Classification of anti-symmetric solutions to the fractional Lane-Emden system

We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian (-Delta)(s) (0 < s < 1). First we obtain a Liouville type theorem in the often-used defining space L-2s. An interesting lower bound on the solutions is derived to estimate the Lipschitz coefficient in the sub-linear cases. Considering the anti-symmetric property, one can naturally extend the defining space from L-2s to L2s+1. Surprisingly, with this extension, we show the existence of non-trivial solutions. This is very different from the previous results of the Lane-Emden system.