Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space

We construct nontrivial unbounded domains $\mathrm{\Omega }$ in the hyperbolic space ${\mathbb{H}}^N$, $N\in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -Δ_{$\mathbb{H}$^N} u+u-u^p=0,, $\text{in}$,,Ω, ,, u=0,,,\partial_νu=$\text{const}$,,$\text{on}$,,\partialΩ\nonumber \end{equation} has a positive bounded solution in $C^{2,\alpha }\left(\mathrm{\Omega }\right)\cap H^1\left(\mathrm{\Omega }\right)$. We also give a condition under which this construction holds for larger dimensions $N$. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space.