  We investigate the nonexistence and existence of nontrivial positive
solutions to $\Delta_m u+u^p|\nabla u|^q\leq0$ on noncompact geodesically
complete Riemannian manifolds, where $m>1$, and $(p,q)\in \mathbb{R}^2$.
According to classification of $(p, q)$, we establish different volume growth
conditions to obtain Liouville's theorems for the above quasilinear
differential inequalities, and we also show these volume growth conditions are
sharp in most cases. Moreover, the results are completely new for $(p, q)$ of
negative pair, even in the Euclidean space.
