Sharp Liouville type results for semilinear elliptic inequalities involving gradient terms on weighted graphs

Abstract: We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms [ \Delta u+u^p\left|\nabla u\right|^q\leq0, ] on weighted graphs, where $(p,q)\in\mathbb{R}^2$. We give a complete classification of $(p,q)$ under which sharp volume growth assumptions are established.