Does LW actually open & run for a while before it shuts down? Or does it start loading & then shut down? Are you getting any error messages?
Try removing your Java version & reinstalling using the offline installer (point 1 below)
Delete your LW preferences folder when LW's closed (points 2 & 3 below)
Fixes for LimeWire
Reinstall LW using it's offline installer
http://www.limewire.com/english/content/download.shtml
Are you running any antispyware programs? Sometimes people have trouble when an antispy's recognising LW as a threat. Try adding LW to the list of exceptions/allowed apps or temporarily disable the antispyware.