  Liouville theorems for scaling invariant nonlinear elliptic systems (saying
that the system does not possess nontrivial entire solutions) guarantee a
priori estimates of solutions of related, more general systems. Assume that
$p=2q+3>1$ is Sobolev subritical, $n\le3$ and $\beta\in{\mathbb R}$. We first
prove a Liouville theorem for the system $$\left.\begin{aligned} -\Delta u
&=|u|^{2q+2}u+\beta|v|^{q+2}|u|^q u, \\ -\Delta v
&=|v|^{2q+2}v+\beta|u|^{q+2}|v|^q v, \end{aligned}\ \right\}
\quad\hbox{in}\quad {\mathbb R}^n,$$ in the class of radial functions $(u,v)$
such that the number of nodal domains of $u,v,u-v,u+v$ is finite. Then we use
this theorem to obtain a priori estimates of solutions to related elliptic
systems. In the cubic case $q=0$, those solutions correspond to the solitary
waves of a system of Schr\"odinger equations, and their existence and
multiplicity have been intensively studied by various methods. One of those
methods is based on a priori estimates of suitable global solutions of
corresponding parabolic systems. Unlike the previous studies, our Liouville
theorem yields those estimates for all $q\geq0$ which are Sobolev subcritical.
