Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

Abstract

Let (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ0 with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem $$ (P_{\epsilon })\qquad -\mathcal {L}_{g} u+\epsilon u=u^{\frac {N+2}{N-2}} \text { in } (M,g) . $$ We prove that for any k ∈ ℕ, there exists ek > 0 such that for all e ∈ (0, ek) the problem (P𝜖) has a symmetric solution ue, which looks like the superposition of k positive bubbles centered at the point ξ0 as e → 0. In particular, ξ0 is a towering blow-up point.