Existence and orbital stability of standing waves to nonlinear Schrodinger system with partial confinement

We are concerned with the existence of solutions to the nonlinear Schrodinger system in R-3: -Delta u(1) + (x(1)(2) + x(2)(2))u(1) = lambda(1)u(1) + mu(1) |u(1)|(p1-2)u(1) + beta r(1) |u(1)|(r1-2)u(1) |u(2)|(r2) and -Delta u(2) + (x(1)(2) + x(2)(2))u(2) = lambda(2)u(2) + mu(2)|u(2)|(p2-2)u(2) + beta r(2) |u(1)|(r1) |u(2)|(r2-2)u(2) under the constraint integral(R3) |u(1)|(2) dx = a(1) > 0, integral(R3) |u(2)|(2) dx = a(2) > 0, where mu(1), mu(2), beta > 0, 2 < p(1), p(2) < 10/3, r(1), r(2) > 1, r(1) + r(2) < 10/3. In the system, the parameters lambda(1), lambda(2) are unknown and appear as Lagrange multipliers. Our solutions are achieved as global minimizers of the underlying energy functional subject to the constraint. Our purpose is to establish the compactness of any minimizing sequence, up to translation. As a by-product, we obtain the orbital stability of the set of global minimizers. Published by AIP Publishing.