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

TL;DR

This paper constructs positive solutions with clustering behavior for a perturbed critical elliptic equation on 4- and 5-dimensional closed manifolds, extending known results from higher dimensions and addressing open cases.

Contribution

It provides the first construction of clustering solutions in dimensions four and five, filling a gap in the existing literature for these lower dimensions.

Findings

Constructed clustering positive solutions in dimensions 4 and 5.

Extended the understanding of solutions for perturbed critical elliptic equations.

Provided new patterns for the Lin--Ni problem on closed manifolds.

Abstract

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.