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 $$h \in C^1(M)$$ and $$\xi _0$$ is a non-degenerate critical point of the mass function we prove the existence, for any $$ k \ge 1$$ of a positive blowing-up solution $$u_\varepsilon $$ of $$\begin{aligned} \triangle _g u_\varepsilon +\big ( c_n S_g +\varepsilon h\big ) u_\varepsilon = u_\varepsilon ^{2^*-1} \end{aligned}$$ that blows up, as $$\varepsilon \rightarrow 0$$ , 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 Brézis–Nirenberg problem $$\begin{aligned} \triangle _{\xi } u_\varepsilon - \varepsilon u_\varepsilon = |u_\varepsilon |^{\frac{4}{n-2}} u_\varepsilon \text { in } \Omega , \quad u_\varepsilon = 0 \text { on } \partial \varOmega \end{aligned}$$ on a smooth bounded open set $$\varOmega \subset {\mathbb {R}}^n$$ , $$n \ge 7$$ , that look like the superposition of k positive bubbles of alternating sign as $$\varepsilon \rightarrow 0$$ .