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)) = vᵖ(x) + k₁(x)uʳ(x), & x ∈ RN, G_β (v(x)) = uq(x) + k₂(x)vˢ(x), & x ∈ RN . 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.