There's not enough information here.
1. Is this a mod_perl application or a CGI script?
2. Have you
checked the mycert.p12 file with (say) openssl to make sure it hasn't been inadvertently corrupted?
3. Have you tried to connect to the remote service using curl (or some other web tool that supports HTTPS connections) with your certificate to ensure that the problem is with your perl libraries and not your certificates?
4. What is version of LWP that SOAP::Lite is using? The semantics of handling SSL certificates changed in LWP-6 and later
Mark