Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds

摘要

In this paper we obtain essentially sharp generalized Keller-Osserman conditions for wide classes of differential inequalities of the form Lu >= b(x)f(u)l(vertical bar del u vertical bar) and Lu >= b(x)f(u)l(vertical bar del u vertical bar) - g(u)h(vertical bar del u vertical bar) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions b and l. A weak maximum principle which extends and improves previous results valid for the phi-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry-Emery Ricci tensor, are presented.