The data volume is exploding due to various newly-developing applications that call for stringent communication requirements towards 5th generation wireless systems. Fortunately, mobile edge computing makes it possible to relieve the heavy computation pressure of ground users and decrease the latency and energy consumption. What is more, the unmanned aerial vehicle has the advantages of agility and easy deployment, which gives the unmanned aerial vehicle enabled mobile edge computing system opportunities to fly towards areas with communication demand, such as hotspot areas. However, the limited endurance time of unmanned aerial vehicle affects the performance of mobile edge computing services, which results in the incomplete mobile edge computing services under the time limit. Consequently, this paper concerns the energy-efficient scheme design of the unmanned aerial vehicle while providing high-quality offloading services for ground users, particularly in the regions where the ground communication infrastructures are overloaded or damaged after natural disasters. Firstly, the model of energy-efficient design of the unmanned aerial vehicle is set up taking the constraints of the energy limitation of the unmanned aerial vehicle, the data causality, and the speed of the unmanned aerial vehicle into account. Subsequently, aiming at maximizing the energy efficiency of the unmanned aerial vehicle in the unmanned aerial vehicle enabled mobile edge computing system, the bits allocation in each time slot and the trajectory of the unmanned aerial vehicle are jointly optimized. Secondly, a successive convex approximation based alternating algorithm is brought forward to deal with the non-convex energy efficiency maximization problem. Finally, it is proved that the proposed energy efficient scheme design of the unmanned aerial vehicle is superior to other benchmark schemes by the simulation results. Besides, how the performance of proposed scheme design change under different parameters is discussed.