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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.

(a). In this paper, we present a sufficient condition for the existence of solutions to semilinear elliptic inequality on a large class of manifolds. That is, if there exist constants r 0 and C 0 satisfying ∫ r 0 + ∞ t 2 p − 1 V x ( t ) p − 1 d t ⩽ C 0 , ∀ x ∈ M , then there exists a positive solution to Δ u + u p ⩽ 0 , where V x 0 ( r ) is the volume of geodesic ball of radius r centered at x 0 . (b). In Grigor’yan and Sun (2014), Grigor’yan and Sun proved that if V x 0 ( r ) ⩽ r α 1 ( ln r ) α 2 then the only nonnegative weak solution of Δ u + u p ⩽ 0 on a complete Riemannian manifold is identically 0, here V x 0 ( r ) is the volume of geodesic ball of radius r centered at x 0 , and α 1 = 2 p p − 1 , α 2 = 1 p − 1 ; moreover, parameters α 1 and α 2 are sharp that if α 2 > 1 p − 1 then there exists a manifold admiting a positive weak solution. In this paper, we present a sufficient condition for the existence of solutions on a large class of manifolds. That is, if there exist positive constants r 0 and C 0 satisfying ∫ r 0 + ∞ t 2 p − 1 V x ( t ) p − 1 d t ⩽ C 0 , ∀ x ∈ M , then there exists a positive solution to Δ u + u p ⩽ 0 .

This paper determines when the Cauchy problem $$\begin{aligned} \left\{ \begin{array}{ll} {{\partial _t u} = \Delta u -Vu+ Wu^p} &{}\quad \text{ in } M \times (0, \infty ) \\ {u(\cdot ,0)= {u_0(\cdot )}} &{}\quad \text{ in } M \end{array} \right. \end{aligned}$$ has no global positive solution on a connected non-compact geodesically complete Riemannian manifold for a given triple (V, W, p). As the principal result of this paper, Theorem 1.1 optimally extends in a unified way most of the previous results in this subject (cf. Ishige in J Math Anal Appl 344:231–237, 2008; Pinsky in J Differ Equ 246(6):2561–2576, 2009; Zhang in Duke Math J 97:515–539, 1999; Zhang in J Differ Equ 170:188–214, 2001).

In this paper we deal with the uniqueness of nonnegative solutions to systems of quasilinear elliptic differential inequalities on Riemannian manifolds, and we obtain the Liouville type theorems only in terms of the volume growth of geodesic ball. By applying the perturbation and gluing techniques, a counterexample is constructed to show the sharpness of the volume growth condition.