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Frontiers of Mathematics in China

, Volume 13, Issue 6, pp 1501–1514 | Cite as

Positive solutions of p-th Yamabe type equations on graphs

  • Xiaoxiao Zhang
  • Aijin LinEmail author
Research Article

Abstract

Let G = (V,E) be a finite connected weighted graph, and assume 1 ⩽ αpq. In this paper, we consider the p-th Yamabe type equation ―∆pu+huq―1 = λfuα―1 on G, where ∆p is the p-th discrete graph Laplacian, h < 0 and f > 0 are real functions defined 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 ∈ ℝ. In particular, when q = p, our result generalizes Ge’s main theorem from the case of αp > 1 to the case of 1 ⩽ αp, It is interesting that our new approach can also work in the case of αp > 1.

Keywords

p-th Yamabe type equation graph Laplacian positive solutions 

MSC

34B35 35B15 58E30 

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Notes

Acknowledgements

The first author would like to thank Professor Yanxun Chang for constant guidance and encouragement. The second author would like to thank Professor Gang Tian and Huijun Fan for constant encouragement and support. Both authors would also like to thank Professor Huabin Ge for many helpful conversations. The first author was supported by the National Natural Science Foundation of China (Grant Nos. 11471138, 11501027, 11871094) and the 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).

References

  1. 1.
    Aubin T. The scalar curvature. In: Cahen M, Flato M, eds. Differential Geometry and Relativity: A Volume in Honour of André Lichnerowicz on His 60th Birthday. Math Phys Appl Math, Vol 3. Dordrecht: Reidel, 1976, 5–18CrossRefGoogle Scholar
  2. 2.
    Bauer F, Hua B, Jost J. The dual cheeger constant and spectra of infinite graphs. Adv Math, 2014, 251(1): 147–194MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chung F R K. Spectral Graph Theory. Providence: Amer Math Soc, 1997zbMATHGoogle Scholar
  4. 4.
    Chen W, Li C. A note on the Kazdan-Warner type conditions. J Differential Geom, 1995, 41: 259–268MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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: 642–652MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Frank B, Hua B, Yau S-T. Sharp Davies-Gaffney-Grigor’Yan lemma on graphs. Math Ann, 2017, 368: 1429–1437MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grigor’yan A, Lin Y, Yang Y. Kazdan-Warner equation on graph. Calc Var Partial Differential Equations, 2016, 55(4): 92 (13pp)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grigor’yan A, Lin Y, Yang Y. Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci China Math, 2017, 60: 1311–1324MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grigor’yan A, Lin Y, Yang Y. Yamabe type equations on graphs. J Differential Equations, 2016, 261: 4924–4943MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ge H. The p-th Kazdan-Warner equation on graphs. Commun Contemp Math (to appear)Google Scholar
  11. 11.
    Ge H. Kazdan-Warner equation on graph in the negative case. J Math Anal Appl, 2017, 453(2): 1022–1027MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ge H. A p-th Yamabe equation on graph. Proc Amer Math Soc, 2018, 146(5): 2219–2224MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Haeseler S, Keller M, Lenz D, Wojciechowski R. Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions. J Spectr Theory, 2012, 2(4): 397–432MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Han Z. A Kazdan-Warner type identity for the σ k curvature. C R Acad Sci Paris, 2006, 342: 475–478MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lin Y, Wu Y. Blow-up problems for nonlinear parabolic equations on locally finite graphs. Acta Math Sci Ser B Engl Ed, 2018, 38(3): 843–856MathSciNetCrossRefGoogle Scholar
  16. 16.
    Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature. J Differential Geom, 1984, 20: 479–495MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Trudinger N. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann Sc Norm Super Pisa, 1968, 3: 265–274MathSciNetzbMATHGoogle Scholar
  18. 18.
    Wang Y, Zhang X. A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds. Sci China Ser A, 2008, 51(6): 1111–1118MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yamabe H. On a deformation of Riemannian structures on compact manifolds. Osaka Math J, 1960, 12: 21–37MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zhang X, Lin A. Positive solutions of p-th Yamabe equation on infinite graphs. Proc Amer Math Soc, https://doi.org/10.1090/proc/14362

Copyright information

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

Authors and Affiliations

  1. 1.Institute of MathematicsBeijing Jiaotong UniversityBeijingChina
  2. 2.Department of MathematicsNational University of Defense TechnologyChangshaChina

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