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

We describe the asymptotic behavior of positive solutions uε of the equation ´∆uà u " 3 u 5´ε in Ω Ă 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 uε 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 Brézis and Peletier (1989).Similar results are also obtained for solutions of the equation ´∆u `pa `εV qu " 3 u 5 in Ω.