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

TL;DR

This paper constructs complex solutions with multiple bubbles for Yamabe-type and Brézis-Nirenberg problems in high dimensions, revealing intricate blow-up behaviors and solution structures.

Contribution

It introduces a novel method combining finite-dimensional reduction and sharp analysis to produce multi-bubble solutions in high-dimensional geometric PDEs.

Findings

Existence of multi-bubble blowing-up solutions for Yamabe equations.

Construction of sign-changing solutions for the Brézis-Nirenberg problem.

Solutions concentrate at a critical point of the mass function.

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 $\varepsilon >0$, of a positive blowing-up solution $u_{\varepsilon}$ of $$\triangle_g u_{\varepsilon} +\big( c_n S_g +\varepsilon h\big) u_{\varepsilon} = u_{\varepsilon}^{2^*-1},$$ 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 $u_{\varepsilon}$ for the Br\'ezis-Nirenberg problem $$ \triangle_{\xi} u_{\varepsilon} - \varepsilon u_{\varepsilon} = |u_{\varepsilon}|^{\frac{4}{n-2}} u_{\varepsilon} \textrm{ in } \Omega, \quad u_{\varepsilon} = 0 \textrm{ on } \partial \Omega $$ 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.