Nash-Moser iteration approach to the logarithmic gradient estimates and Liouville Properties of quasilinear elliptic equations on manifolds

2023/11/05 by Jie He, He, Jie, Jingchen Hu +3 · 1 citation
Computer Science · Mathematics · #35B45 #35J92 #58J05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2311.02568

openalex publication_date 2023/11/05 · openalex created_date 2023/11/08 · openalex updated_date 2026/07/28

Abstract

In this paper, we provide a new routine to employ the Nash-Moser iteration technique to analyze the local and global properties of positive solutions to the equation Δ_pv + a|∇ v|^qv^r =0 on a complete Riemannian manifold with Ricci curvature bounded from below, where p>1, q, r and a are some real constants. Assuming certain conditions on a, p, q and r, we can derive universal and succinct Cheng-Yau type logarithmic gradient estimates for such solutions. In particular, we give the obvious expressions of constants in the logarithmic gradient estimate for entire solutions to the above equation (see \thmreft10). The gradient estimates enable us to obtain some Liouville-type theorems, Harnack inequalities and some local estimates near singularities for positive solutions. Some of our results are new even in the case the domain is an Euclidean space and p=2.

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• Classification of positive solutions to a class of Laplace equations with a gradient term

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