Classification of anti-symmetric solutions to nonlinear fractional Laplace equations

We study anti-symmetric solutions to nonlinear equations (-Delta)(s)u = u(p) involving fractional Laplacian operators of order 2s. First, in the often used defining space L-2s, we establish a Liouvill type theorem. Some suitable forms of maximum principles and some subtle lower bounds of the solutions are the key ingredients here. Second, observing the anti-symmetric property of the solutions, we extend the usual defining space L2s for (-Delta)(s) to L2s+1. It is very interesting to see the existence of solutions in the expanded and somewhat more natural space. In fact, we show the existence of non-trivial solutions when p + 2s < 1 and the nonexistence when p + 2s > 1. Finding a suitable super-solution is an important step in constructing non-trivial solutions.