Some results on the weighted Yamabe problem with or without boundary

Let $(M^n,g,e^{-\varphi }d{V}_g,e^{-\varphi }d{A}_g,m)$ be a compact smooth metric measure space with boundary with $n⩾3$. In this article, we consider several Yamabe-type problems on a compact smooth metric measure space with or without boundary: uniqueness problem on the weighted Yamabe problem with boundary, characterization of the weighted Yamabe solitons with boundary and the existence of positive minimizers in the weighted Escobar quotient.