Ideally you'd want the installer to work I guess. I'm not sure what it's problem is. However, since you already have JRE 1.3.1 installed there's an alternative installation method. I used this method to install 1.7 on my RedHat 7.1 system with the Java 1.3.1 SDK.
1. Go to the LimeWire download page (at
http://www.limewire.com) and download the Solaris package *without* the installer included. It's 621kb and for some reason is called LimeWireLinux.tgz.
2. Extract the LimeWire files from the archive with this command: tar -zxvf LimeWireLinux.tgz
That's it. A directory called LimeWire is created in the same directory as LimeWireLinux.tgz and it contains LimeWire's classes and the runlime.sh script. Assuming your java installation is okay, you can now change to the new LimeWire directory and run LimeWire with ./runlime.sh
[ Update: I just noticed that LimeWireLinux.tgz is actually around 662 KB in size. The download page claims it's 621 KB. :-) ]