On elliptic systems with Sobolev critical growth

this paper, we study the following Dirichlet problem with Sobolev critical exponent {-Delta u=vertical bar u vertical bar(2*-2)u+alpha/2*vertical bar u vertical bar(alpha-2)vertical bar v vertical bar(beta)u, x is an element of Omega, -Delta v=vertical bar v vertical bar(2*-2)v+beta/2*vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2)v, x is an element of Omega, where alpha, beta > 1, alpha + beta = 2* := 2N/N-2 (N >= 3) and Omega = R-N or Omega is a smooth bounded domain in R-N. When Omega = R-N, we obtain a uniqueness result on the least energy solutions and show that a manifold of the synchronized type of positive solutions is non-degenerate for the above system for some ranges of the parameters alpha, beta, N. Our analysis also yields non uniqueness of positive vector solutions for other parameters. Moreover, we establish a global compactness result and we extend a classical result of Coron on the existence of positive solutions of scalar equations with critical exponent on domains with nontrivial topology to the above elliptic system.