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

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- by Xiaoxiao Zhang and Aijin Lin
- Proc. Amer. Math. Soc.
**147**(2019), 1421-1427 - DOI: https://doi.org/10.1090/proc/14362
- Published electronically: December 19, 2018
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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 $G$.## References

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## Bibliographic 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 1421-1427 - MSC (2010): Primary 35A15, 35J05, 35J60, 46E39
- DOI: https://doi.org/10.1090/proc/14362
- MathSciNet review: 3910409