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
$$\left\{\begin{array}{l} -\Delta u_i+\lambda_i u_i = \partial_i G(u) \quad\text{in }\mathbb{R}^N,\ N\ge3, \\ u_i\in H^1(\mathbb{R}^N), & & \\ \textstyle\int_{\mathbb{R}^N}|u_i|^2\,dx\le \rho_i^2 \end{array} i\in\{1,\ldots,K\}\right.$$
with $G\ge0$, where $\rho_i>0$ is prescribed and $(\lambda_i,u_i)\in\mathbb{R}\times H^1(\mathbb{R}^N)$ is to be determined, $i\in\{1,\ldots,K\}$. 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 on $G$ 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 Sobolev subcritical growth at infinity. The more assumptions we make on $G$, $N$, and $K$, the more can be said about the minimizers of the energy functional. In particular, if $K=2$, $N\in\{3,4\}$, and $G$ satisfies further assumptions,
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then $u=(u_1, u_2)$ is normalized, i.e., $\int_{\mathbb{R}^N}|u_i|^2\,dx = \rho_i^2$ for $i \in\{1, 2\}$.