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 satisfieswhere 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.