The subject of my talk is a recent result on the asymptotic
behavior of positive solutions $u_\epsilon$ of the equation $-\Delta u +
au = 3\,u^{5-\epsilon}$ in $\Omega\subset\R^3$ 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
$\Omega$.

This is joint work with Rupert Frank (LMU Munich / Caltech) and Hynek
Kovarik (Brescia).