Blowing-up solutions for second-order critical elliptic equations: the impact of the scalar curvature

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 $(M,g)$