Hai, I just installed (free version of) Limewire 5. There is a full connection, but my Search does not give any results. Does anyone know why?

Could be for various reasons such as how you are searching. Could also be due to ultrapeers you are connected to. Remember searching in LW is not like Google.
Try using the Advanced search tools found under the Tools menu. Example for music, see tips here (also applies to video): How to find music

And if LOTR's link doesn't fix your problem, also try:

1. In LW go tools > options > security. Click on the Settings button, next to Warning Messages > untick the box referring to license > ok. Then click ok on the main options page.

2. Problems with the search

3. Delete LW's preferences folder when LW's properly closed. See point 3 here & also point 2 if you've used older versions of LW in the past (there could be a folder in either location).
Fixes for LimeWire
This will change all of your settings within LW's options, you'll have to go back through afterwards & reset things to the way you like them.
Vista users: LimeWire Pref Folder in Vista

4. In LW tools > options > advanced > super really advanced > performance. Disable ultrapeer capabilities, connecting to DHT & also TLS > ok.