Summary: We study anti-symmetric solutions to nonlinear equations \((-\Delta)^su=u^p\) 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 \(L_{2s}\) for \((-\Delta)^s\) to \(L_{2s+1}\). 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.