The Ottoman Empire didn't really react at all. Information likely would've only become widespread with the publication of Waldseemueller's map in 1507. By this time the Ottomans were staring down a religious revolt in eastern Anatolia by the Qizilbashi and Shahkulu. After this revolt flared up and was put down, the empire spiraled into a succession crisis. By the time that was settled, and Selim I had ousted his father, it was only a short time before Selim channeled all his fury towards Ismail I of Persia. And even as that war ended, another war, this time with the Mamluks, flared up. Selim went to war with the argument of "he who aids the heretic [Ismail's predominantly Shia state of Persia] is himself a heretic," met an unexpected success, and would extend that campaign long enough to annex the entirety of the Mamluk sultanate.

This massive expansion, and Selim's rather early death shortly thereafter, forced his successor, Suleyman, to spend years dealing with internal issues -- an uprising in Syria, the codification of laws, the attempt to balance the sultan's new religious responsibilities with his secular responsibilities, and, of course, the numerous campaigns that Suleyman would embark upon within his first two decades of rule.

There was a somewhat similar topic a short while ago that I also responded to. You might find that comment chain interesting.