In this paper, we study an unmanned-aerial-vehicle (UAV) based full-duplex (FD) multi-user communication network, where a UAV is deployed as a multiple-input–multiple-output (MIMO) FD base station (BS) to serve multiple FD users on the ground. We propose a multi-objective optimization framework which considers two desirable objective functions, namely sum uplink (UL) rate maximization and sum downlink (DL) rate maximization while providing quality-of-service to all the users in the communication network. A novel resource allocation multi-objective-optimization-problem (MOOP) is designed which optimizes the downlink beamformer, the beamwidth angle, and the 3D position of the UAV, and also the UL power of the FD users. The formulated MOOP is a non-convex problem which is generally intractable. To handle the MOOP, a weighted Tchebycheff method is proposed, which converts the problem to the single-objective-optimization-problem (SOOP). Further, an alternative optimization approach is used, where SOOP is converted in to multiple sub-problems and optimization variables are operated alternatively. The numerical results show a trade-off region between sum UL and sum DL rate, and also validate that the considered FD system provides substantial improvement over traditional HD systems.