The advancement of the Internet of Things (IoT) and the availability of wide cloud services have led to the horizon of edge computing paradigm which demands for processing the data at the edge of the network. The development of 5G technology has led to the increased usage of IoT-based devices and the generation of a large volume of data followed by increased data traffic, which is difficult to process by the mobile edge computing (MEC) platform. The latest inventions related to unmanned aerial vehicles (UAVs) helps to assist and replace the edge servers used for MEC. In the present work, the objective is to develop self-adaptive trajectory optimization algorithm (STO) which is a multi-objective optimization algorithm used to solve the vital objectives associated with the above scenario of a UAV-assisted MEC system. The objectives identified are minimizing the energy consumed by the MEC and minimizing the process emergency indicator, where the process emergency indicator implies the urgency level of a particular process. Finding the optimal values for these conflicting objectives will help to further efficiently apply UAV for MEC systems. A self-adaptive multi-objective differential evolution-based trajectory optimization algorithm (STO) is proposed, where a pool of trial vector generation strategies is extended. The strategies and the crossover rate associated with a differential evolution (DE) algorithm are self-adapted using fuzzy systems to improve the population diversity. The experimentation is planned to be conducted on hundreds of IoT device instances considered to be fixed on the ground level and to evaluate the performance of the proposed algorithm for a single unmanned aerial vehicle-assisted mobile edge computing system.