Unmanned aircraft vehicle (UAV) -assisted mobile edge computing (MEC) is an effective way to alleviate the lack of energy and computing power of the Internet of Things (IoT) devices in remote areas. This paper considers the competition among multiple aerial service providers (ASPs) to provide UAV-assisted MEC services to multiple ground network operators (GNOs). We first quantify the conflicting interests of the ASPs and GNOs by different profit functions. Then, the system-wide UAV scheduling and resource allocation is formulated as a multi-objective optimization problem, where an ASP aims to reap the most profits from providing MEC services to the GNOs, while a GNO aims to seek the service of a certain ASP to meet its performance requirements. This problem is a mixed-integer nonlinear programming (MINLP), and we propose a matching theory based algorithm to solve it. We first investigate the UAV trajectory planning and resource allocation between a single UAV and a single service area, and solve it using the Lagrange relaxation and successive convex optimization (SCA) methods. According to the obtained results, the GNOs and ASPs are associated in the framework of the matching theory, which results in a weak Pareto optimality. Simulation results show that the proposed matching theory based algorithms achieve the considerable performance.