Update your Java (point 1 below). Use the offline JRE installer & remove your old version before installing the new one.
Fixes for LimeWire

Delete LW's preferences folder (points 2 & 3 above). Make sure LW's properly closed first.

Try changing compatibility (point 5 above).

Also, these might help

click LW icon & it begins to initialize but then stops and nothing happens
The way to fix the Blank screen on startup
GeoForce 3D setting