Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrodinger systems

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrodinger system with critical exponent: {-Delta u + lambda(1)u = vertical bar u vertical bar(2*-2)u + v alpha vertical bar u vertical bar(alpha-2)vertical bar v vertical bar(beta)u, in R-N, -Delta V + lambda(2)v = vertical bar v vertical bar(2*-2)v + v beta vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2) v, in R-N, integral(RN) u(2) =a(2), integral(RN) v(2) = b(2), where N = 3, 4, alpha, beta > 1, 2 < alpha + beta < 2* = 2N/N-2. We prove that a normalized ground state does not exist for. < 0. When. > 0 and alpha+ beta <= 2 + 4N, we show that the system has a normalized ground state solution for 0 < v < v(0), the constant v(0) will be explicitly given. In the case alpha + beta > 2 + 4/N we prove the existence of a threshold v(1) >= 0 such that a normalized ground state solution exists for v > v(1), and does not exist for v < v(1). We also give conditions for v(1) = 0. Finally we obtain the asymptotic behavior of the minimizers as v -> 0(+) or v ->+ infinity.