The only thing that seems to have kept the Ottomans alive past the 18th century were the French and British who saw it as a buffer against Austria and particularly Russia. By 1914, the empire had a population of 21 million, similar to that of Spain (coincidentally another former great power). Of those 21 million, perhaps 11 million were ethnically Turkish, and 15.5 million were Muslim. In the age of ethnic-nationalism, keeping multi-ethnic/multi-religious empire was a difficult task at best.
Perhaps an early POD where nationalism doesn't take off in Europe and the empire continues some sort of prosperity. By 1900, it seems that the population was overwhelmingly illiterate, especially in the Balkans. The economy remained stuck in a mediaeval state into the 20th century and this would need to change.