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Positive solutions of $p$-th Yamabe type equations on infinite graphs


Authors: Xiaoxiao Zhang and Aijin Lin
Journal: Proc. Amer. Math. Soc. 147 (2019), 1421-1427
MSC (2010): Primary 35A15, 35J05, 35J60, 46E39
DOI: https://doi.org/10.1090/proc/14362
Published electronically: December 19, 2018
MathSciNet review: 3910409
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Abstract: Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, and let $\Delta _p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation \begin{equation*} -\Delta _pu+h|u|^{p-2}u=gu^{\alpha -1} \end{equation*} on $G$, where $h$ and $g$ are known, $2<\alpha \leq p$. The prototype of this equation comes from the smooth Yamabe equation on an open manifold. We prove that the above equation has at least one positive solution on $G$.


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Additional Information

Xiaoxiao Zhang
Affiliation: Institute of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
MR Author ID: 1276479
Email: xiaoxiaozhang0408@bjtu.edu.cn

Aijin Lin
Affiliation: College of Science, National University of Defense Technology, Changsha 410073, People’s Republic of China
MR Author ID: 1049517
Email: aijinlin@pku.edu.cn

Received by editor(s): September 1, 2017
Published electronically: December 19, 2018
Additional Notes: The first author was supported by the National Natural Science Foundation of China (Grants No. 11471138, 11501027, and 11871094) and Fundamental Research Funds for the Central Universities (Grant No. 2017JBM072).
The second author was supported by the National Natural Science Foundation of China (Grant No. 11401578).
Communicated by: Lei Ni
Article copyright: © Copyright 2018 American Mathematical Society