Gradient estimates for $\Delta_b u + au^{p+1} = 0$ on pseudo-Hermitian manifolds

TL;DR

This paper establishes gradient estimates for positive solutions of a nonlinear PDE on pseudo-Hermitian manifolds and derives Liouville-type theorems under certain curvature conditions.

Contribution

It provides new gradient estimates for a class of nonlinear equations on pseudo-Hermitian manifolds, extending previous results to noncompact and Sasakian-type cases.

Findings

Gradient estimates for solutions of the PDE on pseudo-Hermitian manifolds

Liouville-type theorems for Sasakian-type manifolds with nonnegative Ricci curvature

Applicability to equations with different signs of parameters a and p

Abstract

In this paper, we derive the gradient estimates for the positive solutions of the equation $\Delta_b u + au^{p+1} = 0$ on complete noncompact pseudo-Hermitian manifolds, where $a > 0$ and $p \leq 0$ or $a < 0$ and $p > 0$ are two constants. As an application, we will obtain a Liouville-type theorem when the manifolds are Sasakian-type with nonnegative pseudo-Hermitian Ricci curvature.