When I launch a research on this new version of Limewire, the results disappear from the screen, appear when I clic on em, nothing is available in the research tab and eventually limewire crashes and is closed down.
Can my system handle this version? How can I fix Limewire? Where could I eventually find older versions of Limewire? (never had a problem before)

The newer versions of LW require higher version Java. That's why your problems. If you still have an older version LW, then trash the new one & reinstall the older one. But you will need to double-click the harddrive icon, open the Library, open the Receipts folder, then find the file: Install LimeWire.pkg, delete it before installing an older version LW. Don't remove any other receipt files, the system needs them.

Also you should delete your LW prefs folder found here: /Users/your_username/Library/Preferences/LimeWire

Whilst I personally have numerous versions of LW, you could step back to version 4.8.1 ... check the Sticky's at top of section or other mac stickies. You will definitely need to remove your prefs folder before installing that version because it would behave erratically & not very well with prefs from newer versions. It was a very good version (better than 4.9 or some of the 4.10 versions.)

The only alternative I can think of with your present version is changing skins to see if that improves things. Another tip is Tip: Purge the Downloads Queue! which improves performance; freshens it up again.