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

Abstract

For dimensions $N \geq 4$, we consider the Brézis-Nirenberg variational problem of finding \[ S(εV) := \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω|\nabla u|^2 \, dx +ε\int_ΩV\, |u|^2 \, dx}{\left(\int_Ω|u|^q \, dx \right)^{2/q}}, \] where $q=\frac{2N}{N-2}$ is the critical Sobolev exponent and $Ω\subset \mathbb{R}^N$ is a bounded open set. We compute the asymptotics of $S(0) - S(εV)$ to leading order as $ε\to 0+$. 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 $N = 3$.