This study presents a mobile edge computing (MEC)-enabled UAV communication system, where a number of UAVs are served by terrestrial base stations (TBSs) equipped with computation resource in the non-orthogonal multiple access manner. Each UAV has to offload its computing tasks to the proper TBS due to the limited energy supply. For this, the authors aim at minimising the sum of transmission energy of UAVs and computation energy of TBSs through jointly optimising the UAV transmit power, computation resource allocation, and UAV grouping. Considering the non-convexity of this optimisation problem, they obtain the optimal solution in the coupled steps: the convex resource allocation optimisation and the combinatorial UAV grouping optimisation. By exploiting the convex nature of the resource allocation optimisation problem, they obtain the optimal transmit power and computation allocation based on the KKT conditions and the idea of gradient descent method when considering a single TBS. Then, they adopt the simulated annealing to obtain the optimal UAV grouping and TBS selection based on the proposed resource allocation optimisation algorithm. Finally, simulation results show that the proposed joint optimisation of transmit power, computation resource allocation, and UAV grouping can effectively reduce the energy consumption of MEC-aware UAV communication system.