This is only an educated guess here, as I don't have the facilities currently to test, nor the time to read docs to understand what's happening. Can you extract the object, check for defined, then act appropriately?

Instead of:

my $abstr = $dom1->at('div.abstract-content > p' || 'not provided')->t
+ext;
[download]

Will something like this example work?:

my $obj   = $dom1->at('div.abstract-content > p' || 'not provided');
my $abstr = defined $obj ? $obj->text : 'Object undefined';
[download]

If it does work, you could elaborate on it a little, to say log the entries that are broken.