  In this paper we establish existence and nonexistence results concerning
fully nontrivial minimal energy solutions of the nonlinear Schr\"odinger system
  \begin{align*}
  \begin{gathered}
  -\Delta u + \, u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad\text{in}\R^n,
  -\Delta v + \omega^2 v = |v|^{2q-2}v + b|u|^q|v|^{q-2}v\quad\text{in}\R^n.
  \end{gathered}
  \end{align*}
  We consider the repulsive case $b<0$ and assume that the exponent $q$
satisfies $1<q<\frac{n}{n-2}$ in case $n\geq 3$ and $1<q<\infty$ in case $n=1$
or $n=2$. For space dimensions $n\geq 2$ and arbitrary $b<0$ we prove the
existence of fully nontrivial nonnegative solutions which converge to a
solution of some optimal partition problem as $b\to -\infty$. In case $n=1$ we
prove that minimal energy solutions exist provided the coupling parameter $b$
has small absolute value whereas fully nontrivial solutions do not exist if
$1<q\leq 2$ and $b$ has large absolute value.
