I installed the Java j2re 1.4.1-01, it installed to a folder at /user/java/j2re1.4.1_01 (this location is used in the PATH=$PATH: )
then, I opened my Terminal (DO NOT install as root, so don't use Super User Mode Terminal)
then cd to the folder where I downloaded LimeWireLinux.bin, then I typed in:

PATH=$PATH:/usr/java/j2re1.4.1_01/bin sh ./LimeWireLinux.bin