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

Abstract

In this paper, we consider nonlinear problems involving nonlocal Monge-Ampère operators. By using a sliding method, we establish monotonicity of positive solutions for nonlocal Monge-Ampère 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-Ampère 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.