Feasibility of Nash-Moser iteration for Cheng-Yau-type gradient estimates of nonlinear equations on complete Riemannian manifolds

In this manuscript, we employ the Nash-Moser iteration technique to determine a condition under which the positive solution u of the generalized nonlinear Poisson equation div (φ(|∇ u|²)∇ u) + ψ(u²)u = 0, on a complete Riemannian manifold with Ricci curvature bounded from below can be shown to satisfy a Cheng-Yau-type gradient estimate. We define a class of φ-Laplacian operators by Δφ(u):=div (φ(|∇ u|²)∇ u), where φ is a C² function under some certain growth conditions. This can be regarded as a natural generalization of the p-Laplacian, the $(p,q)$-Laplacian and the exponential Laplacian, as well as having a close connection to the prescribed mean curvature problem. We illustrate the feasibility of applying the Nash-Moser iteration for such Poisson equation to get the Cheng-Yau-type gradient estimates in different cases with various φ and ψ. Utilizing these estimates, we proves the related Harnack inequalities and a series of Liouville theorems. Our results can cover a wide range of quasilinear Laplace operator (e.g. p-Laplacian for φ(t)=tp/2-1), and Lichnerowicz-type nonlinear equations (i.e. ψ(t) = Atᵖ + Btq + Ctlog t + D).