Proveďte dotaz. Ve vašem kódu $qry
není proveden
$db_connection = mysqli_connect("localhost", "username", "password", "database");
$qry = "UPDATE papers INNER JOIN users ON papers.user_id = users.user_id SET p_name = '$paper', p_authors = '$author', p_corresponding = '$corresponding', p_email = '$mail', p_cauthor = '$coauthors', p_abstract = '$abstract', p_keywords = '$keywords', p_jname = '$journal', p_date = '$date' WHERE p_id = '{$_GET['id']}' AND username = '{$_SESSION['user']}'";
mysqli_query($db_connection, $qry);