A direct method of moving planes for the fractional Laplacian

作者：Chen, Wenxiong; Li, Congming^*; Li, Yan

来源：Advances in Mathematics, 2017, 308: 404-437.

DOI：10.1016/j.aim.2016.11.038

• The fractional Laplacian
• Maximum principles for anti-symmetric functions
• Narrow region principle
• Decay at infinity
• Method of moving planes
• Radial symmetry
• Monotonicity
• Non-existence of positive solutions

In this paper, we develop a direct method of moving planes for the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre, we work directly on the non-local operator. Using the integral defining the fractional Laplacian, by an elementary approach, we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space, such as strong maximum principles for anti-symmetric functions, narrow region principles, and decay at infinity. Then, using simple examples, semi-linear equations involving the fractional Laplacian, we illustrate how this new method of moving planes can be employed to obtain symmetry and non-existence of positive solutions. @@@ We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities.

• 出版日期2017-2-21
• 单位上海交通大学