Frontiers of Mathematics in China >
Positive solutions of p-th Yamabe type equations on graphs
Received date: 20 Nov 2017
Accepted date: 16 Oct 2018
Published date: 02 Jan 2019
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Let G = (V,E) be a nite connected weighted graph, and assume . In this paper, we consider the p-th Yamabe type equation on G, where 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 . In particular, when q = p; our result generalizes Ge's main theorem from the case of to the case of . It is interesting that our new approach can also work in the case of .
Key words: p-th Yamabe type equation; graph Laplacian; positive solutions
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
1 |
Aubin T. The scalar curvature. In: Cahen M, Flato M, eds. Differential Geometry and Relativity: A Volume in Honour of Andre Lichnerowicz on His 60th Birthday. Math Phys Appl Math, Vol 3. Dordrecht: Reidel, 1976, 5–18
|
2 |
Bauer F, Hua B, Jost J. The dual cheeger constant and spectra of innite graphs. Adv Math, 2014, 251(1): 147–194
|
3 |
Chung F R K. Spectral Graph Theory. Providence: Amer Math Soc, 1997
|
4 |
Chen W, Li C. A note on the Kazdan-Warner type conditions. J Differential Geom, 1995, 41: 259–268
|
5 |
Chung Y-S, Lee Y-S, Chung S-Y. Extinction and positivity of the solutions of the heat equations with absorption on networks. J Math Anal Appl, 2011, 380: 642–652
|
6 |
Frank B, Hua B, Yau S-T. Sharp Davies-Gaffney-Grigor'Yan lemma on graphs. Math Ann, 2017, 368: 1429–1437
|
7 |
Grigor'yan A, Lin Y, Yang Y. Kazdan-Warner equation on graph. Calc Var Partial Differential Equations, 2016, 55(4): 92–13pp)
|
8 |
Grigor'yan A, Lin Y, Yang Y. Existence of positive solutions to some nonlinear equations on locally nite graphs. Sci China Math, 2017, 60: 1311–1324
|
9 |
Grigor'yan A, Lin Y, Yang Y. Yamabe type equations on graphs. J Differential Equations, 2016, 261: 4924–4943
|
10 |
Ge H. The p-th Kazdan-Warner equation on graphs. Commun Contemp Math (to appear)
|
11 |
Ge H. Kazdan-Warner equation on graph in the negative case. J Math Anal Appl, 2017, 453(2): 1022–1027
|
12 |
Ge H. A p-th Yamabe equation on graph. Proc Amer Math Soc, 2018, 146(5): 2219–2224
|
13 |
Haeseler S, Keller M, Lenz D, Wojciechowski R. Laplacians on innite graphs: Dirichlet and Neumann boundary conditions. J Spectr Theory, 2012, 2(4): 397–432
|
14 |
Han Z. A Kazdan-Warner type identity for the σk curvature. C R Acad Sci Paris, 2006, 342: 475–478
|
15 |
Lin Y, Wu Y. Blow-up problems for nonlinear parabolic equations on locally nite graphs. Acta Math Sci Ser B Engl Ed, 2018, 38(3): 843–856
|
16 |
Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature. J Differential Geom, 1984, 20: 479–495
|
17 |
Trudinger N. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann Sc Norm Super Pisa, 1968, 3: 265–274
|
18 |
Wang Y, Zhang X. A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds. Sci China Ser A, 2008, 51(6): 1111{1118
|
19 |
Yamabe H. On a deformation of Riemannian structures on compact manifolds. Osaka Math J, 1960, 12: 21–37
|
20 |
Zhang X, Lin A. Positive solutions of p-th Yamabe equation on innite graphs. Proc Amer Math Soc, https://doi.org/10.1090/proc/14362
|
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