We investigate a joint energy-aware power and rate control scheme for overlay spectrum sharing cognitive radio networks, and we expect to achieve an optimal performance tradeoff between individual fairness and network energy efficiency. Therefore, a quality of energy efficiency-based Nash bargaining cooperative game (QoE2-NBCG) is formulated with the aid of a newly-built ‘energy-per-bit’ individual utility function in this paper. Mathematically, a multi-constrained optimisation problem is formulated with the novel individual QoE2 utility function. To reduce the implementation complexity, we utilise decomposition techniques to decouple this multi-constrained problem. Meanwhile, an iterative dynamic bargaining algorithms (IDBA) with limited information exchange is developed based on the derived closed-form of power and rate solutions. In addition, the existence and uniqueness of socially optimal Nash bargaining solutions are validated via theorems. Numerical results verify the convergence of the proposed algorithm, the overall performance improvement and the optimal tradeoff between transmission rate and ‘energy-per-bit’ efficiency.