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

Liouville's theorems to quasilinear differential inequalities involving gradient nonlinearity term on manifolds