Least energy solutions to a cooperative system of Schrodinger equations with prescribed L2-bounds: at least L2-critical growth

We look for least energy solutions to the cooperative systems of coupled Schrodinger equations {-Delta u(i) + lambda(i)u(i) = partial derivative(i)G(u) in R-N, N >= 3, u(i) is an element of H-1(R-N), i is an element of {1, ..., K} integral(RN) vertical bar u(i)vertical bar(2) dx <= rho(2)(i) with G >= 0, where rho(i) > 0 is prescribed and (lambda(i), u(i)) is an element of R x H-1(R-N) is to be determined, i is an element of {1, ..., K}. Our approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohozaev constraints intersected with the product of the closed balls in L-2(R-N) of radii rho(i), which allows to provide general growth assumptions about G and to know in advance the sign of the corresponding Lagrange multipliers. We assume that G has at least L-2-critical growth at 0 and admits Sobolev critical growth. The more assumptions we make about G, N, and K, the more can be said about the minimizers of the corresponding energy functional. In particular, if K = 2, N is an element of {3, 4}, and G satisfies further assumptions, then u = (u(1), u(2)) is normalized, i.e., integral(RN) vertical bar u(i)vertical bar(2) dx = rho(2)(i) for i is an element of {1, 2}.