The positive solutions to quasilinear elliptic inequalities on model Riemannian manifolds

We investigate the problem of implementation of Liouville type theorems on the existence of positive solutions to some quasilinear elliptic inequalities on model (spherically symmetric) Riemannian manifolds. In particular, we find exact conditions for the existence and nonexistence of entire positive solutions to the studied inequalities on the Riemannian manifolds. The method is based on study of radially symmetric solutions to an ordinary differential equation generated by the basic inequality and establish the relationship of the existence of entire positive solutions to quasilinear elliptic inequalities and solvability of the Cauchy problem for this equation. Moreover, in the paper we apply classical methods of the theory of elliptic equations and inequalities the second order (the maximum principle, the principle of comparison, etc.). The results generalize similar results, obtained previously by Y. Naito and H. Usami for Euclidean space $\mathbf R^n$, as well as some earlier results of the papers by A. G. Losev and E. A. Mazepa.