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In this work we consider the weakly coupled Schr\"odinger cubic system \[\begin{cases} \displaystyle -\Delta u_i+\lambda_i u_i= \mu_i u_i^{3}+u_i\sum_{j\neq i}b_{ij} u_j^2 \\ u_i\in H^1(\mathbb{R}^N;\mathbb{R}), \quadi=1,\ldots, d, \end{cases} \] where $1\leq N\leq 3$, $\lambda_i,\mu_i >0$ and$b_{ij}=b_{ji}>0$ for $i\neq j$. This system admits semitrivial solutions, thatis solutions $\mathbf{u}=(u_1,\ldots, u_d)$ with null components. We provideoptimal qualitative conditions on the parameters $\lambda_i,\mu_i$ and $b_{ij}$under which the ground state solutions have all components nontrivial, or,conversely, are semitrivial.  This question had been clarified only in the $d=2$ equations case. For $d\geq3$ equations, prior to the present paper, only very restrictive results wereknown, namely when the above system was a small perturbation of thesuper-symmetrical case $\lambda_i\equiv \lambda$ and $b_{ij}\equiv b$. We treatthe general case, uncovering in particular a much more complex and richerstructure with respect to the $d=2$ case.

Semitrivial vs. fully nontrivial ground states in cooperative cubic Schr\"odinger systems with $d\ge3$ equations