  The existence and nonexistence of $\lambda$-harmonic functions in unbounded
domains of $\mathbb{H}^n$ are investigated. We prove that if the $(n-1)/2$
Hausdorff measure of the asymptotic boundary of a domain $\Omega$ is zero, then
there is no bounded $\lambda$-harmonic function of $\Omega$ for $\lambda \in
[0,\lambda_1(\mathbb{H}^n)]$, where $\lambda_1(\mathbb{H}^n)=(n-1)^2/4$. For
these domains, we have comparison principle and some maximum principle.
Conversely, for any $s>(n-1)/2,$ we prove the existence of domains with
asymptotic boundary of dimension $s$ for which there are bounded
$\lambda_1$-harmonic functions that decay exponentially at infinity.
