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A $ p$-th Yamabe equation on graph


Author: Huabin Ge
Journal: Proc. Amer. Math. Soc. 146 (2018), 2219-2224
MSC (2010): Primary 34B35, 35B15, 58E30
DOI: https://doi.org/10.1090/proc/13929
Published electronically: January 8, 2018
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Abstract: Assume $ \alpha \geq p>1$. Consider the following $ p$-th Yamabe equation on a connected finite graph $ G$:

$\displaystyle \Delta _p\varphi +h\varphi ^{p-1}=\lambda f\varphi ^{\alpha -1},$    

where $ \Delta _p$ is the discrete $ p$-Laplacian, $ h$ and $ f>0$ are known real functions defined on all vertices. We show that the above equation always has a positive solution $ \varphi $ for some constant $ \lambda \in \mathds {R}$.

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Additional Information

Huabin Ge
Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
Email: hbge@bjtu.edu.cn

DOI: https://doi.org/10.1090/proc/13929
Received by editor(s): December 11, 2016
Received by editor(s) in revised form: August 16, 2017
Published electronically: January 8, 2018
Additional Notes: The research is supported by National Natural Science Foundation of China under Grant No.11501027.
Communicated by: Goufang Wei
Article copyright: © Copyright 2018 American Mathematical Society

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