Abstract: Let（M,g） be a smooth compact Riemannian manifold of dimension n.Denote△f=△-▽f.▽ the weighted Laplacian operator,where f is a smooth real valued function on M.When N is finite and the N-Bakry-Emery Ricci tensor is bounded from below by a constant,we establish local gradient estimates for positive solutions of the following simple Lichnerowicz equation△fu+cu^-α=0 on a compact Riemannian manifold,where α is a positive constant and c is a smooth function. 

Key words: gradient estimate, Lichnerowicz equation, weighted Laplacian