Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2021; 58(3): 703-722

Published online May 1, 2021 https://doi.org/10.4134/JKMS.j200221

Copyright © The Korean Mathematical Society.

Existence of global solutions to some nonlinear equations on locally finite graphs

Yanxun Chang, Xiaoxiao Zhang

Beijing Jiaotong University; Beijing Wuzi University

Abstract

Let $G=(V,E)$ be a connected locally finite and weighted graph, $\Delta_p$ be the $p$-th graph Laplacian. Consider the $p$-th nonlinear equation $$-\Delta_pu+h|u|^{p-2}u=f(x,u)$$ on $G$, where $p>2$, $h,f$ satisfy certain assumptions. Grigor'yan-Lin-Yang \cite{GLY2} proved the existence of the solution to the above nonlinear equation in a bounded domain $\Omega\subset V$. 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 \cite{GLY2}. To the $m$-order differential operator $\mathcal{L}_{m,p}$, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.

Keywords: Fr\'{e}chet derivative, graph, nonlinear equation

MSC numbers: 05C22, 35J05, 35J60

Supported by: The authors would like to thank Professor Huabin Ge for his helpful discussions. The first author is supported by National Natural Science Foundation of China under Grant No. 11971053. The second author is supported by National Natural Science Foundation of China under Grant No. 11871094