Tilde is a feature of the shell which Perl is probably not able to expand for you. You can substitute it with something like this:

use lib "$ENV{HOME}/public_html/PERL_TEST";

There's also no guarantee that you're going to have a HOME environment variable either.

A more reliable method might be something like this:

BEGIN
{
        my $home = (getpwuid($<))[7];   # Current user

        use lib "$home/public_html/PERL_TEST";
}
[download]

Update:
The above was merely given as an example of how you can replace tilde for the current user, figuring it was fairly obvious how to retool it to be more generic. What I probably should've put was something like this:

BEGIN
{
        my $user = 'someuser';
        my ($home) = (getpwnam($user))[7];  # Specific user

        die "Can't find user $user home directory\n" unless($home);

        use lib "$home/bin";

        print $home;
}
[download]

While having a hard-coded path is one way to ensure that the application will work, having an application use libraries from the home directory of a particular user isn't all that unusual, although it can be confusing if not documented clearly.