Generally it's a good idea to 3-bet pre-flop with AK because it gives you more information. You force the initial raiser to give you more information about his hand, i.e. does he fold, flat your 3-bet, or 4-bet you, and based on this and anything else you know you can start to narrow his range a bit. It also discourages speculative hands in later position or in the blinds from coming along because of pot odds
, especially since the opener just min-raised.
Usually (but not always) people in later position or in the blinds who throw in a "cold 4-bet" pre-flop have very strong hands. In this case, I would have folded because in general you're not going to be any kind of favorite with that much action going on pre-flop. If you have AK and you're against say AQ and any PP going to the showdown, you're not going to come out ahead the majority of the time and 39BB is too much to commit here, given that you don't know if the button is going to force you to go all-in pre-flop if you just call.