Given (M, g) a compact Riemannian manifold of dimension n ≥ 3, we are interested in the existence of blowing-up sign-changing families (u ϵ)ϵ>0 ∈ C 2, θ(M), θ ∈ (0, 1), of solutions to where Δ g : = −div g (∇) and h ∈ C 0, θ(M) is a potential. Assuming the existence of a nondegenerate solution to the limiting equation (which is a generic assumption), we prove that such families exist in two main cases: in small dimension n ∈ {3, 4, 5, 6} for any potential h or in dimension 3 ≤ n ≤ 9 when . These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet [11] and Khuri et al. [19].