This paper investigates the admission control problem on the satellite multi-beam networks with non-orthogonal multiple access (NOMA). The goal is to maximize the number of supported users on the premise of ensuring the quality of service (QoS) by optimizing the subchannel and power allocation. We provide the system model and then formulate the admission control problem as a mixed integer non-convex optimization problem. The non-convexity and existence of integer variable make the optimal solution difficult to get. Therefore, we propose a joint subchannel matching and power allocation algorithm to obtain the suboptimal solution so as to reduce the computation complexity. The proposed algorithm can be used for both NOMA and orthogonal frequency division multiplexing access (OFDMA). Specifically, the subchannel matching problem is solved by a two-stage matching process where users are accessed to subchannel dynamically. The power allocation problem is modeled as a super-modular game where the existence and uniqueness of Nash equilibrium (NE) are analyzed. Moreover, an iterative power allocation algorithm is proposed based on the NE searching method. Finally, simulation results are provided for demonstrating the effectiveness and feasibility of the proposed algorithm. INDEX TERMS Admission control, matching theory, multi-beam satellite system, non-orthogonal multiple access, subchannel, power allocation.