Working on the assumption that the references will only find one PDF (which I'm not entirely convinced of), the following code should give you a starting point.

#!/usr/bin/env perl

use strict;
use warnings;

use LWP::UserAgent;
use URI::Escape;
use File::Basename;

our $VERSION = '0.001';

my $agent_name = join '/' => basename($0), $VERSION;
my $query_base = 'https://duckduckgo.com/html/?q=';
my $pdf_re = qr{href="([^"]+\.pdf)"};

my $ua = LWP::UserAgent->new(agent => $agent_name);

while (<DATA>) {
    chomp;
    my $req = HTTP::Request->new(GET => $query_base . uri_escape($_));
    $req->content_type('text/html');

    my $res = $ua->request($req);

    if ($res->is_success) {
        print "Search successful.\n";

        if ($res->content =~ $pdf_re) {
            my $pdf_url = $1;
            print "PDF found: $pdf_url\n";
            process_pdf_url($pdf_url);
        }
        else {
            print "PDF not found!\n";
        }
    }
    else {
        print $res->status_line, "\n";
    }
}

sub process_pdf_url {
    my $pdf_url = shift;

    print "Stub - download $pdf_url,\n\trename, upload to database, et
+c.\n";

    return;
}

__DATA__
1.    Abilez O, Benharash P, Mehrotra M, Miyamoto E, Gale A, Picquet J
+, Xu C, Zarins C (2006) A novel culture system shows that stem cells 
+can be grown in 3D and under physiologic pulsatile conditions for tis
+sue engineering of vascular grafts. J Surg Res 132:170-178.
[download]

Output:

$ pm_web_search_pdf.pl
Search successful.
PDF found: http://med.stanford.edu/arts/arts_students/CVs/CV_abilez_09
+2007.pdf
Stub - download http://med.stanford.edu/arts/arts_students/CVs/CV_abil
+ez_092007.pdf,
    rename, upload to database, etc.
[download]