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:Δu+aulogu+bu=0inM, where a<0,ba<0,b are real constants, M   is a complete non-compact Riemannian manifold. Our result is optimal in the sense when (M,g)(M,g) is a complete non-compact expanding gradient Ricci soliton. By definition, (M,g)(M,g) is called an expanding gradient Ricci soliton if for some constant c<0c<0, it satisfies thatRc=cg+D2f,Rc=cg+D2f, where Rc   is the Ricci curvature, and D2fD2f is the Hessian of the potential function f on M  . We show that for a complete non-compact Riemannian manifold (M,g)(M,g), the local gradient bound of the function f=loguf=logu, where u is a positive solution to the equation above, is well controlled by some constants and the lower bound of the Ricci curvature.