A Liouville theorem for the CR Yamabe type equation on Sasakian manifolds

In this paper, we study the CR Yamabe type equation \begin{align} Δ_b u+F(u)=0 \nonumber \end{align} on complete noncompact $(2n+1)$-dimensional Sasakian manifolds with nonnegative curvature. Under some assumptions, we prove a rigidity result, that is, the manifold is CR isometric to Heisenberg group ${\mathbb{H}}^n$. The proofs are based on the Jerison-Lee’s differential identity combining with integral estimates.