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

Abstract

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 $$\mathrm{div}(\phi (|\nabla u{|}^2)\nabla u)+\psi (u^2)u=0,$$ 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 $\phi$-Laplacian operators by ${\Delta }_{\phi }(u):=\mathrm{div}(\phi (|\nabla u{|}^2)\nabla u)$, where $\phi$ is a $C^2$ 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 $\phi$ and $\psi$. 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 $\phi (t)=t^{p/2-1}$), and Lichnerowicz-type nonlinear equations (i.e. $\psi (t)=A{t}^p+B{t}^q+Ct\log t+D$).