Towering phenomena for the Yamabe equation on symmetric manifolds

2016/03/04 by Morabito, Filippo, Pistoia, Angela, Vaira, Giusi · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.01538

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_ε) -\mathcal L_g u+εu=u^{N+2\over N-2} \hboxin (M,g) . We prove that for any k∈ \mathbb N, there exists ε_k>0 such that for all ε∈ (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 ξ₀ as ε→ 0. In particular, ξ₀ is a \em towering blow-up point.

Cited by

• Multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

Related

• Clustering phenomena for linear perturbation of the Yamabe equation
• Blow-up solutions for linear perturbations of the Yamabe equation
• Blow-up phenomena for linearly perturbed Yamabe problem on manifolds\n with umbilic boundary
• Compactness of solutions to the Yamabe problem. III
• On Yamabe type problems on Riemannian manifolds with boundary