Existence of global solutions to some nonlinear equations on locally finite graphs
J. Korean Math. Soc. 2021 Vol. 58, No. 3, 703-722
https://doi.org/10.4134/JKMS.j200221
Published online April 2, 2021
Printed May 1, 2021
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
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