Blowing-up Solutions for 2nd-Order Critical Elliptic Equations: The Impact of the Scalar Curvature

Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$ \begin{align*} &\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0 \ \textrm{in }M\end{align*}$$is that $h_0\in C^1(M)$ touches the Yamabe potential 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 to introduce a suitable mass that is defined only where Druet’s condition holds. This mass carries global information both on $h_0$ and $(M,g)$.