Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions

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• 作者：Wenxiong Chen ; Congming Li ; Guanfeng Li
• 关键词：Mathematics Subject Classification35J60
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：56
• 期：2
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0835
• 卷排序：56

文摘

In this paper, we consider equations involving fully nonlinear non-local operators $$\begin{aligned} F_{\alpha }(u(x)) \equiv C_{n,\alpha } PV \int _{{\mathbb {R}}^n} \frac{G(u(x)-u(z))}{|x-z|^{n+\alpha }} dz= f(x,u). \end{aligned}$$We prove a maximum principle and obtain key ingredients for carrying on the method of moving planes, such as narrow region principle and decay at infinity. Then we establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in a unit ball and in the whole space, as well as non-existence of solutions on a half space. We believe that the methods developed here can be applied to a variety of problems involving fully nonlinear nonlocal operators. We also investigate the limit of this operator as \(\alpha {\rightarrow }2\) and show that $$\begin{aligned} F_{\alpha }(u(x)) {\rightarrow }a(-\bigtriangleup u(x)) + b |{\bigtriangledown }u(x)|^2. \end{aligned}$$Mathematics Subject Classification35J60Communicated by F. H. Lin.