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We look for least energy solutions to the cooperative systems of coupled Schrödinger equations    $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i + \lambda _i u_i = \partial _iG(u)\quad \mathrm {in} \ {\mathbb {R}}^N, \ N \ge 3,\\ u_i \in H^1({\mathbb {R}}^N), \\ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \le \rho _i^2 \end{array} \right. i\in \{1,\ldots ,K\} \end{aligned}$$  - Δ u i + λ i u i = ∂ i G ( u ) in R N , N ≥ 3 , u i ∈ H 1 ( R N ) , ∫ R N | u i | 2 d x ≤ ρ i 2 i ∈ { 1 , … , K } with    $$G\ge 0$$  G ≥ 0 , where    $$\rho _i&gt;0$$  ρ i &gt; 0 is prescribed and    $$(\lambda _i, u_i) \in {\mathbb {R}}\times H^1 ({\mathbb {R}}^N)$$  ( λ i , u i ) ∈ R × H 1 ( R N ) is to be determined,    $$i\in \{1,\dots ,K\}$$  i ∈ { 1 , ⋯ , K } . Our approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints intersected with the product of the closed balls in    $$L^2({\mathbb {R}}^N)$$  L 2 ( R N ) of radii    $$\rho _i$$  ρ 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    $$L^2$$  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    $$K=2$$  K = 2 ,    $$N\in \{3,4\}$$  N ∈ { 3 , 4 } , and  G  satisfies further assumptions, then    $$u=(u_1,u_2)$$  u = ( u 1 , u 2 ) is normalized, i.e.,    $$\int _{{\mathbb {R}}^N} |u_i|^2 \, dx=\rho _i^2$$  ∫ R N | u i | 2 d x = ρ i 2 for    $$i\in \{1,2\}$$  i ∈ { 1 , 2 } .

Least energy solutions to a cooperative system of Schrödinger equations with prescribed $$L^2$$-bounds: at least $$L^2$$-critical growth