Using Mandrake 8.1
I do everything as root here...

I downloaded the Newest JDK from java.sun.com (1.3)
it comes in a .bin file, chmod it to 777 then install it.
I then went into the /root/.bashrc file and added the path to the bin to it "/java/jdk1.3.1_01/bin"
then at the command line type . .bashrc to reload it.
then type java in a terminal, if the path works you should see it, if it didn't it will tell you it didn't find it.
then download Limewire, I installed it using all default setting which included the install directory of /root/LimeWire...the first time I tried to change that to /var/Limewire and it wouldn't work. So stick with the default settings.

then just go into the /root/LimeWire Directory and type ./LimeWire18c
and it should start right up

Hope this helps