Solutions of an elliptic system with a nearly critical exponent

الأصل · EN

Consider the problem eqnarray* -Δu &=& vᵖ v>0 in Ω, -Δv &=& u u>0 in Ω, u&=&v=0 on ∂ Ω, eqnarray* where Ω is a bounded convex domain in ⁿ, N>2, with smooth boundary ∂ Ω. Here p,q>0, and equation* ε:=N/p+1+N/q+1-(N-2). equation* This problem has positive solutions for >0 (with pq>1) and no non-trivial solution for ≤ 0. We study the asymptotic behaviour of least energy solutions as → 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.