Infinitely many solutions for a H\'enon-type system in hyperbolic space

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.