Gradient estimates and Liouville theoremsfor a class of elliptic equation on Riemannian manifolds

In this paper we consider the gradient estimates on positive solutions to the following semi-linear elliptic equation defined on a complete Riemannian manifold $ (M, \, g) $:$ \Delta u + uh(lnu) = 0, $where $ h $ is a smooth function.We employ the Nash-Moser iteration technique to obtain some refined gradient estimates of the solutions to the above equation, if $ (M, \, g) $ satisfies $ Ric \geq -(n-1)\kappa $, where $ n $ is the dimension of $ M $ and $ \kappa $ is a nonnegative constant. By the obtained gradient estimates we also derive a Liouville type theorem for the above equation under some suitable geometric and analysis conditions. And as applications, we further derive a Cheng-Yau's type gradient estimate for the solution to the $ 2 $-dimensional Einstein-scalar field Lichnerowicz equation.