@Kevin_Z 

For some reason it worked for about two days now. But today the same problem occured.  I'm not the best in software.. maybe you can explain, how to do the ping test to my router?

I posted some pictures. The IP is set manually at the moment, but I tried automatic of course too. It says, that my phone is connected, also in the settings of my Iphone, but there is no internet connection I can use. It means that there is no WIFI sign in the top corner of my phone and I cannot get any internet connection. Does this explanation make sense? 

It also keeps spamming "sent a detection for DHCP". (picture 2)

I also tried the Theter App. The position of the router is perfect.

Thanks for your help!!

@Binderli 

I forgot to mention, that I have a Speedport Smart, which was given to us by the German Telekom! And the Repeater is a RE200, which I purchased 3 Weeks ago. 

@Binderli The same happens to me, the repeater stops giving access to internet after 2 or 3 days. I found several other people stating the same with no response from TpLink

@Sergioms I really feel you... unfortunately the community had no real solution and the tp link support didn't help either. It's seems they know about the problem but basically just ignore it. The only solution for me was returning it and buying one from another brand. 
Have a nice day!