Symmetry of Components for Semilinear Elliptic Systems

Source abstract

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 Schrödinger 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.