Gradient estimates for the $p$-Laplacian Lichnerowicz equation on smooth metric measure spaces
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Abstract:
In this paper, we consider the weighted $p$-Laplacian Lichnerowicz equation \begin{equation*} \triangle _{p,f} u+cu^{\sigma }=0 \end{equation*} 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.References
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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
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5451-5461
- MSC (2010): Primary 58J05; Secondary 58J35
- DOI: https://doi.org/10.1090/proc/13997
- MathSciNet review: 3866881