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

For dimensions $N geq 4$, we consider the Brezis-Nirenberg variational problem of finding [ S(epsilon V) := inf_{0 otequiv uin H^1_0(Omega)} frac{int_Omega | abla 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 $N = 3$.