Blow-up of solutions of critical elliptic equations in three dimensions

We describe the asymptotic behavior of positive solutions u 6 of the equation - A u + au = 3u 5 - 6 in Q c 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 6 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 Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation - A u + ( a + E V ) u = 3u 5 in Q .