Nonlinear PDEs were mentioned above, but there is also an old example from linear PDEs (I learned about this from Boris Paneah). In a 1939 paper (link http://www.ams.org/journals/bull/1939-45-12/S0002-9904-1939-07103-6/S0002-9904-1939-07103-6.pdf) Bourgin and Duffin study the Dirichlet problem for the wave equation on a rectangle with sides A and B. They show that the uniqueness of the problem depends on the ratio A/B being irrational. More strikingly, they show that the existence of a solution (with certain regularity) depends on how difficult it is to approximate A/B by rational numbers.
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