Bubbling above the threshold of the scalar curvature in dimensions four and five

Abstract

On any closed manifold $$(M^n,g)$$ of dimension $$n\in \{4,5\}$$ we exhibit new blow-up configurations for perturbations of a purely critical stationary Schrodinger equation. We construct positive solutions which blow-up as the sum of two isolated bubbles, one of which concentrates at a point $$\xi $$ where the potential k of the equation satisfies $$\begin{aligned} k(\xi ) > \frac{n-2}{4(n-1)} S_g(\xi ), \end{aligned}$$ where $$S_g$$ 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.