  We consider two cases of the asymptotically flat scalar-flat Yamabe problem
on a non-compact manifold with boundary, in dimension $n\geq3$. First,
following arguments of Cantor and Brill in the compact case, we show that given
an asymptotically flat metric $g$, there is a conformally equivalent
asymptotically flat scalar-flat metric that agrees with $g$ on the boundary. We
then replace the metric boundary condition with a condition on the mean
curvature: Given a function $f$ on the boundary that is not too large, we show
that there is an asymptotically flat scalar-flat metric, conformally equivalent
to $g$ whose boundary mean curvature is given by $f$. The latter case involves
solving an elliptic PDE with critical exponent using the method of sub- and
supersolutions. Both results require the usual assumption that the Sobolev
quotient is positive.
