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

We look for least energy solutions to the cooperative systems of coupled Schrödinger equations
$\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i + \lambda _i u_i = \partial _iG(u)\quad \mathrm {in} \ {\mathbb {R}}^N, \ N \ge 3,\\ u_i \in H^1({\mathbb {R}}^N), \\ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \le \rho _i^2 \end{array} \right. i\in \{1,\ldots ,K\} \end{aligned}$
with 𝐺≥0, where $\rho _i>0$ is prescribed and $(\lambda _i, u_i) \in {\mathbb {R}}\times H^1 ({\mathbb {R}}^N)$ is to be determined, $i\in \{1,\dots ,K\}$. 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 $L^2({\mathbb {R}}^N)$ 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 ∈ {3,4}, and G satisfies further assumptions, then $u=(u_1,u_2)$ is normalized, i.e., $\int _{{\mathbb {R}}^N} |u_i|^2 \, dx=\rho _i^2$ for 𝑖 ∈ {1,2}.