Energy asymptotics in the three-dimensional Brezis--Nirenberg problem

For a bounded open set $Omegasubsetmathbb R^3$ we consider the minimization problem $$ S(a+epsilon V) = inf_{0 otequiv uin H^1_0(Omega)} frac{int_Omega (| abla u|^2+ (a+epsilon V) |u|^2),dx}{(int_Omega u^6,dx)^{1/3}} $$ involving the critical Sobolev exponent. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on $a$ and $V$ we compute the asymptotics of $S(a+epsilon V)-S$ as $epsilonto 0+$, where $S$ is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to $a$ and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $S(a+epsilon V)<S$ for all sufficiently small $epsilon>0$.