Regularity and classification of solutions to static Hartree equations involving fractional Laplacians

In this paper, we are concerned with the fractional order equations (1) with Hartree type $ \dot{H}^{\frac{α}{2}} $-critical nonlinearity and its equivalent integral equations (3). We first prove a regularity result which indicates that weak solutions are smooth (Theorem 1.2). Then, by applying the method of moving planes in integral forms, we prove that positive solutions $ u $ to (1) and (3) are radially symmetric about some point $ x_{0}∈\mathbb{R}^{d} $ and derive the explicit forms for $ u $ (Theorem 1.3 and Corollary 1). As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities (Corollary 2).