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.
Categories: math.AP