A sharp Liouville principle for $$\Delta _m u+u^p|\nabla u|^q\le 0$$ on geodesically complete noncompact Riemannian manifolds

For $$(m,p,q)\in (1,\infty )\times {\mathbb {R}}\times {\mathbb {R}}$$ , this paper establishes a sharp Liouville principle for the weak solutions to the quasilinear elliptic inequality of second order $$\Delta _m u+u^p|\nabla u|^q\le 0$$ on the geodesically complete noncompact Riemannian manifolds, which is new even for the Euclidean spaces.