Let (M, g) be a complete non-compact Riemannian manifold without boundary. In thispaper, we give the gradient estimates on positive solutions to the following elliptic equation withsingular nonlinearity:Δu(x)+cu^(-α)(x)=0 in M,where α> 0, c are two real constants. When c < 0 and M is a bounded smooth domain in R^n, theabove 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 asan supplement of that of J. Li [J. Funct. Anal., 100, 233-256 (1991)], where the nonlinearity is thepositive power of u.
