###### Abstract  
In this paper, we consider a novel cooperative underlay cognitive radio network based on non-orthogonal multiple access (CR-NOMA) with adaptive relay selection and power allocation. In secondary networks, dedicated relay assistance and user assistance are used to achieve communication between the base station and the far (and near) user. Here, a two-stage adaptive relay selection and power allocation strategy is proposed to maximize the achievable data rate of the far user while ensuring the service quality of near user. Furthermore, the closed-form expressions of outage probability of two secondary users are derived, respectively, under interference power constraints, revealing the impact of transmit power, number of relays, interference threshold and target data rate on system outage probability. Numerical results and simulations validate the advantages of the established cooperation and show that  
desired full diversity gain [14; 15]. In [9], a two-stage max-min relay selection strategy based on the cooperative NOMA system was proposed to ensure the quality of service (QoS) requirements of users. However, the outage performance obtained by using the fixed power allocation scheme is not optimal. Authors in [10] developed a two-stage relay selection strategy based on decode-and-forward (DF) and amplify-and-forward (AF) relays with adaptive power allocation. This scheme not only obtained full diversity gain, but also reduced the outage probability of the cooperative NOMA system. Two relay selection strategies named two-stage weighted-max-min (WMM) with fixed power allocation and max-weighted-harmonic-mean (MWHM) with adaptive power allocation were proposed in [11], and research results confirmed that the reasonable power allocation coefficient has an important effect on system performance. To extend the work of [10; 11], researchers in [16] established an optimization model by taking rate fairness and imperfect channel into consideration. However, the above cooperative NOMA systems considered user cooperation and relay cooperation separately. It is obvious that if both user cooperation and dedicated relay cooperation are adopted, diversity gain and overall system performance will be improved. Coordinated direct and relay transmission into NOMA systems was introduced by [17], where the far user obtained the side information from the relay or near user. Compared with non-coordinated direct and relay transmission, the scheme in [17] achieved greater capacity gain. An AF relay selection strategy was proposed for a two-user downlink NOMA system in [18], where the near user was likely to be the optimal relay if it could decode the superimposed signal successfully; otherwise, an optimal relay was selected to serve both users. However, power adaptation was not considered in [17; 18]. Therefore, the authors in [6] extended the model of [18] and proposed a two-stage relay selection scheme with power adaptation to obtain \(N+1\) full diversity gain for users, where \(N\) is the number of relays. In addition, dynamic spectrum or bandwidth allocation effectively improves the utilization of network resources by rationalizing their allocation [19]. For example, cognitive radio is a classical technology to solve the problem of spectrum scarcity. The integration of CRNs and NOMA could further improve spectrum efficiency, and the core idea of CR-NOMA is to provide opportunistic services for secondary users while ensuring the QoS of primary users. Recently, many works have been carried out to evaluate and analyze the performance of cooperative CR-NOMA networks, in which relay selection and power allocation are the focus. Overlay CR-NOMA networks with user cooperation were considered in [20; 21]. In [20], the signals of primary users and secondary users were relayed by secondary transmitters; while in [21], the optimal decoded secondary user was selected to relay the signals of undecoded users.