  This paper is devoted to study the semilinear elliptic system of H\'enon-type
\begin{eqnarray*} -\Delta_{\mathbb{B}^{N}}u= K(d(x))Q_{u}(u,v) \\
-\Delta_{\mathbb{B}^{N}}v= K(d(x))Q_{v}(u,v) \end{eqnarray*} in the hyperbolic
space $\mathbb{B}^{N}$, $N\geq 3$, where $u, v \in
H_{r}^{1}(\mathbb{B}^{N})=\{\phi\in H^1(\mathbb{B}^N): \phi\, \text{is
radial}\}$ and $-\Delta_{\mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator
on $\mathbb{B}^N$, $Q \in C^{1}(\mathbb{R}\times \mathbb{R},\mathbb{R})$ is a
p-homogeneous function, $d(x)=d_{\mathbb{B}^N}(0,x)$ and $K\geq0 $ is a
continuous function.
  We prove a compactness result and together with the Clark's theorem we
establish the existence of infinitely many solutions.
