As far as I can see, I've got the right version of java in the right place:

$ which java
/usr/local/bin/java

$ /usr/local/bin/java -version
java version "1.3.1_02"
Java(TM) 2 Runtime Environment, Standard Edition (build 1.3.1_02-b02)
Java HotSpot(TM) Client VM (build 1.3.1_02-b02, mixed mode)

$ ls -l /usr/local/bin/java
lrwxrwxrwx 1 root root 30 Feb 25 19:43 /usr/local/bin/java -> /usr/java/jre1.3.1_02/bin/java

The LimeWire download page for the basic version says that java1.3.1 (JRE) is needed. I have LimeWirePro and the download page for it says nothing to the contrary.

I don't understand why the installer is looking for java in directory
/usr/bin/jdk-1.1.6/../bin/Linux/green_threads
which resolves to:
/usr/bin/bin/Linux/green_threads