We consider the existence and multiplicity of riemannian metrics of prescribed mean curvature and zero boundary mean curvature on the three dimensional half sphere (S3 +, gc) endowed with its standard metric gc. Due to Kazdan-Warner type obstructions, conditions on the function to be realized as a scalar curvature have to be given. Moreover the existence of critical point at infinity for the associated Euler Lagrange functional, makes the existence results harder to be proved. However it turns out that such noncompact orbits of the gradient can be treated as usual critical point once a Morse Lemma at infinity is performed. In particular their topological contribution to the level sets of the functional can be computed. We prove that, under generic conditions on K, this topology at infinity is a lower bound for the number of metrics in the conformal class of gc having prescribed scalar curvature and zero boundary mean curvature.