In this paper, we propose to utilize multiple aerial intelligent reflecting surfaces (AIRSs) for significantly enhancing the ergodic capacity of a non-orthogonal multiple access (NOMA) network. Unlike previous works where only one AIRS was often utilized to support only two NOMA users, Q AIRSs and L users are applied in the proposed NOMA-AIRS network. We mathematically derive the ergodic capacity (EC) expressions at the multiple users of the proposed NOMA-AIRS network over Nakagami-m channels recommended for use in the fifth and beyond generations (5G & B5G) of mobile communications. Numerical results show that the proposed NOMA-AIRS network can deeply exploit the advantages of AIRSs. In particular, by using Q = 4 AIRSs where each AIRS has 20 reflecting elements (REs), the ECs of the proposed NOMA-AIRS network are significantly higher than the ECs of the conventional NOMA network (without AIRSs). Moreover, the ECs of the proposed NOMA-AIRS network can be two times higher than those of the conventional NOMA network in the low transmit power regime. However, if the transmit power is high enough, the benefits of AIRSs are reduced due to the NOMA features. Besides confirming the advantages of AIRSs, the effects of key parameters such as the number of REs and total of REs in the AIRSs, bandwidth, altitudes of AIRSs, and carrier frequency are thoroughly determined. Based on the achieved observations, we give some recommendations to enhance the EC performance of the proposed NOMA-AIRS network in practice scenarios.INDEX TERMS Non-orthogonal multiple access, aerial intelligent reflecting surface, multiuser systems, moment functions, ergodic capacity.