5G can provide the multiplicative capacity gains needed to support a large number of drones/UAS (Unmanned Aircraft Systems). 5G cellular networks with newly available millimeter wave (mmWave) frequency bands can provide wireless communication links for control as well as data traffic for drones and drone swarms. Drones are becoming increasingly important for commercial uses such as delivery and transportation as well as for public safety search and rescue of natural disaster victims, surveillance of remote critical infrastructure, surveys of environmental quality in protected regions, and detection of threats during major public events. This paper presents research findings in the following areas critical to validating the effectiveness of providing required 5G access to the drones with security, reliability, and spectral efficiency: 1) Radio coverage for the drone corridor by adding a separate set of antennas for coverage in the air while the conventional set of antennas continues to provide coverage on the ground. Beam transmission and validation with ray-tracing simulations are covered. 2) Optimization of uplink communication from a swarm of drones with a single mmWave beam by grouping the drones with power allocations for non-orthogonal multiple access (NOMA). 3) Optimization of the network lifetime of a swarm of drones resulting in suitable trajectories in the presence of interference. 4) Methods including precoding that can enhance physical layer security with channel information about the interference source. The paper concludes with plans for future research to provide further scientific basis for the proposed cellular drone network.