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

J. Korean Math. Soc. Published online April 2, 2021

yanxun chang and xiaoxiao zhang
Beijing Jiaotong 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.