Clustering phenomena for linear perturbation of the Yamabe equation

Let $(M,g)$ be a non-locally conformally flat compact Riemannian manifold with dimension $N\ge7.$ We are interested in finding positive solutions to the linear perturbation of the Yamabe problem $-\mathcal L_g u+εu=u^{N+2\over N-2}\ \hbox{in}\ (M,g)$ where the first eigenvalue of the conformal laplacian $-\mathcal L_g$ is positive and $ε$ is a small positive parameter. We prove that for any point $ξ_0\in M$ 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 $ξ_0$ as $ε$ goes to zero. In particular, $ξ_0$ is a non-isolated blow-up point.

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