Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality

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Asymptotic method of moving planes for fractional parabolic equations (2020, background): “For elliptic equations involving either local or nonlocal operators, a number of systematic approaches have been established to study these qualitative properties, such as the method of moving planes[8, 20, 21, 27, 33, 34, 35, 45], the method of moving spheres[11, 29], and the sliding methods [15, 1”
Maximum principles for the fractional p-Laplacian and symmetry of solutions (2017, methodology): “on 3 and 4 respectively. For more articles concerning the method of moving planes for nonlocal equations and for integral equations, please see [CLv] [FL] [Ha] [HLZ] [HWY] [Lei] [LL] [LLM] [LZ] [LZ1] [LZ2] [MC] [MZ] and the references therein. 2 Various Maximum Principles and a Boundary Estimate Let”
Borderline Variational Problems Involving Fractional Laplacians and Critical Singularities (2015, background): “Asymptotic properties of the positive solutions were given by Y. Lei [26], Lu and Zhu [31], and Yang and Yu [38].”
Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space (2017, methodology): “method of moving planes [12, 13] (see also [3, 6, 17, 20]) in integral forms initially used in [7, 8].”