One major problem the Aztecs dealt with was the spread of diseases. Some of these were known as measles, smallpox, the influenza virus, and mumps.
Other issues include the struggle of such feeding a large population. There were also ritual sacrifices, political problems through rivalry and prejudice and technological problems.
Lastly, the fall of the Aztec empire was because Spanish conquistador, Hernán Cortés, invading the empire. It was done in 1521 and it resulted in the end of the Aztec civilization.
Hope this helped :))