Can bitcoin exist without miners? In the past year only, the bitcoin mining difficulty has increased fivefold. If in the coming years the difficulty increases so much that mining is no longer profitable (i.e. consumes more power than money is earned), can the bitcoin network exist without miners? Can bitcoin supporters or governments have data centers that handle transactions only and do not spend computational resources on mining new bitcoins?
if in the coming years the difficulty increases so much that mining is no longer profitable


That's not really possible. The mining power is set so that the miners need 10 minutes in average to mine a block. If 50% of the miners would disappear because it's not profitable any more, the difficulty would decrease so that it's profitable again. 


  Can the bitcoin network exist without miners?


Yes and no. There is still the decentralized blockchain, so you can still check who has how many bitcoins, and you can view the whole history of transactions (they do still exist), but you could not spend them if nobody would mine the transactions.


  Can bitcoin supporters, or governments have data centers that handle transactions only and do not spend computational resources on mining new bitcoins?


Transactions can't be confirmed if nobody spends the necessary processing power to mine blocks. Confirming transactions requires mining.