# American Institute of Mathematical Sciences

November  2020, 19(11): 5269-5283. doi: 10.3934/cpaa.2020237

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

 School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical, Sciences, Central China Normal University, Wuhan, 430079, China, Department of Mathematical Sciences, Yeshiva University, New York, NY, USA

Received  April 2020 Revised  June 2020 Published  September 2020

Fund Project: This work was supported by Natural Science Foundation of China (Grant No.11771166), Hubei Key Laboratory of Mathematical Sciences, Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46 and China Scholarship Council

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.

Citation: Yahui Niu. Monotonicity of solutions for a class of nonlocal Monge-Ampère problem. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5269-5283. doi: 10.3934/cpaa.2020237
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