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

TL;DR

This work develops Cheng-Yau–type gradient estimates for positive solutions of a generalized nonlinear Poisson equation on complete Riemannian manifolds with a Ricci lower bound, using Nash-Moser iteration. It introduces the φ-Laplacian $Δ_φ u := div(φ(|grad u|^2) grad u)$ to unify a broad class of nonlinear operators (including the p-Laplacian, (p,q)-Laplacian and exponential Laplacian) and studies $Δ_φ u + ψ(u^2)u = 0$, obtaining a gradient bound $|∇u|/u ≤ C(1+√K R)/R$ on balls $B(o,R)$ under structural conditions on $φ$ and $ψ$ expressed via degree functions $δ_φ$ and $δ_ψ$. The analysis yields Harnack inequalities and Liouville-type theorems, with constants depending only on dimension and equation-specific parameters. Applications cover the $p$-Laplacian, weighted $(p_1,...,p_r)$-Laplacians and several nonlinearities (e.g., Lane-Emden, Allen-Cahn, and logarithmic forms), establishing dimension- and structure-dependent Liouville thresholds and extending Cheng-Yau type controls to a wide nonlinear elliptic framework on 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 $$\operatorname{div} (\varphi(|\nabla u|^2)\nabla u) + ψ(u^2)u = 0,$$ 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 $\varphi$-Laplacian operators by $Δ_{\varphi}(u):=\operatorname{div} (\varphi(|\nabla u|^2)\nabla u)$, where $\varphi$ 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 $\varphi$ 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 $\varphi(t)=t^{p/2-1}$), and Lichnerowicz-type nonlinear equations (i.e. $ψ(t) = At^{p} + Bt^{q} + Ct\log t + D$).

Figures (3)

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• Figure 3: The admissible areas for Liouville theorem compared with wang2024gradient

Theorems & Definitions (31)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.2: Harnack's inequality
• Theorem 1.3: Liouville theorem
• Corollary 1.1
• Corollary 1.2
• Remark 1.3
• Definition 2.1
• Lemma 2.1