Towers of Bubbles for Yamabe-Type Equations and for the Brezis-Nirenberg Problem in Dimensions n ≥ 7

Let (M, g) be a closed locally conformally flat Riemannian manifold of dimension n >= 7 and of positive Yamabe type. If h is an element of C-1 (M) and xi(0) is a non-degenerate critical point of the mass function we prove the existence, for any k >= 1 of a positive blowing-up solution u(epsilon) of Delta(g)u(epsilon) + (c(n)S(g) + epsilon h)u(epsilon) = u(epsilon)(2)*(-1) that blows up, as epsilon -> 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(epsilon) for the Brezis-Nirenberg problem Delta(xi)u(epsilon) - epsilon u(epsilon) = vertical bar u(epsilon)vertical bar(4/n)(-)(2) u(epsilon) in Omega, u(epsilon) = 0 on partial derivative Omega on a smooth bounded open set Omega subset of R-n, n >= 7, that look like the superposition of k positive bubbles of alternating sign as epsilon -> 0.