Affiliation: Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China
Email: zhaozongliang09@163.com

Dengyun Yang
Affiliation: College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, People’s Republic of China
Email: yangdengyun@126.com

DOI: https://doi.org/10.1090/proc/13997
Keywords: p-Laplacian, positive solutions, gradient estimate.
Received by editor(s): October 28, 2016
Received by editor(s) in revised form: October 29, 2016, and September 29, 2017
Published electronically: September 17, 2018
Additional Notes: The first author is the corresponding author
Communicated by: Lei Ni
Article copyright: © Copyright 2018 American Mathematical Society

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