Blow-up examples for second order elliptic PDEs of critical Sobolev growth

Let (M, g) be a smooth compact Riemannian manifold of dimension n greater than or equal to 3, and Deltag = - div(g) del be the Laplace-Beltrami operator. Let also 2* be the critical Sobolev exponent for the embedding of the Sobolev space H-1(2) (M) into Lebesgue's spaces, and h be a smooth function on M. Elliptic equations of critical Sobolev growth such as (E) Delta(g)u + hu = u(2*-1) have been the target of investigation for decades. A very nice H-1(2)-theory for the asymptotic behaviour of solutions of such equations has been available since the 1980' s. The C-0-theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of (E). It was used as a key point by Druet to prove compactness results for equations such as (E). An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of (E). We present such examples in this article.