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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?f u+cu?α=0 on a compact Riemannian manifold, whereαis a positive constant and c is a smooth function.