Title: Liouville theorem and a priori estimates of radial solutions for a non-cooperative elliptic system Show Chinese title

Title: Liouville 定理和非合作椭圆系统径向解的先验估计

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. Show Chinese abstract

Abstract: 对于尺度不变的非线性椭圆系统，Liouville定理（即系统不具有非平凡的整体解）保证了相关更一般系统的解的先验估计。 假设$p=2q+3>1$是Sobolev次临界，$n\le3$和$\beta\in{\mathbb R}$。 我们首先在径向函数$(u,v)$的类中证明了系统$$\left.\begin{aligned} -\Delta u &=|u|^{2q+2}u+\beta|v|^{q+2}|u|^q u, \\ -\Delta v &=|v|^{2q+2}v+\beta|u|^{q+2}|v|^q v, \end{aligned}\ \right\} \quad\hbox{in}\quad {\mathbb R}^n,$$的Liouville定理，使得$u,v,u-v,u+v$的节点域数量有限。 然后我们利用这个定理来获得相关椭圆系统的解的先验估计。 在三次方情形$q=0$中，这些解对应于Schrödinger方程组的孤立波，它们的存在性和多重性已经被各种方法深入研究。 其中一种方法是基于相应抛物系统适当全局解的先验估计。 与之前的研究不同，我们的Liouville定理对所有Sobolev次临界的$q\geq0$都给出了这些估计。