Abstract :
In this paper, we study the local gradient estimate for the positive solution to the following equation:
u + aulog u+bu =0 inM,
wherea <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 +D2f,
where Rc is the Ricci curvature, and D2f 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.
© 2006 Elsevier Inc. All rights reserved.
