On the universal local and global properties of positive solutions to Δ_pv+b|∇ v|^q+cv^r=0 on complete Riemannian manifolds

2023/11/22 by He, Jie, Wang, Youde
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.13179

Abstract

In this paper we study the positive solutions to a nonlinear elliptic equation Δ_pv+b|∇ v|^q+cv^r =0 defined on a complete Riemannian manifold (M,g) with Ricci curvature bounded from below, where p>1, q, r, b and c are some real constants. If p>1 is given and bc≥ 0, we provide a new routine to give some regions of (q, r) such that the Cheng-Yau's logarithmic gradient estimates hold true exactly on such given regions. In particular, we derive the upper bounds of the constants c(n, p, q, r) in the Cheng-Yau's gradient estimates for the entire solutions to the above equation. As applications, we reveal some universal local and global properties of positive solutions to the equation. On the other hand, we extend some results due to \citeMR1879326 to the case the domain of the equation is a complete manifold and obtain wider ranges of (q,r) for Liouville properties.

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