Positive solutions of $p$-th Yamabe type equations on infinite graphs
HTML articles powered by AMS MathViewer
- 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
- PDF | Request permission
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
- Alexander Grigor’yan, Yong Lin, and Yunyan Yang, Kazdan-Warner equation on graph, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 92, 13. MR 3523107, DOI 10.1007/s00526-016-1042-3
- Alexander Grigor’yan, Yong Lin, and YunYan Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math. 60 (2017), no. 7, 1311–1324. MR 3665801, DOI 10.1007/s11425-016-0422-y
- Alexander Grigor’yan, Yong Lin, and Yunyan Yang, Yamabe type equations on graphs, J. Differential Equations 261 (2016), no. 9, 4924–4943. MR 3542963, DOI 10.1016/j.jde.2016.07.011
- H. Ge, The $p$-th Kazdan-Warner equation on graphs. To appear in Commun. Contemp. Math.
- Huabin Ge, Kazdan-Warner equation on graph in the negative case, J. Math. Anal. Appl. 453 (2017), no. 2, 1022–1027. MR 3648273, DOI 10.1016/j.jmaa.2017.04.052
- Huabin Ge, A $p$-th Yamabe equation on graph, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2219–2224. MR 3767372, DOI 10.1090/proc/13929
- Huabin Ge, Bobo Hua, and Wenfeng Jiang, A note on Liouville type equations on graphs, Proc. Amer. Math. Soc. 146 (2018), no. 11, 4837–4842. MR 3856150, DOI 10.1090/proc/14155
- Huabin Ge and Wenfeng Jiang, Yamabe equations on infinite graphs, J. Math. Anal. Appl. 460 (2018), no. 2, 885–890. MR 3759076, DOI 10.1016/j.jmaa.2017.12.020
- H. Ge and W. Jiang, The $1$-Yamabe equation on graph, Commun. Contemp. Math., online published, DOI: https://doi.org/10.1142/S0219199718500402.
- H. Ge and W. Jiang, Kazdan-Warner equation on infinite graphs, J. Korean Math. Soc. 55 (2018), no. 5, 1091–1101.
- Huabin Ge, A $p$-th Yamabe equation on graph, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2219–2224. MR 3767372, DOI 10.1090/proc/13929
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