RESEARCH ARTICLE

Positive solutions of p-th Yamabe type equations on graphs

  • Xiaoxiao ZHANG 1 ,
  • Aijin LIN , 2
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  • 1. Institute of Mathematics, Beijing Jiaotong University, Beijing 100044, China
  • 2. Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Received date: 20 Nov 2017

Accepted date: 16 Oct 2018

Published date: 02 Jan 2019

Copyright

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature

Abstract

Let G = (V,E) be a nite connected weighted graph, and assume 1αpq. In this paper, we consider the p-th Yamabe type equation Δpu+huq1=λfuα1 on G, where Δp is the p-th discrete graph Laplacian, h<0 and f>0 are real functions dened on all vertices of G: Instead of H. Ge's approach [Proc. Amer. Math. Soc., 2018, 146(5): 2219–2224], we adopt a new approach, and prove that the above equation always has a positive solution u>0 for some constant λ. In particular, when q = p; our result generalizes Ge's main theorem from the case of αp1 to the case of 1αp. It is interesting that our new approach can also work in the case of αp1.

Cite this article

Xiaoxiao ZHANG , Aijin LIN . Positive solutions of p-th Yamabe type equations on graphs[J]. Frontiers of Mathematics in China, 2018 , 13(6) : 1501 -1514 . DOI: 10.1007/s11464-018-0734-8

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