PHP form not finding empty field

if (empty($name) || empty($email) || empty($message)) 
 {
     $emailErr = "All fields are mendatory";
 }

you can use php function empty or is_null

isset() will always return true in this case, because the variable has been set (To an empty string). To check to see if a variable is empty, you should use what you did with $email and $message:

if ( $name == "" || $email == "" || $message == "" ) {

To demonstrate, you can simply run the following:

$a = ""
var_dump(!isset($a)); // false
var_dump($a == ""); // true

Please try this ,

I check it in my local system its working fine , error and success message is coming as per your requirement

<?php
  if(isset($_POST['submit'])) {
    $name = $_POST['name'];
    $email = $_POST['email'];
    $message = $_POST['message'];
    $emailErr = "";

    if (!isset($name) || $email=="" || $message=="" ) {
      $emailErr = "All fields are mendatory";
    } else {
      $mailTo = "[email protected]";
      mail($mailTo);
      $emailErr = "Email Send";
    }
 }
 ?>
 <form name="contactform" method="post" action="">
  <input type="text" data-placeholder="<?php echo $name; ?>"name="name">
  <input type="text" data-placeholder="<?php echo $email; ?>"name="email">
  <textarea data-placeholder="<?php echo $message; ?>" name="message" rows="10"></textarea>
  <input type="submit" name="submit" class="btn btn-footer" value="ENVOYER" id="btnContact">
  <p class="text desktop"><?php echo $emailErr; ?></p>
</form>

After Form submit, if field name is exists then isset() will return true. But if you check empty value for any field you can use empty($fieldname) or $fieldname == "" or $fieldname == NULL.

Try This:

if(isset($_POST['submit'])) {
  $name = $_POST['name'];
  $email = $_POST['email'];
  $message = $_POST['message'];
  $emailErr = "";

  if (empty($name) || empty($email) || empty($message) ) {
    $emailErr = "All fields are mendatory";
  } else {
    $mailTo = "[email protected]";
    mail($mailTo);
    $emailErr = "Email Send";
  }
}