It seems classically known (and mentioned in several papers without reference) that there exist bounded non-constant harmonic functions on the hyperbolic space $\mathbb{H}^n, n \geq 2$. I am interested in seeing how one can prove this fact. In particular, I can imagine that there could be roundabout proofs (Poisson boundary etc.) or direct constructive proofs (directly solving a pde). In that case, I would be very interested in learning about all of them. Any hints/references would be highly appreciated.