  This paper proposes an energy-efficient RIS-assisted downlink NOMA
communication for LEO satellite networks. The proposed framework simultaneously
optimizes the transmit power of ground terminals of the LEO satellite and the
passive beamforming of RIS while ensuring the quality of services. Due to the
nature of the considered system and optimization variables, the energy
efficiency maximization problem is non-convex. In practice, obtaining the
optimal solution for such problems is very challenging. Therefore, we adopt
alternating optimization methods to handle the joint optimization in two steps.
In step 1, for any given phase shift vector, we calculate satellite transmit
power towards each ground terminal using the Lagrangian dual method. Then, in
step 2, given the transmit power, we design passive beamforming for RIS by
solving the semi-definite programming. We also compare our solution with a
benchmark framework having a fixed phase shift design and a conventional NOMA
framework without involving RIS. Numerical results show that the proposed
optimization framework achieves 21.47\% and 54.9\% higher energy efficiency
compared to the benchmark and conventional frameworks.
