Infinitely many solutions for a Hénon-type system in hyperbolic space

Abstract

This paper is devoted to study the semilinear elliptic system of Hénon-type \begin{eqnarray*} -Δ_{\mathbb{B}^{N}}u= K(d(x))Q_{u}(u,v) \\ -Δ_{\mathbb{B}^{N}}v= K(d(x))Q_{v}(u,v) \end{eqnarray*} in the hyperbolic space ${\mathbb{B}}^N$, $N\ge 3$, where $u,v\in H_r^1({\mathbb{B}}^N)=\{\varphi \in H^1({\mathbb{B}}^N):\varphi \text{is radial}\}$ 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 $K\ge 0$ is a continuous function. We prove a compactness result and together with the Clark's theorem we establish the existence of infinitely many solutions.

12 pages