  Given a closed manifold $(M^n,g)$, $n\geq 3$, Olivier Druet proved that a
necessary condition for the existence of energy-bounded blowing-up solutions to
perturbations of the equation $$\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0\hbox{
in }M$$ is that $h_0\in C^1(M)$ touches the Scalar curvature somewhere when
$n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that
Druet's condition is also sufficient provided we add its natural differentiable
version. For $n\geq 6$, our arguments are local. For the low dimensions
$n\in\{4,5\}$, our proof requires the introduction of a suitable mass that is
defined only where Druet's condition holds. This mass carries global
information both on $h_0$ and $(M,g)$.
