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To cope with the unprecedented surge in demand for data computing for the applications, the promising concept of multi-access edge computing (MEC) has been proposed to enable the network edges to provide closer data processing for mobile devices (MDs). Since enormous workloads need to be migrated, and MDs always remain resource-constrained, data offloading from devices to the MEC server will inevitably require more efficient transmission designs. The integration of nonorthogonal multiple access (NOMA) technique with MEC has been shown to provide applications with lower latency and higher energy efficiency. However, existing designs of this type have mainly focused on the transmission technique, which is still insufficient. To further advance offloading performance, in this work, we propose an application-driven NOMA enabled computation offloading scheme by exploring the characteristics of applications, where the common data of the application is offloaded through multi-device cooperation. Under the premise of successfully offloading the common data, we formulate the problem as the maximization of individual offloading throughput, where the time allocation and power control are jointly optimized. By using the successive convex approximation (SCA) method, the formulated problem can be iteratively solved. Simulation results demonstrate the convergence of our method and the effectiveness of the proposed scheme.