Liouville theorem for the nonlinear Poisson equation on manifolds

In this note, we study a Modica type gradient estimate for smooth solutions to general non-linear Poisson equation Δ u − f ( u ) = 0 , in M n , u : M n → R where ( M , g ) is a complete Riemannian manifold with bounded geometry and non-negative Ricci curvature and f is the derivative of the non-negative smooth function F ( u ) on R. Then we use this gradient estimate to conclude a Liouville theorem.