  We construct solutions of the constraint equation with non constant mean
curvature on an asymptotically hyperbolic manifold by the conformal method. Our
approach consists in decreasing a certain exponent appearing in the equations,
constructing solutions of these sub-critical equations and then in letting the
exponent tend to its true value. We prove that the solutions of the
sub-critical equations remain bounded which yields solutions of the constraint
equation unless a certain limit equation admits a non-trivial solution.
Finally, we give conditions which ensure that the limit equation admits no
non-trivial solution.
