Monotonicity of positive solutions for fractional [formula omitted]-systems in unbounded Lipschitz domains

Abstract

In this paper, without imposing any decay assumptions on the solutions at infinity, we first develop an unbounded narrow region principle for the fractional p-Laplacian systems, an essential ingredient to carry on the direct method of moving planes; then we combine this direct method with the sliding method to obtain the monotonicity of positive bounded solutions for cooperative systems involving the fractional p-Laplacian in unbounded Lipschitz domains. We believe that the new ideas introduced here will become useful tools in studying qualitative properties of solutions for other types of nonlinear nonlocal systems in various unbounded domains.