Towers of Bubbles for Yamabe-Type Equations and for the Brézis–Nirenberg Problem in Dimensions $$n \ge 7$$

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