Existence of normalized solutions for Schrödinger systems with linear and nonlinear couplings

In this paper we study the nonlinear Bose-Einstein condensates Schr & ouml;dinger system {-Delta u(1 )- lambda(1)u(1 )= mu(1)u(1)(3 )+ beta u(1)u(2)(2 )+ kappa(x)u(2 )in R-3, -Delta u(2 )- lambda(2)u(2 )= mu(2)u(2)(3 )+ beta u(1)(2)u(2 )+ kappa(x)u(1 )in R-3, integral(3 )(R)u(1)(2 )= a(1)(2), integral(3)(R) u(2)(2 )= a(2)(2), where a(1), a(2), mu(1), mu(2), kappa = kappa(x) > 0, beta < 0, and lambda(1), lambda(2 )are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive.