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

Abstract

We prove some existence and uniqueness theorems for the solution of the Lichnerowicz equation:\(8\Delta _{g\phi } - R_\phi + M_\phi ^{ - 7} + Q_\phi ^{ - 3} + \tau _\phi ^5 = 0\), 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 assumptionR⩾0,M⩾0,Q⩾0, τ⩾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].