Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds

In this paper, we study the local gradient estimate for the positive solution to the following equation: Delta u + au log u + bu = 0 in M, where a < 0, b are real constants, M is a complete non-compact Riemannian manifold. Our result is optimal in the sense when (M, g) is a complete non-compact expanding gradient Ricci soliton. By definition, (M, g) is called an expanding gradient Ricci soliton if for some constant c < 0, it satisfies that Rc = cg + D(2)f, where Rc is the Ricci curvature, and D(2)f is the Hessian of the potential function f on M. We show that for a complete non-compact Riemannian manifold (M, g), the local gradient bound of the function f = log u, where u is a positive solution to the equation above, is well controlled by some constants and the lower bound of the Ricci curvature. (C) 2006 Elsevier Inc. All rights reserved.