In this talk I will report recent progress on the Yamabe equation defined either on a punctured disk of a smooth manifold or outside a compact subset of $\mathbb R^n$ with an asymptotically flat metric.  What we are interested in is the behavior of solutions near the singularity. It is well known that the study of the Yamabe equation is sensitive to the dimension of the manifold and is closely related to the Positive Mass Theorem. In my recent joint works with Jingang Xiong (Beijing Normal University) and Zhengchao Han (Rutgers) we proved dimension-sensitive results and our work showed connection to other problems.