Question about User php class. Should I connect to DB in this class?

I have small Class which handles user's information. (name, surname, login, hash of password). Also this class can activate, register and login the user. This means I should connect to DB. Is it normal? I'm using Pear MDB2 to connect to DB.

Does it mean that every instance of User will have MDB2 object initialised?

class User {
    private $login;
    private $hash;
    private $name;

    /* some getters and setters */
    //
    // public function get_name() { return $this->name }
    // ..
    //

    private $connect; // MDB2 connection
    private $dsn = ""; // connection config

    public function exists() { // return true if User exists
        require ('MDB2.php');
        $this->connect = MDB2::connect($this->dsn);
        // check
    }
}

Should I make $connect and $dsn variables static? Or may be I should use some pattern for this idea?

Update: (kind of decision)

I store database connection in static variable:

class DataBase {
    // connections options
    static public $connect = NULL;      // MDB2 connection variable
    static public $dsn = "pgsql://user:pass@IP/database";
    static public $connected = false;

    static function reportError() {
        if (PEAR::isError(self::$connect)) {
            $res["success"]=false;
            $res["errors"][] = self::$connect->getMessage();
            echo json_encode($res);
            exit;
        }
    } //: reportError

} //: Class DataBase

So If I want to connect to DB I write:

// connect only if it's not connected
if (!DataBase::$connected) {
    require_once('MDB2.php'); // Pear MDB2 class
    DataBase::$connect = MDB2::factory(DataBase::$dsn); // connection
    DataBase::$connected = true; // Flag that we're already connected
}

It's normal? Or I should make it another way?