Liouville-type theorems for systems of elliptic inequalities involving $ p $-Laplace operator on weighted graphs

In this paper, we study a system of elliptic inequalities involving $ p $-Laplace operator on a connected, locally finite, weighted graph $ (V, E) $$ \begin{equation*} \begin{cases} -\Delta_{p_1}u(x)&\ge \sigma(x)v(x)^r \mbox{ in } V,\\ -\Delta_{p_2}v(x)&\ge \sigma(x)u(x)^s \mbox{ in } V, \end{cases} \end{equation*} $where $ r, s>0 $ and $ \Delta_{p_i} $ is the $ p_i $-Laplacian operator on $ V $ with $ p_i>1 $, $ i = 1,2 $. We establish some non-existence results of positive solutions of the system under assumption on volume growth of a ball. An example is also constructed to show the sharpness of our condition.