The positive solutions to quasilinear elliptic inequalities on model Riemannian manifolds

Abstract

We investigate the problem of implementation of the Liouville type theorems on the existence of positive solutions of 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 of the studied inequalities on the Riemannian manifolds. The method is based on a study of radially symmetric solutions of an ordinary differential equation generated by the basic inequality and establishing the relationship of the existence of entire positive solutions of quasilinear elliptic inequalities and solvability of the Cauchy problem for this equation. In addition, in the paper we apply classical methods of the theory of elliptic equations and second order inequalities (the maximum principle, the principle of comparison, etc.). The obtained results generalize similar results obtained previously by Y. Naito and H. Usami for Euclidean space Rn, as well as some earlier results by A. G. Losev and E. A. Mazepa.