The sixth generation (6G) network is envisioned to cover remote areas, with the help of satellites and unmanned aerial vehicles (UAVs). Considering the vastness of remote areas and the sparsity of users therein, we investigate a cognitive satellite-UAV network, where satellites and UAVs coordinately share spectrum to provide low-rate and high-rate services in a complementary manner. Multiple UAVs form a virtual antenna array to serve unevenly distributed users via multipleinput-multiple-output (MIMO) non-orthogonal multiple access (NOMA). An on-demand coverage framework is proposed so as to dynamically focus the communication resources on target users. In the framework, a radio map recording the slowlyvarying large-scale channel state information (CSI) is utilized. Different from traditional pilot-based approaches, the large-scale CSI is obtained by a lookup in the radio map per the position information of users and UAVs, during the online optimization of the network. In this way, the system overhead could be largely reduced. To explore the potential gain of such a framework, we formulate a joint power allocation problem to maximize the minimum user rate, which is not only non-convex but also with implicit expressions. We recast the problem after uncovering its mathematical characteristics, and derive its locally-optimal solution in an iterative manner. Simulation results corroborate that the proposed framework can significantly improve the coverage performance at a low cost.Index Terms-Cognitive satellite-UAV network, large-scale channel state information (CSI), non-orthogonal multiple access (NOMA), on-demand coverage, unmanned aerial vehicle (UAV).