Liouville Type Theorems Involving the Fractional Laplacian on the Upper Half Euclidean Space

In this paper, we mainly establish Liouville-type theorems for the elliptic semi-linear equations involving the fractional Laplacian on the upper half of Euclidean space. We employ a direct approach by studying an equivalent integral equation instead of using the conventional extension method. Applying the method of moving planes in integral forms, we prove the non-existence of positive solutions under very weak conditions. We also extend the results to a more general equation.