Oh I misunderstood your post. I thought you couldn't open the dmg file. So you can't open the program after installing? And re-installing the Java "Offline" version didn't help? Or if you didn't try that, then do it. Get the java from the above link & re-install it.

Otherwise try deleting LW's preferences whilst LW is CLOSED which are found here:

/Users/your_username/Library/Preferences/LimeWire

OR do you mean you double-click the installer & nothing happens?
If that's the case then try one of Jum's nightly builds: http://www.mager.org/LimeWire/
BTW his builds are CVS beta builds. No installation necessary, just drag & drop the program to your HDD from the dmg file.

BTW LW 4.6 had security issues (there is a very remote chance someone could access files from outside your shared folders -> other areas of your computer. LW 4.8.1 fixed that issue!)