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 you downloaded LimeWireLinux.bin, I typed:

cd downloads <enter>

then I typed in:

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

then the install wizard runs, follow the instructions.