Nonnegative solutions of a coupled k-Hessian system involving different fractional Laplacians

This paper studies the following coupled k-Hessian system with different order fractional Laplacian operators: \(\begin{aligned} {\left\{ \begin{array}{ll} {S_k}({D^2}w(x))-A(x)(-\varDelta )^{\alpha /2}w(x)=f(z(x)),\\ {S_k}({D^2}z(x))-B(x)(-\varDelta )^{\beta /2}z(x)=g(w(x)). \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{S}_k(D^{2}w(x))-A(x){(-\Delta )}^{\alpha /2}w(x)=f(z(x)),\hfill \\ S_k(D^{2}z(x))-B(x){(-\Delta )}^{\beta /2}z(x)=g(w(x)).\hfill \end{array}\right)\end{array}$$ Firstly, we discuss decay at infinity principle and narrow region principle for the k-Hessian system involving fractional order Laplacian operators. Then, by exploiting the direct method of moving planes, the radial symmetry and monotonicity of the nonnegative solutions to the coupled k-Hessian system are proved in a unit ball and the whole space, respectively. We believe that the present work will lead to a deep understanding of the coupled k-Hessian system involving different order fractional Laplacian operators.