user70895: 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?

user30267: 
can the bitcoin network exist without miners?

Not really, the network may still exist reporting historical transactions and collecting unconfirmed transactions but without miners finding blocks transactions will never confirm.

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

No

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)

Understand that difficulty is reactive to the amount of mining power aiming to keep the block rate roughly constant. Increased difficulty is driven by increased mining, similarly decreased mining would result in decreased difficulty. So as long as changes to mining capacity are gradual the number of miners should have little impact on transactions.
Mining in turn is driven by economics, if mining is profitable more mining equipment will be brought online. If mining loses money then mining equipment will be taken offline. An equilibrium should be reached where the cost of mining is comparable to the income (mining rewards and transaction fees) from mining.
However there is a catch, the difficulty only adjusts every 2016 blocks and it can only adjust by a factor of 4 each time. If there is a sudden crash in the value of bitcoin (or in the post-reward era a sudden crash in transaction fees) a large proportion of mining infrastructure may suddenly become unprofitable. If the unprofitable infrastructure is shut down too quickly it could cause major problems for bitcoin.

user5406: Complete loss of mining would shut down transaction confirmation on Bitcoin. Transactions are confirmed by being included in a block of the blockchain. 
As the mining reward is released to the successful miner by the Coinbase transaction in the block they authored, it is not possible to confirm transactions without the expending the work of mining for the block subsidy.
Luckily, the difficulty should never grossly exceed the available hashrate, as the difficulty is a self-regulating system that adjusts every 2016 blocks. At worst, a portion of the hashrate would leave the network, making block intervals longer for some period until the difficulty adjustment normalizes the block intervals.
Herenby, the readjustment scales the difficulty up or down to a limit of factor 4, such that the next 2016 blocks would take 14 days if the hashrate were to stay the same as in the past difficulty period. 
As there were great advances in hashing hardware since Bitcoin emerged, so far the difficulty has been mostly adjusting upwards, and the average block interval is in average closer to nine minutes than ten.

user-1: This is an excellent question. I think when the last bitcoin is mined a century from now, it is conceivable that mining could be replaced by higher transaction fees. So the blockchains can keep growing and cryptocurrencies could live on forever.
Some interesting articles on the topic: 
https://news.bitcoin.com/what-happens-bitcoin-miners-all-coins-mined/
https://cryptocoinmastery.com/what-happens-when-all-bitcoins-have-been-mined/

user69241: 
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.

70895: <p>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?</p>
 30267: <blockquote>
  <p>can the bitcoin network exist without miners?</p>
</blockquote>

<p>Not really, the network may still exist reporting historical transactions and collecting unconfirmed transactions but without miners finding blocks transactions will never confirm.</p>

<blockquote>
  <p>Can bitcoin supporters or governments have data centers that handle transactions only and do not spend computational resources on mining new bitcoins?</p>
</blockquote>

<p>No</p>

<blockquote>
  <p>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)</p>
</blockquote>

<p>Understand that difficulty is reactive to the amount of mining power aiming to keep the block rate roughly constant. Increased difficulty is driven by increased mining, similarly decreased mining would result in decreased difficulty. So as long as changes to mining capacity are gradual the number of miners should have little impact on transactions.</p>

<p>Mining in turn is driven by economics, if mining is profitable more mining equipment will be brought online. If mining loses money then mining equipment will be taken offline. An equilibrium should be reached where the cost of mining is comparable to the income (mining rewards and transaction fees) from mining.</p>

<p>However there is a catch, the difficulty only adjusts every 2016 blocks and it can only adjust by a factor of 4 each time. If there is a sudden crash in the value of bitcoin (or in the post-reward era a sudden crash in transaction fees) a large proportion of mining infrastructure may suddenly become unprofitable. If the unprofitable infrastructure is shut down too quickly it could cause major problems for bitcoin.</p>
 5406: <p>Complete loss of mining would shut down transaction confirmation on Bitcoin. Transactions are confirmed by being included in a block of the blockchain. 
As the mining reward is released to the successful miner by the Coinbase transaction in the block they authored, it is not possible to confirm transactions without the expending the work of mining for the block subsidy.</p>

<p>Luckily, the difficulty should never grossly exceed the available hashrate, as the difficulty is a self-regulating system that adjusts every 2016 blocks. At worst, a portion of the hashrate would leave the network, making block intervals longer for some period until the difficulty adjustment normalizes the block intervals.<br>
Herenby, the readjustment scales the difficulty up or down to a limit of factor 4, such that the next 2016 blocks would take 14 days if the hashrate were to stay the same as in the past difficulty period. </p>

<p>As there were great advances in hashing hardware since Bitcoin emerged, so far the difficulty has been mostly adjusting upwards, and the average block interval is in average closer to nine minutes than ten.</p>
 -1: <p>This is an excellent question. I think when the last bitcoin is mined a century from now, it is conceivable that mining could be replaced by higher transaction fees. So the blockchains can keep growing and cryptocurrencies could live on forever.</p>

<p>Some interesting articles on the topic: </p>

<p><a href="https://news.bitcoin.com/what-happens-bitcoin-miners-all-coins-mined/" rel="noreferrer">https://news.bitcoin.com/what-happens-bitcoin-miners-all-coins-mined/</a></p>

<p><a href="https://cryptocoinmastery.com/what-happens-when-all-bitcoins-have-been-mined/" rel="noreferrer">https://cryptocoinmastery.com/what-happens-when-all-bitcoins-have-been-mined/</a></p>
 69241: <blockquote>
  <p>if in the coming years the difficulty increases so much that mining is no longer profitable</p>
</blockquote>

<p>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. </p>

<blockquote>
  <p>Can the bitcoin network exist without miners?</p>
</blockquote>

<p>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.</p>

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

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