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

Given a closed manifold (M-n, g), n >= 3, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation Delta(g)u + h(0)u = u(n+2/n-2), u > 0 in M is that h(0) is an element of C-1 (M) touches the Yamabe potential somewhere when n >= 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 >= 6, our arguments are local. For the low dimensions n is an element of {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).