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 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.j200221Published online April 2, 2021Printed 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 Downloads: Full-text PDF   Full-text HTML