Conformal scalar curvature rigidity on Riemannian manifolds

Let $(M, \bar g)$ be an $n$-dimensional complete Riemannian manifold. In this paper, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $\Omega$ with $\partial \Omega$, can one find a conformal metric $g$ whose scalar curvature $R[g]\ge R[\bar g]$ on $\Omega$ and the mean curvature $H[g] \ge H[ \bar g]$ on $\partial \Omega$ with $\bar g = g$ on $\partial \Omega$? We prove that $\bar g = g$ on some smooth domains in a general Riemannian manifold, which is an extension of the previous results given by Qing and Yuan, and Hang and Wang.