BUBBLING ABOVE THE THRESHOLD OF THE SCALAR CURVATURE IN DIMENSIONS FOUR AND FIVE

On any closed manifold (M n , g) of dimension n ∈ {4, 5} we exhibit new blow-up configurations for perturbations of a purely critical stationary Schrödinger equation. We construct positive solutions which blow-up as the sum of two isolated bubbles, one of which concentrates at a point ξ where the potential k of the equation satisfies k(ξ) > n − 2 4(n − 1) Sg(ξ), where Sg is the scalar curvature of (M n , g). The latter condition requires the bubbles to blow-up at different speeds and forces us to work at an elevated precision. We take care of this by performing a construction which combines a priori asymptotic analysis methods with a Lyapounov-Schmidt reduction