To improve the radio spectral efficiency for 5G and beyond, novel radio access techniques need to be designed to accommodate unprecedented number of connected devices, and nonorthogonal multiple access (NOMA) has become a promising candidate. Additionally, power allocation and NOMA-secondary user (SU) assignment technology is an efficient way to enhance the resource utilization efficiency at the power domain and the spectral domain for underlaying cognitive NOMA networks. In this article, first, a joint power allocation and SU assignment problem is formulated for the NOMA downlink transmission in an underlaying cognitive radio network. The worst-case achievable rate for the NOMA-SU is maximized. To solve this mixed-integer nonlinear programming problem, we divide the original optimization problem into two subproblems: NOMA-SU assignment and power allocation. Next, a heuristic algorithm is adopted to solve the NOMA-SU assignment subproblem, and successive convex approximation based method is utilized to design a suboptimal power allocation algorithm. Furthermore, an alternative joint NOMA-SU assignment and power allocation scheme are proposed with its average computational complexity analysis given. Finally, numerical results show that the total throughput for the proposed algorithm outperforms more than 30% compared with an existing benchmark scheme at least.