Prescribed Mass Solutions to Schrodinger Systems With linear Coupled Terms

We consider the existence of normalized solutions to the following nonlinear Schrodinger system { - Delta u +lambda(1)u = mu(1)|u|(p-2)u + beta v in R-N, - Delta v +lambda(2)v = mu(2)|v|(q-2)v + beta u in R-N, under the mass constraints integral (N)(R) |u|(2)dx = a(2) (1) and integral (N)(R) |v|(2)dx = a(2) (2), where 2 < p <= 2 + 4/N <= q <= 2*, beta, mu(1), mu(2) > 0, a(1), a(2) > 0, and lambda(1),lambda(2) is an element of R appear as Lagrange multipliers. Under different character on p, q with respect to the mass critical exponent, we prove several existence results and precise asymptotic behavior of these solutions as (a1, a2)-> (0, 0). These cases present substantial differences with respect to purely mass subcritical or mass supercritical situations.