In scenarios where excessive network resources are needed temporarily, such as crowded hotspots, sports events and accidents, existing base stations (BSs) cannot meet the demand of users. Alternatively, due to the easy deployment and bird's-eye view, unmanned aerial vehicles (UAVs) can be deployed as agents between the BS and users to expand the capacity of mobile social networks (MSNs), whereby UAVs download contents from the BS, and deliver contents to mobile users via UAV-user link. However, the introduction of UAVs has brought some challenges, i.e., UAVs are limited by battery capacity and power, and malicious users in MSNs cheat UAVs for selfish benefits. To address these challenges, firstly, we develop a trust mechanism and power allocation scheme for the UAV-assisted system. The trust mechanism is divided into two phases. In the first phase, a UAV, as an agent, removes malicious users from content requesters. In the second phase, the UAV selects the honest users who are eligible to participate in content distribution from the remaining users. Secondly, the power allocation scheme based on the Stackelberg game is designed for UAVs to allocate the transmission power to users, maximizing UAVs' and users' utility functions. Finally, the extensive simulation results demonstrate that the proposed scheme can efficiently improve the utility of UAVs.