Given $Hin [0,1)$ and given a $C^0$ exterior domain $Omega$ in a $H-$hypersphere of $mathbb{H}^3,$ the existence of hyperbolic Killing graphs of CMC $H$ defined in $overline{Omega}$ with boundary $ partial Omega $ included in the $H-$hypersphere is obtained.
It is known that for $Omega subset mathbb{R}^{2}$ an unbounded convex domain and $H>0$, there exists a graph $Gsubset mathbb{R}^{3}$ of constant mean curvature $H$ over $Omega $ with $partial G=$ $partial Omega $ if and only if $Omega $ is included i
n a strip of width $1/H$. In this paper we obtain results in $mathbb{H}^{2}times mathbb{R}$ in the same direction: given $Hin left( 0,1/2right) $, if $Omega $ is included in a region of $mathbb{ H}^{2}times left{ 0right} $ bounded by two equidistant hypercycles $ell(H)$ apart, we show that, if the geodesic curvature of $partial Omega $ is bounded from below by $-1,$ then there is an $H$-graph $G$ over $Omega $ with $partial G=partial Omega$. We also present more refined existence results involving the curvature of $partialOmega,$ which can also be less than $-1.$
A version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol$_3$ is proved. Moreover, we construct examples of admissible domains where the results may be applied.
We investigate, for the Laplacian operator, the existence and nonexistence of eigenfunctions of eigenvalue between zero and the first eigenvalue of the hyperbolic space H^n, for unbounded domains of H^n. If a domain is contained in a horoball, we pro
ve that there is no positive bounded eigenfunction that vanishes on the boundary. However, if the asymptotic boundary of a domain contains an open set of the asymptotic boundary of H^n, there is a solution that converges to 0 at infinity and can be extended continuously to the asymptotic boundary. In particular, this result holds for hyperballs.
