Positive least energy solutions for k-coupled Schrodinger system with critical exponent: the higher dimension and cooperative case

In this paper, we study the following k-coupled nonlinear Schrodinger system with Sobolev critical exponent: {-Delta u(i) + lambda(i)u(i) = mu(i)u(i)(2)*(-1) + Sigma(k)(j=1,j not equal i) beta(ij)u(i)(2)*(/2-1) u(j)(2)*(/2) in Omega, u(i) > 0 in Omega and u(i) = 0 on partial derivative Omega, i = 1, 2, ..., k. Here Omega C R-N is a smooth bounded domain, 2* = 2N/N-2 is the Sobolev critical exponent, -lambda(1)(Omega) < lambda(i) < 0, mu(i) > 0 and beta(ij) = beta(ji) not equal 0, where lambda(1)(Omega) is the first eigenvalue of -Delta with the Dirichlet boundary condition. We characterize the positive least energy solution of the k-coupled system for the purely cooperative case beta(ij) > 0, in higher dimension N >= 5. Since the k-coupled case is much more delicate, we shall introduce the idea of induction. We point out that the key idea is to give a more accurate upper bound of the least energy. It is interesting to see that the least energy of the k-coupled system decreases as k grows. Moreover, we establish the existence of positive least energy solution of the limit system in R-N, as well as classification results.