$('#textbox').keyup(function(){
    $("#display_students").text('');
    var name = $('#textbox').val();
    if($.trim(name) != ''){
        $.post('../core/Query/students/student_search.php', {st_process: name}, function(data){
            $("#display_students").append(data);
        });

    }
});

<?php


if(isset($_POST['st_process']) === true && empty($_POST['st_process']) === false){
    include("../Query-core.php");
    include('../db.php');

        $authentication = new DBRequest($host = 'localhost',
                                $user = 'root',
                                $pass = '',
                                $db = 'contas');

        $selectedstudents = $authentication->selectionQueryLike("alunos", $authentication->e(trim($_POST['st_process'])), "student_process");

        if(mysqli_num_rows($selectedstudents) > 0){
            echo "<table><tr><th>Nome</th><th>NºProcesso</th><th>ID</th><th>Apagar</th></tr>";

            while($row =  $selectedstudents->fetch_assoc()){
               echo "<tr>";
               echo "<td><a href='#'>" . $row["student_name"] . "</a></td>";
               echo "<td>" . $row["student_process"] . "</td>";
               echo "<td>" . $row["student_ID"] . "</td>";
               echo "<td><a href=?usrdelete=" . $row["student_ID"] . ">Delete<a/></td>";

               echo "</tr>";

            }
            echo "</table>";
    }
}

?>