Positive solutions for Schrödinger-Poisson system with singularity on compact Riemannian manifolds

In this paper, we consider the following Schrödinger-Poisson system with singularity$ \begin{cases}\Delta_g u+\beta(x) u+\eta u \phi = \mu u^{-r}, & \text { in } M, \\ \Delta_g \phi+\phi = q u^2, & \text { in } M, \\ u>0, & \text { in } M ,\end{cases} $where $ (M, g) $ is a 3-dimensional compact Riemannian manifold without boundary, $ \Delta_g = -\operatorname{div}_g (\nabla_g\cdot) $ is the Laplace-Beltrami operator, $ \eta = \pm1 $, $ r\in(0, +\infty ) $ is a constant, $ q>0 $ is a positive number, and $ \mu>0 $ is a parameter. The function $ \beta \in C^{2}(M) $ satisfies the condition that $ \Delta_g+\beta $ is coercive. Employing the $ \varepsilon $-approximation techniques and variational methods, we establish the existence, uniqueness, multiplicity, and nonexistence of positive solutions, subject to appropriate conditions.