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

Abstract

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, $n\le 3$ and $\beta \in \mathbb{R}$. We first prove a Liouville theorem for the system $$\begin{array}{rl}-\Delta u& =|u{|}^{2q+2}u+\beta |v{|}^{q+2}|u{|}^{q}u,\\ -\Delta v& =|v{|}^{2q+2}v+\beta |u{|}^{q+2}|v{|}^{q}v,\end{array}\}in{\mathbb{R}}^n,$$ 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 Schrödinger 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 $q\ge 0$ which are Sobolev subcritical.