Symmetric Properties for System Involving Uniformly Elliptic Nonlocal Operators

In this paper, we obtain symmetry and monotonicity of positive solutions for the systems involving uniformly elliptic nonlocal operators in a domain (bounded or unbounded) in R n using a direct method of moving planes. Our results include subcritical case, critical case and supercritical case and seem to be the first symmetric properties of the system involving uniformly elliptic nonlocal operators and containing the gradient of the solutions in the nonlinear terms.