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In the paper, we study the following system of partial differential equations (PDEs) involving fractional Laplacian operators(1)under the boundary conditions, where,,and. In order to overcome the difficulty that there are no corresponding maximum principles for the operatorsandin, we employ the method of moving planes in integral forms to the system of integral equations which is equivalent to System (1). Then, we obtain some Liouville type theorems for a pair of solutionsof System (1) under different assumptions.

Liouville-type theorems for a system of fractional Laplacian equations