In this paper, we study the employment of power-domain non-orthogonal multiple access (NOMA) concepts for a cooperative cognitive relay network (CCRN) downlink system in order to allow a base station (BS) to simultaneously transmit signals to a primary user (PU) and a secondary user (SU). As such, the considered system falls into the field of cognitive radio inspired power-domain NOMA. In this scheme, the interference power constraint of the PU imposed to SUs in conventional underlay CCRNs is replaced by controlling the power allocation coefficients at the BS and relay. Specifically, expressions for the symbol error rates at the PU and SU for different modulation schemes as well as expressions for the achievable rates are derived. On this basis, the effect of system parameters such as total transmit power and power allocation coefficients on the performance of the CCRN with power-domain NOMA is numerically studied. These numerical results provide insights into selecting favorable operation modes of the CCRN employing power-domain NOMA. © 2018 IEEE.