Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes

In this paper, we are concerned with the fractional order static Hartree equations with critical nonlocal nonlinearity. We prove that the positive solutions are radially symmetric about some point in Rd and must assume the certain explicit forms. The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving plane method for fractional Laplacians in [6]. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., Decay at infinity and Narrow region principle (Theorem 2.1, Theorem 2.6).