In this talk, we first consider fractional Laplace systems on two different types of domains, and we get monotonicity of solutions by applying the sliding method. We also study anti-symmetric solutions of nonlinear equations with fractional Laplacian, and we obtain nonexistence of positive solutions in the often used defining space $\mathcal{L}{2s}$ space. Based on the anti-symmetric property, we can extend the defining space from $\mathcal{L}{2s}$ to $\mathcal{L}_{2s+1}$. Moreover, we show the existence of non-trivial solutions in the extended space.