  We describe the asymptotic behavior of positive solutions $u_\epsilon$ of the
equation $-\Delta u + au = 3\,u^{5-\epsilon}$ in $\Omega\subset\mathbb{R}^3$
with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to
be critical in the sense of Hebey and Vaugon and the functions $u_\epsilon$ are
assumed to be an optimizing sequence for the Sobolev inequality. Under a
natural nondegeneracy assumption we derive the exact rate of the blow-up and
the location of the concentration point, thereby proving a conjecture of
Br\'ezis and Peletier (1989). Similar results are also obtained for solutions
of the equation $-\Delta u + (a+\epsilon V) u = 3\,u^5$ in $\Omega$.
