Asymptotics for a variational problem with critical growth and slightly positive Dirichlet data

Let ΩΩ be a bounded smooth domain in RN, N≥3N≥3. We consider the variational problem inf∫Ω|∇u|2inf∫Ω|∇u|2 for the admissible class Aγ,ε={u∈H1(Ω)|u−ε∈H01(Ω),γ=∫Ω|u|p+1>meas(Ω)εp+1} with p=(N+2)/(N−2)p=(N+2)/(N−2), ε>0ε>0. Caffarelli and Spruck [L.A. Caffarelli, J. Spruck, Variational problems with critical Sobolev growth and positive Dirichlet data, Indiana Univ. Math. J. 39 (1990) 1–18] proved the existence of the solution uγ,εuγ,ε satisfying {−Δuγ,ε=λγ,εuγ,εpin Ωuγ,ε=εon ∂Ω for λγ,ε>0λγ,ε>0. We prove that the solution concentrates at exactly one interior point as εε goes to zero. Furthermore we study the exact rate and location of the blowing up.