Referring this question to Dr. Hatem  Al–Haj, he said:

In places where there is no day or night, the Muslims will ‎fast according to the times of the closest city that has day and ‎night.

For those who find it incapacitated by fasting during one of ‎those days with long hours, and it is truly so (Allah is the best ‎observer), they are allowed to break their fast and make that day ‎up at some other time of the year, for Allah doesn’t burden souls ‎beyond their capacity.‎

Allah said what means:‎ “Observing Saum (fasts)] for a fixed number of days, but if‎any of you is ill or on a journey, the same number (should be‎made up) from other days. And as for those who can fast with‎difficulty, (e.g. an old man), they have (a choice either to fast‎or) to feed a Miskin (poor person) (for every day). But whoever‎does good of his own accord, it is better for him. And that you‎fast is better for you if only you know.” [2:184]