Liouville-type theorems for a system of elliptic inequalities on weighted graphs

Let (V,E) be an infinite, connected, locally finite weighted graph. We are concerned with the system of elliptic inequalities \begin{cases}-\Delta u\ge v^{p}\quad\text{in }V,\\ -\Delta v\ge u^{q}\quad\text{in }V,\end{cases} where p are q are real numbers and \Delta is the standard graph Laplacian. For p\leq 0 or q\leq 0 or p,q>0 and pq\leq 1 , we show that the system has no positive solution. Moreover, we also establish some non-existence results of positive solutions of the system in the case where p,q>0 and pq>1 under some assumptions on the volume growth of graph and weight. We also construct an explicit example to show the existence result in the super-critical case. Our result is, in particular, a natural extension of some results in [Gu, Huang, and Sun, Calc. Var. Partial Differential Equations 62 (2023), no. 2, article no. 42] to the system of inequalities.