Existence, symmetry and regularity of ground states of a non linear choquard equation in the hyperbolic space

In this paper, we explore the positive solutions of the following nonlinear Choquard equation involving the green kernel of the fractional operator $(-\Delta_{\mathbb{B}^N})^{-\alpha/2}$ in the hyperbolic space \begin{equation} \begin{aligned} -\Delta_{\mathbb{B}^{N}} u \, - \, \lambda u \, &= \left[(- \Delta_{\mathbb{B}^{N}})^{-\frac{\alpha}{2}}|u|^p\right]|u|^{p-2}u, \end{aligned} \end{equation} where $\Delta_{\mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbb{B}^{N}$, $\lambda \leq \frac{(N-1)^2}{4}$, $1 < p < 2^*_{\alpha} = \frac{N+\alpha}{N-2}$, $0 < \alpha < N$, $N \geq 3$, $2^*_\alpha$ is the critical exponent in the context of the Hardy-Littlewood-Sobolev inequality. This study is analogous to the Choquard equation in the Euclidean space, which involves the non-local Riesz potential operator. We consider the functional setting within the Sobolev space $H^1(\mathbb{B}^N)$, employing advanced harmonic analysis techniques, particularly the Helgason Fourier transform and semigroup approach to 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 above problem in the subcritical case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.