Remarks on semi-linear elliptic problem involving fractional Hardy operators

The purpose of this paper is to investigate various maximal principles and develop a direct moving planes approach for fractional Hardy operators Lμs=(-Δ)s+μ|x|2s with s∈(0,1) and -22s-1Γ(s+12)πΓ(1-s)≤μ≤0. We establish the narrow region principle and the decay at infinity for the fractional Hardy operator with μ≤0 in unbounded domains, without assuming any specific asymptotic behavior of u near infinity. We also extend the direct method of moving planes to semi-linear elliptic problem involving fractional Hardy operators in both bounded and unbounded domains. Moreover, we prove the symmetry and monotonicity of the nonnegative solutions to generalized static fractional Schrödinger-Hartree equations with the Hardy potential and combined nonlinearities. We believe that our theoretical results and methods can be readily applied to investigate other problems that involve fractional Hardy operators with μ≤0.