Symmetry of Components for Semilinear Elliptic Systems
Symmetry of Components for Semilinear Elliptic Systems
复制标题
DOI：
10.1137/11085428x
复制
发表时间：
2012-07
期刊：
影响因子：
--
通讯作者：
P. Quittner;P. Souplet
中科院分区：
文献类型：
--
作者：
P. Quittner;P. Souplet
本文给出了椭圆型方程组在全空间$\mathbb{R}^n$中的任何正古典解$（u，v）$具有对称性$u=v$的充分条件。作为应用，我们显著地改进了Li和Ma [SIAM J. Math. Anal.，40（2008），pp. 1049- 1057]关于Schrodinger型Sobolev临界椭圆方程组解的分类。我们的技术也适用于一些超临界问题。我们还获得了新的Liouville型定理的非合作系统。此外，我们提供了一些反例，表明我们的假设在某种意义上是必要的。我们的证明是基于适当的最大值原理参数，结合超调和函数的球形平均的性质和一些适当的辅助函数。
In this paper, we give sufficient conditions ensuring that any positive classical solution $(u,v)$ of an elliptic system in the whole space $\mathbb{R}^n$ has the symmetry property $u=v$. As an application, we significantly improve the results of Li and Ma [SIAM J. Math. Anal., 40 (2008), pp. 1049--1057] on the classification of solutions of Sobolev-critical elliptic systems of Schrodinger type. Our techniques apply to some supercritical problems as well. We also obtain new Liouville-type theorems for noncooperative systems. Moreover, we provide some counterexamples which indicate that our assumptions are in a sense necessary. Our proofs are based on suitable maximum principle arguments, combined with properties of spherical means of superharmonic functions and on some appropriate auxiliary functions.