Lots of ways to do this. I guess that the authors and titles are being stored in the same table: $DB_TABLE.
You could move the authors to a seperate table with an id column that you store in the other table with the titles. Would make filling the combo box and returning all titles for the author easy. You would have to think about how you handle authors with the same name: there are lots of them about.
With your current setup
$SQL = "SELECT DISTINCT author_last_name, author_first_name FROM $DB_TABLE ORDER BY author_last_name, author_first_name";
while ($ROW = mysql_fetch_array($RESULT))
{
$RECORD = $ROW["author_last_name"] . ', ' . $ROW["author_first_name"];
echo "<OPTION VALUE='$RECORD'>$RECORD</OPTION>";
}
if(isset($_POST['authorwanted']))
{
$RECORD = explode(", ",$_POST['authorwanted']);
$SQL1 = "SELECT * FROM $DB_TABLE WHERE author_last_name = '" . $RECORD[0] . "', author_first_name = '" . $RECORD[1] . "' ORDER BY title";
Remember, somewhere in all of this you are going to have to cater for different authors with the same name, even relying on middle initials will catch you out if it is a big catalogue.