I used LimeWire version 4.18.8 fine with Leopard on iMac. I got new iMac with Snow Leopard and when I search there is not a single result comes out (I test on such popular name as B**tles). I don't want to update to a new version of LimeWire because of some search limitations there. Can 4.18.8 work with Snow Leopard? or what should I do to make it work?
Please help.

Yes LW 4 will work with Snow Leopard. However, this is an issue for LW 4. When this happens in future, quit LW, wait a few mins & re-open. My guess is your ultrapeers are not responding properly to you.
The alternative is to go to Connections window, remove the longest connected Ultrapeer, wait until LW finds a replacement UP & the connection settles. Then try searches again. Repeat this process.

Also, try searching something else. Sometimes searching same term too many times can result in ultrapeers starting to ignore the search query. So change your search term or technique.

Generally when faced with your scenario, this usually only lasts for 1-5 days. I've experienced this issue myself whilst using LW 4.

Sorry for the very late reply btw, not sure how your question escaped us.