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Clustering phenomena for linear perturbation of the Yamabe equation by Pistoia, Angela; Vaira, Giusi is a scholarly article available to read on EtoBox.

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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.

Author

Pistoia, Angela; Vaira, Giusi

Published

2015

Language

EN