Solutions of an Elliptic System with a Nearly Critical Exponent

This problem has positive solutions for ǫ > 0 (with pqǫ > 1) and no non-trivial solution for ǫ ≤ 0. We study the asymptotic behaviour of least energy solutions as ǫ → 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. Résumé. Considéré le problème −∆uǫ = v p ǫ vǫ > 0 en Ω, −∆vǫ = u qǫ ǫ uǫ > 0 en Ω, uǫ = vǫ = 0 sur ∂Ω, où Ω est un domaine convexe et borné de R , N > 2, avec la frontière régulière ∂Ω. Ici p, qǫ > 0, et ǫ := N p+ 1 + N qǫ + 1 − (N − 2).