A Liouville theorem for the higher-order fractional Laplacian

We study Navier problems involving the higher-order fractional Laplacians. We first obtain nonexistence of positive solutions, known as the Liouville-type theorems, in the upper half-space R-+(n) by studying an equivalent integral form of the fractional equation. Then we show symmetry for positive solutions on B-1(0) through a delicate iteration between lower-order differential/pseudo-differential equations split from the higher-order equation.