  In this paper, we consider the problem of existence and multiplicity of
conformal metrics on a riemannian compact $4-$dimensional manifold $(M^4,g_0)$
with positive scalar curvature. We prove new exitence criterium which provides
existence results for a dense subset of positive functions and generalizes
Bahri-Coron and Chang-Gursky-Yang Euler-Hopf type criterium. Our argument gives
estimates on the Morse index of the solutions and has the advantage to extend
known existence results. Moreover it provides, for generic $K$ {\it Morse
Inequalities at Infinity}, which give a lower bound on the number of metrics
with prescribed scalar curvature in terms of the topological contribution of
its "critical points at Infinity" to the difference of topology between the
level sets of the associated Euler-Lagrange functional.
