You should go to

http://java.com/en/download/manual.jsp

and dowload the linux rpm (jre-1_5_0_01-linux-i586-rpm.bin)

download it to your user's home directory
(/home/your-user)

run this line in the console

chmod a+x jre-1_5_0-linux-i586-rpm.bin

and then

./jre-1_5_0-linux-i586-rpm.bin

The installation file creates jre-1_5_0-linux-i586.rpm file in the current directory.

you can click on the rpm , and use the default application to install tthe JVM

or you can install it typing:
rpm -iv jre-1_5_0-linux-i586.rpm

then
PATH=$PATH:/usr/java/j2re1.4.2_06/bin

Finally download LimeWireLinux.bin to

/home/your-user

type
chmod a+x LimeWireLinux.bin

and now you can start the Limewire installation
./LimeWireLinux.bin