A direct method of moving planes for the fractional Laplacian

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• 作者：Wenxiong Chen^a ; ¹ ; ^wchen@yu.edu ; Congming Li^b ; ^c ; ² ; ^{cli@clorado.edu} ; Yan Li^a ; ^{yali3@mail.yu.edu}
• 关键词：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
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：21 February 2017
• 年：2017
• 卷：308
• 期：Complete
• 页码：404-437
• 全文大小：474 K
• 卷排序：308

文摘

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.