Liouville Type Theorems for Nonlinear p-Laplacian Equation on Complete Noncompact Riemannian Manifolds

In this paper,the authors study the gradient estimates for positive weak solutions to the following p-Laplacian equation△_(p)u+au^(σ)=0 on complete noncompact Riemannian manifold,where a,σare two nonzero real constants with p≠2.Using the gradient estimate,they can get the corresponding Lionville theorem.On the other hand,by virtue of the Poincare inequality,they also obtain a Liouville theorem under some integral conditions with respect to positive weak solutions.