abstract: In this talk, we will look at the question of existence of blowing-up solutions for smooth perturbations of critical elliptic second-order equations on a closed manifold. I will discuss the critical role played by the Yamabe potential for this question and also by dimension 6 in the case of solutions with non-trivial weak limits. I will present new existence results on non-spherical manifolds. This is a joint work with Frédéric Robert.