In this paper, we investigate the achievable energy efficiency of cognitive radio networks where two main modes are of interest, namely, spectrum sharing (known as underlay paradigm) and spectrum sensing (or interweave paradigm). In order to improve the energy efficiency, we formulate a new multiobjective optimization problem that jointly maximizes the ergodic capacity and minimizes the average transmission power of the secondary user network while limiting the average interference power imposed on the primary user receiver. The multiobjective optimization will be solved by first transferring it into a single objective problem (SOP), namely, a power minimization problem, by using the ε-constraint method. The formulated SOP will be solved using two different methods. Specifically, the minimum power allocation at the secondary transmitter in a spectrum sharing fading environment are obtained using the iterative search-based solution and augmented Lagrangian approach for single and multiple secondary links, respectively. The significance of having extra side information and also imperfect side information of cross channels at the secondary transmitter are investigated. The minimum power allocations under perfect and imperfect sensing schemes in interweave cognitive radio networks are also found. Our numerical results provide guidelines for the design of future cognitive radio networks.