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

Abstract

We describe the asymptotic behavior of positive solutions uϵ of the equation −Δu + au = 3u^(5−ϵ) in Ω ⊂ ℝ³ 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 Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation −Δu +(a+ϵV)u = 3u⁵ in Ω.