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 $n\ge 6$ (see for instance [8,12,27,29,33]) and not possible in dimension $3$ by [25]. This also provides new patterns for the Lin--Ni [21] problem on closed manifolds and completes results by Br\'ezis and Li [6] about this problem.