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 equationΔu−f(u)=0,inMn,u:Mn→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.