This paper studies the following coupled k-Hessian system with different order fractional Laplacian operators:

$$\begin{aligned} {\left\{ \begin{array}{ll} {S_k}({D^2}w(x))-A(x)(-\varDelta )^{\alpha /2}w(x)=f(z(x)),\\ {S_k}({D^2}z(x))-B(x)(-\varDelta )^{\beta /2}z(x)=g(w(x)). \end{array}\right. } \end{aligned}$$

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.

中文翻译：

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涉及不同分数拉普拉斯的耦合 k-Hessian 系统的非负解

本文研究了以下具有不同阶分数拉普拉斯算子的耦合k -Hessian 系统：

$$\begin{对齐} {\left\{ \begin{array}{ll} {S_k}({D^2}w(x))-A(x)(-\varDelta )^{\alpha /2 }w(x)=f(z(x)),\\ {S_k}({D^2}z(x))-B(x)(-\varDelta )^{\beta /2}z(x )=g(w(x))。 \end{数组}\对。 } \end{对齐}$$

首先，我们讨论涉及分数阶拉普拉斯算子的k -Hessian系统的无穷大衰减原理和窄域原理。然后，利用移动平面的直接方法，分别证明了耦合k -Hessian系统非负解在单位球和整个空间内的径向对称性和单调性。我们相信目前的工作将导致对涉及不同阶分数拉普拉斯算子的耦合k -Hessian 系统的深入理解。