Clustering phenomena for linear perturbation of the Yamabe equation

Abstract

Let $(M,g)$ be a non-locally conformally flat compact Riemannian manifold with dimension $N\ge 7.$ We are interested in finding positive solutions to the linear perturbation of the Yamabe problem $$-{\mathcal{L}}_{g}u+\epsilon u=u^{\frac{N+2}{N-2}}in(M,g)$$ where the first eigenvalue of the conformal laplacian $-{\mathcal{L}}_g$ is positive and $\epsilon$ is a small positive parameter. We prove that for any point ${\xi }_0\in M$ 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 ${\xi }_0$ as $\epsilon$ goes to zero. In particular, ${\xi }_0$ is a non-isolated blow-up point.