How did the Europeans compete with the Asian empires that had already been built?
In general, they competed with the Asian empires by having superior technology. Through the use of this technology, they tended to overwhelm the Asian empires. A good example of this would be the way that the British competed with the Chinese. China had been an empire for a long time, but the British were able to come in and dictate terms to the Chinese because of their superior weaponry. The British were able, for example (in the Opium Wars), to force the Chinese to allow the importation of opium to China. They were able to threaten the Chinese with destruction because of their ships and cannons and other technology. In this way, the Europeans were able to compete with established Asian empires.