Towering phenomena for the Yamabe equation on symmetric manifolds

Let $(M,g)$ be a compact smooth connected Riemannian manifold (without boundary) of dimension $N\ge7$. Assume $M$ is symmetric with respect to a point $ξ_0$ with non-vanishing Weyl's tensor. We consider the linear perturbation of the Yamabe problem $(P_ε)\qquad-\mathcal L_g u+εu=u^{N+2\over N-2}\ \hbox{in}\ (M,g) .$ We prove that for any $k\in \mathbb N$, there exists $ε_k>0$ such that for all $ε\in (0, ε_k)$ the problem $(P_ε)$ has a symmetric solution $u_ε,$ which looks like the superposition of $k$ positive bubbles centered at the point $ξ_0$ as $ε\to 0$. In particular, $ξ_0$ is a {\em towering} blow-up point.

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