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
ξ 0 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 ∈ ?, there exists
ε k > 0 such that for all
ε ∈ (0,
ε k ) the problem (
P ?? ) has a symmetric solution
u ε , which looks like the superposition of
k positive bubbles centered at the point
ξ 0 as
ε → 0. In particular,
ξ 0 is a
towering blow-up point.