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

Abstract

For a bounded set $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ and a perturbation $V\in C^1(\overline{\hat{I}}\text{c◯})$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -Δu_ε+ εV = N(N-2) u_ε^\frac{N+2}{N-2} \] 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's function of $-\hat{I}”$ on $\hat{I}\text{c◯}$. 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.

v3: We correct an inaccuracy related to Prop.s 2.5 and 2.6: To guarantee that the error functions are in $L^{\infty }$, one needs to subtract all partial second derivatives and not only a radial term; see (2.2) and (2.3). The conclusions of the paper are unchanged. The described inaccuracy also persists in the published version; we have submitted an erratum to the journal