  For a bounded open set $\Omega\subset\mathbb R^3$ we consider the
minimization problem $$ S(a+\epsilon V) = \inf_{0\not\equiv u\in H^1_0(\Omega)}
\frac{\int_\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int_\Omega
u^6\,dx)^{1/3}} $$ 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
$\epsilon\to 0+$, 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 $\epsilon>0$.
