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

Abstract

Let $(V,E)$ be an infinite, connected, locally finite weighted graph. We are concerned with the system of elliptic inequalities

where $p$ are $q$ are real numbers and $\Delta$ is the standard graph Laplacian. For $p\le 0$ or $q\le 0$ or $p,q>0$ and $pq\le 1$, 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.