Symmetry of Positive Solutions for Fully Nonlinear Nonlocal Systems

Abstract

In this paper, we consider the nonlinear systems involving fully nonlinear nonlocal operators

$$\left\{ {\matrix{{{F_\alpha }(u(x)) = {v^p}(x) + {k_1}(x){u^r}(x),} & {x \in {\mathbb{R}^N},} \cr {{G_\beta }(v(x)) = {u^q}(x) + {k_2}(x){v^s}(x),} & {x \in {\mathbb{R}^N}} \cr } } \right.$$

and

$$\left\{ {\matrix{{{F_\alpha }(u(x)) = {{{v^p}(x)} \over {|x{|^a}}} + {{{u^r}(x)} \over {|x{|^b}}},} & {x \in {\mathbb{R}^N}\backslash \{ 0\} ,} \cr {{G_\beta }(v(x)) = {{{u^q}(x)} \over {|x{|^c}}} + {{{v^s}(x)} \over {|x{|^d}}},} & {x \in {\mathbb{R}^N}\backslash \{ 0\} ,} \cr } } \right.$$

where k_i(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.