Liouville theorems involving the fractional Laplacian on a half space

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Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions (2016, methodology): “One can also use the integral equations method, such as the method of moving planes in integral forms and regularity lifting to investigate equations involving fractional Laplacian by first showing that they are equivalent to the corresponding integral equations [2,6,7].”
A direct method of moving planes for the fractional Laplacian (2014, background): “1 in [CFY]) Assume that u ∈ Lα is a locally bounded positive solution of”
Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space (2017, methodology): “Indeed, our approach also gives a simpler proof in the single equation case considered in [5].”