Clustering phenomena for linear perturbation of the Yamabe equation

TL;DR

This paper constructs solutions with multiple peaks for a linear perturbation of the Yamabe problem on certain manifolds, revealing complex blow-up phenomena as the perturbation parameter approaches zero.

Contribution

It demonstrates the existence of multi-peak solutions collapsing at specific points on non-locally conformally flat manifolds, extending understanding of blow-up behavior in perturbed Yamabe problems.

Findings

Existence of solutions with k peaks collapsing at a point as epsilon approaches zero.

Identification of non-isolated blow-up points related to Weyl tensor properties.

Construction of solutions on manifolds with positive conformal Laplacian eigenvalues.

Abstract

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 $$-\mathcal L_g u+\epsilon u=u^{N+2\over N-2}\ \hbox{in}\ (M,g) $$ 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.