By fully exploiting the mobility of unmanned aerial vehicles (UAVs), UAV-based aerial base stations (BSs) can move closer to ground users to achieve better communication conditions. In this paper, we consider a scenario where an aerial BS is dispatched for satisfying the data request of a maximum number of ground users, weighted according to their data demand, before exhausting its on-board energy resources. The resulting trajectory optimization problem is a mixed integer non-linear problem (MINLP) which is challenging solve. Specifically, there are coupling constraints which cannot be solved directly. We exploit a penalty decomposition method to reformulate the optimization formulation into a new form and use block coordinate descent technique to decompose the problem into sub-problems. Then, successive convex approximation technique is applied to tackle non-convex constraints. Finally, we propose a double-loop iterative algorithm for the UAV trajectory design. In addition, to achieve a better coverage performance, the problem of designing the initial trajectory for the UAV trajectory is considered. In the results section, UAV trajectories with the proposed algorithm are shown. Numerical results show the coverage performance with the proposed schemes compared to the benchmarks.