Monotonicity of solutions for a class of nonlocal Monge-Ampère problem

In this paper, we consider nonlinear problems involving nonlocal Monge-Ampere operators. By using a sliding method, we establish monotonicity of positive solutions for nonlocal Monge-Ampere problems both in an infinite slab and in an upper half space. During this process, an important idea we applied is to estimate the singular integrals defining the nonlocal Monge-Ampere operator along a sequence of approximate maximum points. It allows us to assume weaker conditions on nonlinear terms. Another idea is to employ a generalized average inequality which plays an important role and greatly simplify the process of the sliding.