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

The first author was supported by the NNSF of China (No. 11501021), the second author was supported by the NNSF of China (No. 11301166)

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  • 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).

    Mathematics Subject Classification: Primary: 35R11, 35J91; Secondary: 35B06, 35B65.


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