A note on Liouville type equations on graphs

Ge, Huabin; Hua, Bobo; Jiang, Wenfeng

doi:10.1090/proc/14155

A note on Liouville type equations on graphs
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by Huabin Ge, Bobo Hua and Wenfeng Jiang PDF

Proc. Amer. Math. Soc. 146 (2018), 4837-4842 Request permission

Abstract:

In this note, we study the Liouville equation $\Delta u=-e^u$ on a graph $G$ satisfying a certain isoperimetric inequality. Following the idea of W. Ding, we prove that there exists a uniform lower bound for the energy, $\sum _G e^u,$ of any solution $u$ to the equation. In particular, for the 2-dimensional lattice graph $\mathbb {Z}^2,$ the lower bound is given by $4.$

References

William Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), no. 1, 213–242. MR 1230930, DOI 10.2307/2946638
Wen Xiong Chen and Congming Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622. MR 1121147, DOI 10.1215/S0012-7094-91-06325-8
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, Wenfeng Jiang, Kazdan-Warner equation on infinite graphs. arXiv:1706.08698.
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, Yamabe type equations on graphs, J. Differential Equations 261 (2016), no. 9, 4924–4943. MR 3542963, DOI 10.1016/j.jde.2016.07.011
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
Matthias Keller and Michael Schwarz, The Kazdan-Warner equation on canonically compactifiable graphs, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 70, 18. MR 3776360, DOI 10.1007/s00526-018-1329-7
Yong Lin and Yiting Wu, The existence and nonexistence of global solutions for a semilinear heat equation on graphs, Calc. Var. Partial Differential Equations 56 (2017), no. 4, Paper No. 102, 22. MR 3688855, DOI 10.1007/s00526-017-1204-y
Yong Lin, Yiting Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, arXiv:1704.05702.
Russell Lyons, Yuval Peres, Probability on Trees and Networks, available at http://mypage.iu.edu/ rdlyons/prbtree/prbtree.html.
Guofang Wang, Moser-Trudinger inequalities and Liouville systems, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 10, 895–900 (English, with English and French summaries). MR 1689869, DOI 10.1016/S0764-4442(99)80293-6
Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR 1743100, DOI 10.1017/CBO9780511470967
Yunyan Yang, An Interpolation of Hardy Inequality and Trudinger-Moser Inequality in $\mathbb {R}^N$ and Its Applications, J. Funct. Anal. 14 (1973), 349-381.