  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.
