The ? is a regex metacharacter. If you want to match it, match on "\?".
Otherwise, you should be checking for string equality using $a eq $b.

#!/usr/local/perl

$a = "http://www.myurl.com/e.php?a";
$b = qr|http://www.myurl.com/e.php\?a|;

if ($a =~ /$b/) {
        print "FOUND\n";
} else {
        print "NOT FOUND\n";
}

$a = "http://www.myurl.com/e.php";
$b = "http://www.myurl.com/e.php";

if ($a =~ /$b/) {
        print "FOUND\n";
} else {
        print "NOT FOUND\n";
}

$a = "http://www.myurl.com/e.php";
$b = "http://www.myurl.com/e.php";

if ($a eq /$b/) {
        print "FOUND\n";
} else {
        print "NOT FOUND\n";
}
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