Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
7222671 | Nonlinear Analysis: Theory, Methods & Applications | 2018 | 28 Pages |
Abstract
We study the asymptotic behavior of a sequence of positive solutions (uϵ)ϵ>0 as ϵâ0 to the family of equations Îuϵ+a(x)uϵ=uϵ2â(sϵ)â1|x|sϵ in Ωuϵ=0 on âΩ.where (sϵ)ϵ>0 is a sequence of positive real numbers such that limϵâ0sϵ=0, 2â(sϵ)â2(nâsϵ)nâ2 and ΩâRn is a bounded smooth domain such that 0ââΩ. When the sequence (uϵ)ϵ>0 is uniformly bounded in Lâ, 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 aâ¡0 then blow-up can occur only at an interior point of Ω or the point 0ââΩ.
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Authors
Saikat Mazumdar,