Solutions of an elliptic system with a nearly critical exponent

Abstract

Consider the problem−Δuɛ=vɛp,vɛ>0inΩ,−Δvɛ=uɛqɛ,uɛ>0inΩ,uɛ=vɛ=0on∂Ω, where Ω is a bounded convex domain in RN, N>2, with smooth boundary ∂Ω. Here p,qɛ>0, andɛ:=Np+1+Nqɛ+1−(N−2). This problem has positive solutions for ɛ>0 (with pqɛ>1) and no non-trivial solution for ɛ⩽0. We study the asymptotic behavior of least energy solutions as ɛ→0+. These solutions are shown to blow-up at exactly one point, and the location of this point is characterized. In addition, the shape and exact rates for blowing up are given.