In this paper, we are concerned with the following nonlinear equations involving the fractional Hardy operator
\({\cal{L}}_{\mu}^{s}u(x)=f(x,u,\nabla u),\)
where \({\cal{L}}_{\mu}^{s}=(-\Delta)^{s}+{\mu\over{\vert x\vert^{2s}}}\) with s ∈ (0, 1) and μ ≥ 0. We first establish various maximum principles for fractional Hardy operator \({\cal{L}}_{\mu}^{s}\) in bounded or unbounded domains. As applications, we extend the direct method of moving planes and sliding methods for the fractional Hardy problem, and discuss how they can be used to establish symmetry, monotonicity, and uniqueness results for solutions in various domains, including bounded domain, unbounded domain, ℝn, ℝ + n and an epigraph Ω in ℝn, respectively. To our best knowledge, it is the first time to apply the direct sliding method to deal with the fractional Hardy problem with gradient terms. We believe that our theory and methods can be conveniently applied to study other problems involving the fractional Hardy operator.
