Power domain Non-orthogonal multiple access (NOMA) is a candidate multiple access technique for future wireless networks, which can improve spectrum efficiency and user connectivity. NOMA requires superposition coding (SC) at the transmitter side and successive interference cancellation (SIC) at the receiver side that can be achieved with reasonable system complexity. A heterogeneous network (Het-net) assisted with unmanned aerial vehicles (UAVs) is considered an indispensable part of the future wireless networks. Hence, weighted max-min fairness (MMF) data rate-based power allocation optimization for UAV-assisted networks in a heterogeneous cellular system is studied. Since power allocation for weighted MMF-based optimization is a non-convex problem, we characterized Perron-Frobenius (PF) theory-based optimization algorithm that converges very fast. The theoretical base and simulation results with imperfect SIC conditions and different weights for user data rates are presented.