Local and Global Log-Gradient estimates of solutions to Δ_pv+bv^q+cv^r =0 on manifolds and applications

2024/04/22 by Jie He, He, Jie, Yuanqing Ma +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2405.00703

openalex publication_date 2024/04/22 · openalex created_date 2024/05/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation Δ_pv+bv^q+cv^r =0 on complete Riemannian manifolds with Ricci curvature bounded from below, where b, c∈\mathbb R, p>1, and q≤ r are some real constants. Assuming certain conditions on b, c, p, q and r, we derive succinct Cheng-Yau type gradient estimates for positive solutions, which is of sharp form. These gradient estimates allow us to obtain some Liouville-type theorems and Harnack inequalities. Our Liouville-type results are novel even in Euclidean spaces. Based on the local gradient estimates and a trick of Sung and Wang, we also obtain the global gradient estimates for such solutions. As applications we show the uniqueness of positive solutions to some generalized Allen-Cahn equation and Fisher-KPP equation.

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