即使我输入了有效的表和列，表单也没有收集用户输入并且查询无法正常工作

I'm trying to validate the user input and query to delete the record which have the same name. I'm using phpStorm for coding

I have tried to go over the typo, format of the code and check the query in phpAdmin and it's working fine

<?php
/**
 * Created by PhpStorm.
 * User: Administrator
 * Date: 3/24/2019
 * Time: 4:38 PM
 */

// Include config file
require_once "config.php";

$product_name= '';
$product_name_err = '';

// Processing form data when form is submitted
if($_SERVER["REQUEST_METHOD"] == "POST")
{
    if(empty(trim($_POST["product_name"]))){
        $product_name_err = "Please enter the product name.";
    } else{
        $product_name = trim($_POST["product_name"]);
    }

    //Delete the data in the product table
    $sql = "DELETE FROM `products` WHERE `name` = '$product_name'";
    if ($product_name_err =''){
        mysqli_query($link,$sql);
    }

}

?>
<?php include "header_admin.php"?>
<div class="wrapper">
    <form action="<?php echo htmlspecialchars($_SERVER["PHP_SELF"]); ?>" method="post">
        <div class="form-group" <?php echo (!empty($product_name_err)) ? 'has-error' : ''; ?>>
            <label>Product name</label>
            <input type="text" name="product_name" class="form-control" >
            <span class="help-block"><?php echo $product_name_err; ?></span>
        </div>

        <div class="form-group">
            <input type="submit" class="btn btn-primary" value="Delete Item">
        </div>
    </form>
</div>
<?php include "footer.php"?>

I expect the preceding code to check if the field is blank and the query delete that matched record on the database but either of them seem to work properly. The $product_name seem not received any value at all.

The below code should work correctly. There was a typo in your confition: if ($product_name_err =''){ should be if ($product_name_err ==''){

Also, your code was vunerable to injection attacks, which is fixed below by using the mysqli_escape_string function.

if($_SERVER["REQUEST_METHOD"] == "POST") {

    $product_name = mysqli_escape_string($link, trim($_POST["product_name"]));

    if(empty($input)){
        $product_name_err = "Please enter the product name.";
    }

    //Delete the data in the product table
    $sql = "DELETE FROM `products` WHERE `name` = '$product_name'";

    if($product_name_err == ''){
        mysqli_query($link,$sql);
    }

}

Your if condition is incorrect, use '==' instead of '='.

if ($product_name_err ==''){
        mysqli_query($link,$sql);
 }

also you should really consider using prepared statements to prevent sql injection attacks and it does other nice things for you like you not having to escape ' or " characters from your strings.

More info on prepared statements

Php code check will help to check your PHP code. May be it will help you.

#PHP Code Checker

An advanced, custom PHP code checker that searches your code for common, hard to find typos and mistakes; includes a syntax check.