Positive Solutions of Quasilinear Elliptic Inequalities on Noncompact Riemannian Manifolds

In this paper, we consider the generalized solutions of the inequality
$$ -\operatorname{div}(A(x,u,\nabla u)\nabla u)\ge F(x,u,\nabla u)u^q,\qquad q>1, $$
on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville's theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality $-\Delta u\ge u^q$, $q>1$, on spherically symmetric noncompact Riemannian manifolds.