synchronizing takes weeks and more than 120 gb Do I have to download all 120 gb? 
I sent myself 100 euro in bitcoin weeks ago. Waited till bitcoin core synchronised to receive them. but it took weeks. And no money in my wallet. And my pc is out of free space.
For the "out of free space" issue you could use pruning mode. For the "no money in my wallet", you'll have to tell us a little more. Is the transaction confirmed?

also see: 


Cannot access my bitcoins because my wallet is still synchronising. What can I do to spend them immediately?
How can I run bitcoind in pruning mode?