Uniqueness and monotonicity of solutions for fractional equations with a gradient term

In this paper, we consider the following fractional equation with a gradient term su(x) = f(x, u(x), ?u(x)), in bounded domain and upper half space. Firstly, prove monotonicity uniqueness of solutions to by sliding method. order obtain maximum principle on unbounded domain, need estimate singular integrals define Laplacians along sequence approximate points using generalized average inequality. Then Rn + solve difficulties caused term, some new techniques are developed. The paper may be considered as an extension Berestycki Nirenberg [J. Geom. Phys. 5(1988), 237–275].