Positive clusters for smooth perturbations of a critical elliptic equation in dimensions four and five

We construct clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension n=4,5. Such a construction is already available in the literature in dimensions n≥6 (see for instance [10], [14], [30], [32], [36]) and not possible in dimension 3 by [25]. This also provides new patterns for the Lin-Ni [26] problem on closed manifolds and completes results by Brézis and Li [8] about this problem.