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Maximum principles for a fully nonlinear nonlocal equation on unbounded domains

  • *The corresponding author

    *The corresponding author 

X. He was supported by NSFC Grants (11771468, 11971027, 11271386), and W. Zou by NSFC Grants (11771234, 11926323, 11371212)

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  • In this paper, we study equations involving fully nonlinear nonlocal operators

    $ \mathcal {F}_{\alpha}(u(x)) = C_{n, \alpha}P.V.\int_{ \mathbb R^n}\frac{G(u(x)-u(z))}{|x-z|^{n+\alpha}}dz = f(u(x)), \; \; \; x\in \mathbb R^n. $

    We shall establish a maximum principle for anti-symmetric functions on any half space, and obtain key ingredients for proving the symmetry and monotonicity for positive solutions to the fully nonlinear nonlocal equations. Especially, a Liouville theorem is derived, which will be useful in carrying out the method of moving planes on unbounded domains for a variety of problems with fully nonlinear nonlocal operators.

    Mathematics Subject Classification: Primary: 35J60; 35J20; Secondary: 35R11; 35S15.


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