In this paper, we prove the existence of a sequence of nonnegative (weak) solutions for the following problem $\left\{\begin{array}{cc}\hfill & -{\Delta }_{H}u+u=\lambda \alpha \left(\sigma \right)f\left(u\right)\mathrm{in}{\mathbb{B}}^N\hfill \\ \hfill & u\in H^{1,2}\left({\mathbb{B}}^N\right),\hfill \end{array}\right.$where ${\Delta }_H$ denotes the Laplace–Beltrami operator on the Poincaré ball model ${\mathbb{B}}^N$ (with $N\ge 3$) of the hyperbolic space ${\mathbb{H}}^N$, $\alpha \in L^1\left({\mathbb{B}}^N\right)\cap L^{\infty }\left({\mathbb{B}}^N\right)$ is a nonnegative and not identically zero radially symmetric potential, $f$ is a suitable continuous function, and $\lambda$ is a positive real parameter. The analysis developed in this paper combines a compactness embedding result due to Skrzypczak and Tintarev [(2013), Theorem 1.3 and Proposition 3.1], some group-theoretical arguments on the Poincaré ball model ${\mathbb{B}}^N$, and variational methods for smooth functionals defined on the Sobolev space $H^{1,2}\left({\mathbb{B}}^N\right)$ associated to the homogeneous Hadamard manifold ${\mathbb{B}}^N$.

在本文中，我们证明了以下问题的一系列非负（弱）解的存在 $\left\{\begin{array}{cc}\hfill & -{\Delta }_Hü+ü=\lambda \alpha （\sigma ）F（ü）\mathrm{在}{\mathbb{乙}}^{ñ}\hfill \\ \hfill & ü\in H^{\mathrm{1个}，2}（{\mathbb{乙}}^{ñ}），\hfill \end{array}\right.$哪里 ${\Delta }_H$ 表示Poincaré球模型上的Laplace–Beltrami运算符 ${\mathbb{乙}}^{ñ}$ （与 $ñ\ge 3$的双曲空间 ${\mathbb{H}}^{ñ}$， $\alpha \in {\mathrm{大号}}^{\mathrm{1个}}（{\mathbb{乙}}^{ñ}）\cap {\mathrm{大号}}^{\infty }（{\mathbb{乙}}^{ñ}）$ 是一个非负且不完全相同的零径向对称电势， $F$ 是合适的连续函数，并且 $\lambda$是一个正实参数。本文开发的分析结合了Skrzypczak和Tintarev [（2013），定理1.3和命题3.1]，关于庞加莱球模型的一些群理论论证的紧致嵌入结果。${\mathbb{乙}}^{ñ}$以及Sobolev空间上定义的平滑函数的变分方法 $H^{\mathrm{1个}，2}（{\mathbb{乙}}^{ñ}）$ 与均质Hadamard流形相关 ${\mathbb{乙}}^{ñ}$。