The problem is that the Limewire installer does not know where to find your java installation. The way to fix this is to find where you installed java and to add it to the PATH environment variable.

If you installed java using the .rpm from an official Sun website, it is most likely in "/usr/java/". Open a terminal and type

cd /usr/java
ls

You should see a directory named "j2re1.4.2_04" or something similar. Then, in order to let the Limewire installer know that java is installed there type the following:

export PATH=$PATH:/usr/java/j2re1.4.2_04/bin

(replace the 1.4.2_04 numbers with whatever the name of the directory you found above is). To verify the above worked, type

java -version

and you should see a response from java which tells you that you're running version 1.4.2

Once you do this, try running the installer from the same terminal by going to the directory where you downloaded it and typing

sh LimeWireLinux.bin

and that will get you going.