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

We describe the asymptotic behavior of positive solutions $u_{𝜀}$ of the equation $-\mathrm{\Delta }u+au=3u^{5-𝜀}$ in $\mathrm{\Omega }\subset {ℝ}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon, and the functions $u_{𝜀}$ 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 $-\mathrm{\Delta }u+(a+𝜀V)u=3u^5$ in $\mathrm{\Omega }$.