Multiple positive normalized solutions for nonlinear Schrodinger systems

We consider the existence of multiple positive solutions to the nonlinear Schrodinger systems set on H-1(R-N) x H-1(R-N), {-Delta u(1) = lambda(1)u(1) + mu(1)vertical bar u(1)vertical bar(p1-2)u(1) + beta r(1)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) + beta r(2)vertical bar u(1)vertical bar(r1)vertical bar u(2)vertical bar(r2-2)u(2), under the constraint integral(RN) vertical bar u(1)vertical bar(2) dx = a(1), integral(RN) vertical bar u(2)vertical bar(2) dx = a(2). Here a(1), a(2) > 0 are prescribed, mu(1), mu(2), beta > 0, and the frequencies lambda(1), lambda(2) are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when N >= 1, 2 < p(1), p(2) < 2 + 4/N, r(1), r(2) > 1, 2 + 4/N < r(1) + r(2) < 2*, the second when N >= 1, 2 + 4/N < p(1), p(2) < 2*, r(1), r(2) > 1, r(1) + r(2) < 2 + 4/N. In both cases, assuming that beta > 0 is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.