  We consider the problem of finding a metric in a given conformal class with
prescribed non-positive scalar curvature and non-positive boundary mean
curvature on an asymptotically Euclidean manifold with inner boundary. We
obtain a necessary and sufficient condition in terms of a conformal invariant
of the zero sets of the target curvatures for the existence of solutions to the
problem and use this result to establish the Yamabe classification of metrics
in those manifolds with respect to the solvability of the prescribed curvature
problem.
