The sliding methods for the fractional p-Laplacian

In this paper, we develop a sliding method for the fractional p-Laplacian. We first obtain the key ingredient needed in the sliding method in a bounded domain the narrow region principle. Then using nonlinear equations involving the fractional p-Laplacian in both bounded domains and in the whole space, we illustrate how this new sliding method can be employed to obtain monotonicity of solutions. During these processes, we introduce a new idea estimating the singular integrals defining the fractional p-Laplacian along a sequence of approximate maximum points. We believe that the new ideas and methods employed here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities. (C) 2019 Elsevier Inc. All rights reserved.