Gradient estimates for the equation .u + cu .. = 0 on Riemannian manifolds

Let (M, g) be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity: .u(x)+cu..(x)=0inM , where . > 0, c are two real constants. When c < 0 and M is a bounded smooth domain in .n, the above equation is known as the thin film equation, which describes a steady state of the thin film (see Guo-Wei [Manuscripta Math., 120, 193.209 (2006)]). The results in this paper can be viewed as an supplement of that of J. Li [J. Funct. Anal., 100, 233.256 (1991)], where the nonlinearity is the positive power of u.