Energy asymptotics in the Brezis-Nirenberg problem. The higher-dimensional case

For dimensions $N \geq 4$, we consider the Br\'ezis-Nirenberg variational problem of finding $$S(\epsilon V) := \inf_{0\not\equiv u\in H^1_0(\Omega)} \frac{\int_\Omega |\nabla u|^2 \, dx +\epsilon \int_\Omega V\, |u|^2 \, dx}{\left(\int_\Omega |u|^q \, dx \right)^{2/q}},$$ where $q=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\Omega \subset \mathbb{R}^N$ is a bounded open set. We compute the asymptotics of $S(0) - S(\epsilon V)$ to leading order as $\epsilon \to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case $N = 3$.