Clustering phenomena for linear perturbation of the Yamabe equation, Partial differential equations arising from physics and geometry

Let (M,g) be a non-locally conformally flat compact Riemannian manifold with dimension N≥7. We are interested in finding positive solutions to the linear perturbation of the Yamabe problem −Lgu+εu=uN+2N−2 in (M,g) where the first eigenvalue of the conformal laplacian −Lg is positive and ε is a small positive parameter. We prove that for any point ξ0∈M which is non-degenerate and non-vanishing minimum point of the Weyl's tensor and for any integer k there exists a family of solutions developing k peaks collapsing at ξ0 as ε goes to zero. In particular, ξ0 is a non-isolated blow-up point