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

Abstract

Abstract This paper is devoted to studying the semilinear elliptic system of Hénon type {−ΔBNu=K(d(x))Qu(u,v),−ΔBNv=K(d(x))Qv(u,v),u,v∈Hr1(BN),N≥3, $$ \textstyle\begin{cases} -\Delta _{\mathbb{B}^{N}}u= K(d(x))Q_{u}(u,v), \\ -\Delta _{\mathbb{B}^{N}}v= K(d(x))Q_{v}(u,v), \\ \quad u, v\in H_{r}^{1}(\mathbb{B}^{N}),\quad N\geq 3, \end{cases} $$ in the hyperbolic space BN $\mathbb{B}^{N}$, where Hr1(BN)={u∈H1(BN):u is radial} $H_{r}^{1}(\mathbb{B} ^{N})=\{u\in H^{1}(\mathbb{B}^{N}): u \text{ is radial}\}$ and −ΔBN $-\Delta _{\mathbb{B}^{N}}$ denotes the Laplace–Beltrami operator on BN $\mathbb{B}^{N}$, d(x)=dBN(0,x) $d(x)=d_{\mathbb{B}^{N}}(0,x)$, Q∈C1(R×R,R) $Q \in C^{1}( \mathbb{R}\times \mathbb{R},\mathbb{R})$ is p-homogeneous, and K≥0 $K\geq 0 $ is a continuous function. We prove a compactness result and, together with Clark’s theorem, we establish the existence of infinitely many solutions.

Keywords

• Hyperbolic space
• Hénon equation
• Variational methods