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

Abstract

In this paper, the authors study the gradient estimates for positive weak solutions to the following p-Laplacian equation ${\Delta _p}u + a{u^\sigma } = 0$ on complete noncompact Riemannian manifold, where a, σ are two nonzero real constants with p ≠ 2. Using the gradient estimate, they can get the corresponding Liouville theorem. On the other hand, by virtue of the Poincaré inequality, they also obtain a Liouville theorem under some integral conditions with respect to positive weak solutions.

Keywords

p-Laplacian / Liouville theorem / Positive weak solution

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