In this paper, we analyze the performance of an underlay downlink non-orthogonal multiple access (NOMA) network with multiple near and far-users. Different from cognitive radio (CR) inspired CR-NOMA where cognitive principles are used to guarantee performance of one selected user, all nodes here are in the underlay mode, and re-use the spectrum of the primary network. Use of NOMA is counter-intuitive in an underlay network since the transmit powers are constrained to ensure that the interference temperature limit is met. We show in this paper that switching carefully between NOMA, NOMA with relaying, and orthogonal multiple access (OMA) allows good performance to be attained in such networks. We further propose a user selection scheme, and analyze its performance. We derive approximate closed-form expressions for the near and far-user throughput. Simulations are carried out to verify the derived results.