Abstract
In this paper, we consider equations involving fully nonlinear non-local operators
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
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Communicated by F. H. Lin.
Wenxiong Chen: Partially supported by the Simons Foundation Collaboration Grant for Mathematicians 245486.
Congming Li: Partially supported by NSFC 11571233 and NSF DMS-1405175.
Guanfeng Li: Partially supported by the Fundamental Research Funds for the Central Universities (PIRS OF HIT 201614).
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Chen, W., Li, C. & Li, G. Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions. Calc. Var. 56, 29 (2017). https://doi.org/10.1007/s00526-017-1110-3
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DOI: https://doi.org/10.1007/s00526-017-1110-3