Blowing-up solutions for the Yamabe equation

Let (M, g) be a smooth, compact Riemannian manifold of dimension N >= 3. We consider the almost critical problem (P-epsilon) -Delta(g)u + N - 2/4(N - 1) Scal(g) u = u((N + 2)/(N - 2)+epsilon) in M, u > 0 in M, where Delta(g) denotes the Laplace-Beltrami operator, Scal(g) is the scalar curvature of g and epsilon is an element of R is a small parameter. It is known that problem (P-epsilon) does not have any blowing-up solutions when epsilon NE arrow 0, at least for N <= 24 or in the locally conformally flat case, and this is not true anymore when epsilon SE arrow 0. Indeed, we prove that, if N >= 7 and the manifold is not locally conformally flat, then problem (P-epsilon) does have a family of solutions which blow-up at a maximum point of the function xi -> vertical bar Weyl(g)(xi)vertical bar(g) as epsilon SE arrow 0. Here Weyl(g) denotes the Weyl curvature tensor of g.