Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities

In this paper, we investigate the following fractional Sobolev critical Nonlinear Schrodinger coupled systems: {(-Delta)(s) u = mu(1)u + vertical bar u vertical bar(2s)*(-2)u + eta(1)vertical bar u vertical bar(p-2) u + gamma alpha vertical bar u vertical bar(alpha-2)u vertical bar v vertical bar(beta) in R-N, (-Delta)(s) v = mu(2)v + vertical bar v vertical bar(2s)*(-2)v + eta(2)vertical bar v vertical bar(q-2)v + gamma beta vertical bar v vertical bar(alpha)vertical bar v vertical bar(beta-)(2)v in R-N, parallel to u parallel to(2)(L2) = m(1)(2) and parallel to v parallel to(2)(L2) = m(2)(2), where (-Delta)(s) is the fractional Laplacian, N > 2s, s is an element of (0, 1), mu(1), mu(2) is an element of R are unknown constants, which will appear as Lagrange multipliers, 2(s)* is the fractional Sobolev critical index, eta(1), eta(2), gamma, m(1), m(2) > 0, alpha > 1, beta > 1, p, q, alpha + beta is an element of (2 + 4s/N, 2(s)*]. Firstly, if p, q, alpha + beta < 2(s)*, we obtain the existence of positive normalized solution when gamma is big enough. Secondly, if p= q = alpha + beta = 2(s)*, we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle.