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

TL;DR

This paper proves the existence of multi-bubble solutions for the Brezis-Nirenberg problem in four-dimensional domains, addressing a longstanding open problem and providing examples of domains with multiple blow-up solutions.

Contribution

It demonstrates the existence of multi-bubble solutions in 4D, filling a gap in the understanding of the Brezis-Nirenberg problem and extending previous asymptotic analyses.

Findings

Existence of multi-bubble solutions in 4D domains

Examples of domains with multiple blow-up points

Extension of asymptotic analysis results

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. Poincar\`e C Anal. Non Lin\`eaire, 2024.