OOP MySQLi singleton
Well, although this code belongs to Codereview@stackexchenge, but as nobody cares to follow any rules here - so I would.

There are too much problems with your code, ut to outline most:


It seems you don't quite understand what singleton is and why use it.
There is no point in extending application class from database handler class. 
trigger_error('Unknown Method ') is apparently redundant. PHP can handle absent methods as well.
Never use die() in production code. 
This whole class makes very little sense as it don't help you to handle mysqli even a bit. 


Frankly, exactly the same result you can have without any "singleton" class but just by creating mysqli instance and making it

public function __construct() {
    global $mysqli;
    $this-&gt;mysqli = $mysqli;
}