Existence and orbital stability of standing waves for nonlinear Schrodinger systems

In this paper we investigate the existence of solutions in H-1(R-N) x H-1(R-N) for nonlinear Schrodinger systems of the form { -Delta u(1) = lambda(1)u(1) + mu(1)vertical bar u(1)vertical bar(p1-2) u(1) + r(1)beta vertical bar u(1)vertical bar(r1-2) u(1)vertical bar u(2)vertical bar(r2), -Delta u(2)= lambda(2)u(2) + mu(2)vertical bar u(2)vertical bar(p2-2) u(2) + r(2)beta vertical bar u(1)vertical bar(r1) u(2)vertical bar u(2)vertical bar(r2-2) u(2,) under the constraints integral(RN) vertical bar u(1)vertical bar(2) dx = a(1) > 0, integral(RN) vertical bar u(2 vertical bar) dx = a(2) > 0. Here N >= 1, beta > 0, mu(i) > 0, r(i) > 1, 2 < p(i) < 2 + 4/N for i = 1, 2 and r(1) + r(2) < 2 + 4/N. This problem is motivated by the search of standing waves for an evolution problem appearing in several physical models. Our solutions are obtained as constrained global minimizers of an associated functional. Note that in the system lambda(1) and lambda(2) are unknown and will correspond to the Lagrange multipliers. Our main result is the precompactness of the minimizing sequences, up to translation. Assuming the local well posedness of the associated evolution problem we then obtain the orbital stability of the standing waves associated to the set of minimizers. (C) 2016 Elsevier Ltd. All rights reserved.