Abstract In this study, we explore the positive solutions of a nonlinear Choquard equation involving the Green kernel of the fractional operator (− Δ B N) − α ⁄ 2 (-{ {₁^N}) }^- /2 in the hyperbolic space, where Δ B N {₁^N} represents the Laplace-Beltrami operator on B N {B}^N, with α ∈ (0, N) (0, N) and N ≥ 3 N 3. This study is analogous to the Choquard equation in the Euclidean space, which involves the nonlocal Riesz potential operator. We consider the functional setting within the Sobolev space H 1 (B N) H^1 ({B}^N), employing advanced harmonic analysis techniques, particularly the Helgason Fourier transform and semigroup approach to the fractional Laplacian. Moreover, the Hardy-Littlewood-Sobolev inequality on complete Riemannian manifolds, as developed by Varopoulos, is pivotal in our analysis. We prove an existence result for the problem (F α, λ F, ) in the subcritical case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.
Gupta et al. (Wed,) studied this question.