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http://dx.doi.org/10.4134/JKMS.j200221

EXISTENCE OF GLOBAL SOLUTIONS TO SOME NONLINEAR EQUATIONS ON LOCALLY FINITE GRAPHS  

Chang, Yanxun (Institute of Mathematics Beijing Jiaotong University)
Zhang, Xiaoxiao (School of Information Beijing Wuzi University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.3, 2021 , pp. 703-722 More about this Journal
Abstract
Let G = (V, E) be a connected locally finite and weighted graph, ∆p be the p-th graph Laplacian. Consider the p-th nonlinear equation -∆pu + h|u|p-2u = f(x, u) on G, where p > 2, h, f satisfy certain assumptions. Grigor'yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V. In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m-order differential operator 𝓛m,p, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.
Keywords
Frechet derivative; graph; nonlinear equation;
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