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Liouville theorems for scaling invariant nonlinear elliptic systems (sayingthat the system does not possess nontrivial entire solutions) guarantee apriori 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 firstprove 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 usethis theorem to obtain a priori estimates of solutions to related ellipticsystems. In the cubic case $q=0$, those solutions correspond to the solitarywaves of a system of Schr\"odinger equations, and their existence andmultiplicity have been intensively studied by various methods. One of thosemethods is based on a priori estimates of suitable global solutions ofcorresponding parabolic systems. Unlike the previous studies, our Liouvilletheorem yields those estimates for all $q\geq0$ which are Sobolev subcritical.

Liouville theorem and a priori estimates of radial solutions for a non-cooperative elliptic system