Low earth orbit (LEO) satellites are a key technology to enable connectivity for rural and remote users. Communication satellites in LEO can provide coverage to much larger areas than terrestrial or aerial systems, while offering improved data rates when compared with geostationary systems. However, a major challenge with LEO satellite communications is the high mobility of the satellite, which results in a rapidly changing communication channel. Due to this, it is challenging to fairly allocate communication resources to multiple users in the system. This work proposes an Adaptive Power Allocation and Scheduling Scheme (APASS) to ensure user fairness in the downlink of a LEO satellite system serving mobile ground users. First, a novel channel and transmission model is introduced to capture the variability in channel statistics due to the satellite's trajectory. Then, a non-convex optimization problem is formulated to maximize the minimum rate across all ground users over a fixed set of time slots. To solve this problem, the proposed APASS dynamically allocates power and schedules transmissions based on predicted future channel gains. Numerical results show that APASS achieves strong performance even with substantial prediction errors, faring close to an upper bound that assumes perfect future channel knowledge. Furthermore, it improves the minimum user rate by a factor of 2.98 compared to equal-power allocation and maintains user fairness with a Jain's fairness index of well above 0.99.