Global solutions of the Lichnerowicz equation in General Relativity on an asymptotically Euclidean complete manifold

We prove some existence and uniqueness theorems for the solution of the Lichnerowicz equation:, on an asymptotically Euclidian manifold. This equation governs the conformai factor of a metric solution of the constraints in General Relativity. In the first part we prove existence and uniqueness under the simple assumptionR0,M0,Q0, 0, which insures the monotony of the non-differentiated part. In the second part we obtain an existence theorem under more general hypothesis on the coefficients, by use of the Leray-Schauder degree theory. The results of this paper have been announced in [4].