Multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

Abstract

The paper addresses the existence of multi-bubble solutions for the well-known Brezis-Nirenberg problem. Although there is extensive literature on the subject, the existence of solutions that blow up at multiple points in a 4D bounded domain remains an open problem. The goal of the present paper is to resolve this longstanding issue. In particular, we exhibit examples of domains where a large number of multi-bubble solutions exist. Our result can also be seen as the counterpart of the asymptotic analysis carried out by Konig and Laurin in Ann. Inst. H. Poincarè C Anal. Non Linèaire, 2024.