Einstein constraints on asymptotically Euclidean manifolds

We consider the Einstein constraints on asymptotically Euclidean manifolds M of dimension $n>~3$ with sources of both scaled and unscaled types. We extend to asymptotically Euclidean manifolds the constructive method of proof of existence. We also treat discontinuous scaled sources. In the last section we obtain new results in the case of non-constant mean curvature.