Some monotonicity results for the fractional Laplacian in unbounded domain

In this paper, we develop a direct method of moving planes in Rn without any decay conditions at infinity for solutions for fractional Laplacian. We first prove a monotonicity result for semi-linear equations involving the fractional Laplacian equation in Rn, and we also derive a one-dimensional symmetry result, which indicates that fractional De Giorgi conjecture is valid under some conditions. During these processes, we introduce some new ideas: (i) estimating the singular integrals defining the fractional Laplacian along a sequence of approximate maximum; (ii) analyzing the fractional equations along a sequence of approximate maximum, and then by making translation and taking the limit to derive a limit equation.