### Motivation, Contribution, and Organization  
The 3 GPP committed to developing 6G specifications from December 2023 onward. Heterogeneous network paradigms such as D2D communications are expected to expand significantly in the 6G era [25]. However, introducing new technologies like terahertzcommunication will substantially increase network infrastructures and connected devices, which will, in turn, drive up energy costs [29]. Moreover, there is no doubt that 6G will have increasingly diversified and stringent requirements for the QoS of smartphones, whose standby time is limited by the capacity of lithium-ion batteries. Thus, it is crucial to reduce and balance users' energy consumption while ensuring their QoS requirements in energy-limited communication systems. Firstly, contrary to the concept of green communication, some studies have placed a one-sided emphasis on maximizing SE, whether in terms of system throughput or user sum rates, e.g., [12; 13; 17; 18; 19; 20; 21]. Secondly, schemes maximizing EE or WSEE under the FP framework require stringent prerequisites and result in high computational complexity, e.g., [14; 15; 16]. Thirdly, although complex FP problems such as MIFP can achieve near-optimal solutions by leveraging DL, ML tasks depend on extensive training data and considerable computing resources. These demands can result in increased energy costs, e.g., [27; 28]. Finally, a considerable portion of the research effort primarily focuses on maximizing the performance of the secondary system while guaranteeing PUs' minimum QoS requirements, e.g., [12; 14; 16; 19; 27]. In the upcoming 6G era, the primary focus will be reducing and balancing terminal energy consumption while satisfying users' QoS requirements, thereby extending network lifetime as much as possible [1; 10]. This paper proposes weighted sum power (WSP) as a novel objective function for overlay spectrum sharing. WSP is defined as the sum of weighted users' power consumption, addressing resource allocation challenges while aligning with 6G's EE goals. Unlike WSEE in multiple-ratio FP form, WSP minimization can typically be converted into a convex form. Additionally, because WSP is directly related to users' power consumption, it allows for effectively managing their power budgets by dynamically adjusting the corresponding weights. To sum up, the trade-offs between WSP minimization, EE maximization, WSEE maximization, and other energy-related objectives are indeed critical in 6G green communication systems. WSP minimization focuses on reducing and balancing absolute power consumption across users. This aligns with 6G's emphasis on extending terminal battery life and managing energy costs in dense networks. Its linear formulation enables low-complexity convex optimization, making it suitable for dynamic resource allocation in heterogeneous networks. EE maximization focuses on system EE, which optimizes the ratio of throughput to power consumption. WSEE maximization prioritizes EE per user, which ensures fairness in energy utilization across diverse user requirements (e.g., high-rate vs. low-power devices). Their FP nature introduces non-convexity, leading to higher computational complexity and stringent prerequisites for convergence. In scenarios with strict latency requirements (e.g., industrial IoT), WSP minimization ensures timely transmissions by prioritizing power budget adherence, whereas EE or WSEE maximization might sacrifice latency to maintain high EE ratios. Therefore, WSP minimization is particularly advantageous for 6G's energy-limited, latency-sensitive, and heterogeneous environments. While EE or WSEE maximization remains popular for most use cases, the simplicity, scalability, and alignment with 6G's green communication goals make WSP minimization a pragmatic choice. With WSP as the objective function, we jointly optimize the performance of both PUs and SUs.