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

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$, and $\Delta_g = -div_g\nabla$ be the Laplace-Beltrami operator. Let also $2^\star$ 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

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.