Gradient estimates for ${\mathrm{\Delta }}_{b}u+a{u}^{p+1}=0$ on pseudo-Hermitian manifolds

In this paper, we derive the gradient estimates for the positive solutions of the equation ${\mathrm{\Delta }}_{b}u+a{u}^{p+1}=0$ on complete noncompact pseudo-Hermitian manifolds, where $a>0$ and $p\le 0$ 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.