Sliding methods for the higher order fractional laplacians

Abstract

Abstract In this paper, we develop a sliding method for the higher order fractional Laplacians. We first obtain the key ingredients to obtain monotonicity of solutions, such as narrow region maximum principles in bounded or unbounded domains. Then we introduce a new idea of estimating the singular integrals defining the fractional Laplacian along a sequence of approximate maximum points and illustrate how this sliding method can be employed to obtain monotonicity of solutions. We believe that the narrow region maximum principles will become useful tools in analyzing higher order fractional equations.