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

Abstract

For a bounded set $\Omega \subset {\mathbb{R}}^N$ and a perturbation $V\in C^1(\bar{\Omega })$, we analyze the concentration behavior of a blow-up sequence of positive solutions to $-\Delta u_{\epsilon }+\epsilon V{u}_{\epsilon }=N(N-2)u_{\epsilon }^{\frac{(N+2)}{(N-2)}}$ for dimensions $N\ge 4$, 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 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, 2022] in dimension $N=3$, gives a complete picture of blow-up phenomena in the Brezis–Nirenberg framework.