Maximum principles for a fully nonlinear nonlocal equation on unbounded domains.

In this paper, we study equations involving fully nonlinear nonlocal operators Fα(u(x)) = Cn, αP.V.∫RnG(u(x) − u(z))|x − z|n + αdz = ƒ(u(x)),x∈Rn. 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. [ABSTRACT FROM AUTHOR]