Let (M, g) be a smooth compact n-dimension Riemannian manifold. We are concerned with the existence of sign-changing blow-up solutions to the following elliptic problem Delta(g)u + hu = vertical bar u vertical bar(4/n-2-epsilon) u, in M, where Delta(g) = -div(g)(del) is the Laplace-Beltrami operator on M, h is a C-1 function on M, epsilon is a small real parameter such that epsilon goes to 0.