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In this talk, we will consider the question of existence of positive blowing-up solutions to a class of conformally invariant elliptic equations of second order on a closed Riemannian manifold. A result of Olivier Druet provides necessary conditions for the existence of blowing-up solutions whose energy is a priori bounded. Essentially, these conditions say that for such solutions to exist, the potential function in the limit equation must coincide, up to a constant factor, at least at one point, with the scalar curvature of the manifold and moreover, in low dimensions, the weak limit of the solutions must be identically zero. I will present new existence results showing the optimality of these conditions. I will discuss in particular the case of low dimensions and the important role of the mass of the Green's function in this case. This is a joint work with Fr\'ed\'eric Robert (Universit\'e de Lorraine).