  Based on the relations between scattering operators of asymptotically
hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate
elliptic boundary value problems, we formulate fractional Yamabe problems that
include the boundary Yamabe problem studied by Escobar. We observe an
interesting Hopf type maximum principle together with interplays between
analysis of weighted trace Sobolev inequalities and conformal structure of the
underlying manifolds, which extend the phenomena displayed in the classic
Yamabe problem and boundary Yamabe problem.
