Positive solutions of p-th Yamabe type equations on graphs

Abstract

Let G = (V,E) be a nite connected weighted graph, and assume \mathrm{1}\le \alpha \le p\le q. In this paper, we consider the p-th Yamabe type equation -{\Delta p}_{}u+{huq-\mathrm{1}}^{}={\lambda fu\alpha -\mathrm{1}}^{} on G, where {\Delta p}_{} 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 \lambda \in ℝ. In particular, when q = p; our result generalizes Ge's main theorem from the case of \alpha \ge p\rangle \mathrm{1} to the case of \mathrm{1}\le \alpha \le p. It is interesting that our new approach can also work in the case of \alpha \ge p\rangle \mathrm{1}.

Keywords

p-th Yamabe type equation / graph Laplacian / positive solutions

References

[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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