What problem did the Austro-Hungarian Empire and the Ottoman Empire and subsequently the Soviet Union faced?

The three empires were composed of many different ethnicities that were often at odds with each other.

Explanation:

The Austro-Hungarian empire included Austrians and Hungarians, who dominated politically, but also included Croats, Serbs, Romanians, Bulgarians, among others. These groups had ethnic tensions, for language, religion, or cultural reasons.

The Ottomans comprised a Turkish elite who governed from Istanbul over Arabs, Levantines, Kurds, Egyptians, and Magrebies. Everyone knows that the Middle-East is a region brimming with ethnic conflict and the Ottoman years were not the exception.

The Soviet Union faced the same problem. For example, the Baltic States are inhabited by Baltic People, not Slavic People, and as a consequence of their invasion and annexation into the Soviet Union during the Second World War, they still hold grudges against Russians (the Soviet Union was dominated by Russians). Same applies for the Turkic Peoples of Central Asia, or the muslim minorities in the Caucasus (for example, the Chechens, who even fought a brief war against Russia in the 2000s).