  We prove a necessary and sufficient condition for an asymptotically Euclidean
manifold to be conformally related to one with specified nonpositive scalar
curvature: the zero set of the desired scalar curvature must have a positive
Yamabe invariant, as defined in the article. We show additionally how the sign
of the Yamabe invariant of a measurable set can be computed from the sign of
certain generalized "weighted" eigenvalues of the conformal Laplacian. Using
the prescribed scalar curvature result we give a characterization of the Yamabe
classes of asymptotically Euclidean manifolds. We also show that the Yamabe
class of an asymptotically Euclidean manifold is the same as the Yamabe class
of its conformal compactification.
