Symmetry of Positive Solutions for Fully Nonlinear Nonlocal Systems.

In this paper, we consider the nonlinear systems involving fully nonlinear nonlocal operators { F α (u (x)) = v p (x) + k 1 (x) u r (x) , x ∈ ℝ N , G β (v (x)) = u q (x) + k 2 (x) v s (x) , x ∈ ℝ N and { F α (u (x)) = v p (x) | x | a + u r (x) | x | b , x ∈ ℝ N \ { 0 } , G β (v (x)) = u q (x) | x | c + v s (x) | x | d , x ∈ ℝ N \ { 0 } , where ki(x) ≥ 0, i = 1, 2, 0 < α, β < 2, p, q, r, s > 1, a, b, c, d > 0. By proving a narrow region principle and other key ingredients for the above systems and extending the direct method of moving planes for the fractional p-Laplacian, we derive the radial symmetry of positive solutions about the origin. During these processes, we estimate the local lower bound of the solutions by constructing sub-solutions. [ABSTRACT FROM AUTHOR]