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

Yamabe Classification and Prescribed Scalar Curvature in the Asymptotically Euclidean Setting