Abstract
We prove the symmetry of components and some Liouville-type theorems for, possibly sign changing, entire distributional solutions to a family of nonlinear elliptic systems encompassing models arising in Bose-Einstein condensation and in nonlinear optics. For these models we also provide precise classification results for non-negative solutions. The sharpness of our results is also discussed.
Highlights
We prove the symmetry of components and some Liouville-type theorems for, possibly sign changing, entire distributional solutions to a family of nonlinear elliptic systems encompassing models arising in Bose-Einstein condensation and in nonlinear optics
For these models we provide precise classification results for non-negative solutions
We shall prove that any entire distributional solution (u, v) of system (1.1), has the symmetry property u = v
Summary
The aim of this paper is to classify entire solutions (u, v) of the following nonlinear elliptic systems arising in Bose-Einstein condensation and in nonlinear optics (see for instance [7], [14], [19] and the references therein) :. We shall prove that any entire distributional solution (u, v) of system (1.1), has the symmetry property u = v (symmetry of components). We use this result to establish some new Liouville-type theorems as well as some classification results. Our method is different (and complementary) from the one used in [21] It exploits the attractive character of the interaction between the two states u and v. It applies to any distributional entire solution, possibly sign-changing and without any other restriction.
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