J. Korean Math. Soc. 2021; 58(3): 703-722
Online first article April 2, 2021 Printed May 1, 2021
https://doi.org/10.4134/JKMS.j200221
Copyright © The Korean Mathematical Society.
Yanxun Chang, Xiaoxiao Zhang
Beijing Jiaotong University; Beijing Wuzi University
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
2001; 38(5): 1069-1105
2007; 44(6): 1213-1231
2008; 45(6): 1613-1622
2010; 47(5): 967-983
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