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^-a=0 in M,where a 〉 0, c are two real constants. When c 〈 0 and M is a bounded smooth domain in R^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.
