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

TL;DR

This paper proves a Liouville theorem for a class of radial solutions to a nonlinear elliptic system, leading to a priori estimates for solutions, applicable to subcritical cases including solitary waves of Schrödinger systems.

Contribution

It introduces a Liouville theorem for radial solutions of a nonlinear elliptic system, enabling a priori estimates for all Sobolev subcritical cases, extending previous methods.

Findings

Liouville theorem established for radial solutions with finite nodal domains.

A priori estimates derived for solutions of related elliptic systems.

Results applicable to solitary waves in Schrödinger equations for all subcritical q.

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, $n\le3$ 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.