Normalized solutions for nonlinear Schrodinger systems

We consider the existence of normalized solutions in H-1(R-N) x H-1(R-N) for systems of nonlinear Schrodinger equations, which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz, one is led to coupled systems of elliptic equations of the form -Delta u(1) = lambda(1)u(1) + f(1)(u(1)) + partial derivative F-1(u(1), u(2)), -Delta u(2) = lambda(2)u(2) + f(2)(u(2)) + partial derivative F-2(u(1), u(2)), u(1), u(2) is an element of H-1(R-N), N >= 2, and we are looking for solutions satisfying integral(RN) vertical bar u(1)vertical bar(2) = a(1), integral(RN) vertical bar u(2)vertical bar(2) = a(2), where a(1) > 0 and a(2) > 0 are prescribed. In the system, lambda(1) and lambda(2) are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e. f(i)(u(i)) - mu(i)vertical bar u(i)vertical bar(pi-1)u(i), F(u(1), u(2)) - beta vertical bar u(1)vertical bar(r1) vertical bar u(2)vertical bar(r2), with positive constants beta, mu(i), p(i), r(i). The exponents are Sobolev subcritical but may be L-2-supercritical. Our main result deals with the case in which 2 < p(1) < 2 + 4/N < p(2), r(1) + r(2) < 2* in dimensions 2 <= N <= 4. We also consider the cases in which all of these numbers are less than 2 + 4/N or all are bigger than 2 + 4/N.