Hardy–Sobolev equations with asymptotically vanishing singularity: Blow-up analysis for the minimal energy

Abstract

We study the asymptotic behavior of a sequence of positive solutions (uϵ)ϵ>0 as ϵ→0 to the family of equations Δuϵ+a(x)uϵ=uϵ2∗(sϵ)−1|x|sϵ in Ωuϵ=0 on ∂Ω.where (sϵ)ϵ>0 is a sequence of positive real numbers such that limϵ→0sϵ=0, 2∗(sϵ)≔2(n−sϵ)n−2 and Ω⊂Rn is a bounded smooth domain such that 0∈∂Ω. When the sequence (uϵ)ϵ>0 is uniformly bounded in L∞, then up to a subsequence it converges strongly to a minimizing solution of the stationary Schrödinger equation with critical growth. In case the sequence blows up, we obtain strong pointwise control on the blow-up sequence, and then using the Pohozaev identity localize the point of singularity, which in this case can at most be one, and derive precise blow-up rates. In particular when n=3 or a≡0 then blow-up can occur only at an interior point of Ω or the point 0∈∂Ω.