The rapid advancement of information and communication technology (ICT) has made the industry a significant contributor to global carbon emissions. As the foundation of ICT, next-generation mobile communication networks aim to be more powerful and energy-efficient. However, optimizing energy efficiency in real networks is challenging due to large-scale, multi-layer control variables and the dynamic environment. This paper addresses the energy efficiency optimization problem from a network perspective by controlling cross-layer variables including both cell activation status and cell priority, to minimize overall network energy consumption while ensuring user quality-of-experience, which poses an NP-hard mixed-integer nonlinear programming problem. To tackle this, we propose a non-cooperative gam where each cell acts as a player, optimizing its activation status and reference signal transmission power (determining its priority). We show that the game is a potential game, guaranteeing the existence of Nash equilibrium and the convergence of simple distributed algorithms towards Nash equilibrium. We further show that the Nash equilibrium points of the game can (but not always) reach the global optimal energy efficiency. Simulation results show that our proposed method can reduce the total system cost (including both energy consumption cost and user experience loss) by up to 28% compared to existing methods in the literature. Moreover, the performance loss of our proposed method, compared to the global optimal solution, is less than 13.7%. In summary, this work offers a realistic network model, introduces a novel game-based method, and provides extensive performance evaluation, making a significant contribution to both industry and academia.