Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

TL;DR

This paper studies the detailed behavior of solutions that blow up in the higher-dimensional Brezis-Nirenberg problem, identifying concentration points and their properties for dimensions N ≥ 4.

Contribution

It characterizes the concentration points as critical points of a function related to the Green's function and provides the leading order of the concentration speed.

Findings

Concentration points are isolated.

Concentration points are critical points of a Green's function derived function.

Provides the leading order expression of concentration speed.

Abstract

For a bounded set $\Omega \subset \mathbb R^N$ and a perturbation $V \in C^1(\overline{\Omega})$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -\Delta u_\epsilon + \epsilon V = N(N-2) u_\epsilon^\frac{N+2}{N-2} \] for dimensions $N \geq 4$, which are non-critical in the sense of the Brezis--Nirenberg problem. For the general case of multiple concentration points, we prove that concentration points are isolated and characterize the vector of these points as a critical point of a suitable function derived from the Green's function of $-\Delta$ on $\Omega$. Moreover, we give the leading order expression of the concentration speed. This paper, with a recent one by the authors (arXiv:2208.12337) in dimension $N = 3$, gives a complete picture of blow-up phenomena in the Brezis-Nirenberg framework.