Towers of bubbles for Yamabe-type equations and for the Brézis-Nirenberg problem in dimensions $n\ge 7$

Abstract

Let $(M,g)$ be a closed locally conformally flat Riemannian manifold of dimension $n\ge 7$ and of positive Yamabe type. If ${\xi }_0$ denotes a non-degenerate critical point of the mass function we prove the existence, for any $k\ge 1$ and $\epsilon >0$, of a positive blowing-up solution $u_{\epsilon }$ of $${△}_g{u}_{\epsilon }+(c_n{S}_g+\epsilon h)u_{\epsilon }=u_{\epsilon }^{2^{\ast }-1},$$ that blows up like the superposition of $k$ positive bubbles concentrating at different speeds at ${\xi }_0$. The method of proof combines a finite-dimensional reduction with the sharp pointwise analysis of solutions of a linear problem. As another application of this method of proof we construct sign-changing blowing-up solutions $u_{\epsilon }$ for the Brézis-Nirenberg problem $${△}_{\xi }{u}_{\epsilon }-\epsilon u_{\epsilon }=|u_{\epsilon }{|}^{\frac{4}{n-2}}{u}_{\epsilon }\text{ in }\Omega ,u_{\epsilon }=0\text{ on }\partial \Omega$$ on a smooth bounded open set $\Omega \subset {\mathbb{R}}^n$, $n\ge 7$, that look like the superposition of $k$ positive bubbles of alternating sign.