Summary: In this paper, we consider a nonlinear Schrödinger equation involving the fractional Laplacian with Dirichlet condition: \[\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u+A(x)u=f(x,u,\nabla u) \text{ in } \Omega ,\\ u>0\text{ in } \Omega,\quad u\equiv 0 \text{ in } \mathbb{R}^n\setminus \Omega , \end{cases}\] where \(\Omega\) is a domain (bounded or unbounded) in \(\mathbb{R}^n\) which is convex in \(x_1\)-direction. By using some ideas of maximum principle and the direct moving plane method, we prove that the solutions are strictly increasing in \(x_1\)-direction in the left half domain of \(\Omega \). Symmetry of some solutions are also proved. Meanwhile, we obtain a Liouville type theorem on the half space \(\mathbb{R}^n_+\).