I'm using John Resig's simple OOP Class that is adapted to use "use strict" and taken from SO post.
In all examples I see the usage of Class.extend like so:
var MyObj = Class.extend({
    init:function(){},
    prop: "Property"
});
But I found a large disadvantage for me of using it in such way - I cannot have "private" variables, so I cannot store reference to this like var $this = this;.
I found the solution for my case, and now I using the Class.extend in following way: 
var MyObj = Class.extend(new function(){
    var $this = this;
    this.init = function(){};
    this.prop = "Property";
});
Everything works in my case, but I want to know if there some things that can cause me problems in a long run?
Does this way my application will consume much more memory in browser?
What alternative ways I have to implement my needs?
Note: I need to store $this, because I use heavily events and callbacks, so I want to refer "original" this easy to have access to all methods and properties on object.
EDIT: As requested, this is my code example:
(function () {
    "use strict";
    window.QuickPlay = Class.extend(new function () {
        var $this = this;
        this.init = function (initData) {
            $this.elementsToHide.push(initData.el);
            $(function () {
                playProcessStart();
                Sys.Application.add_load(function () {
                    $find("ctl00_ContentPlaceHolderMain_ctrlPlayPopup1").add_closed(function () { $this.setElementsVisibility(""); });
                });
                $this.setElementsVisibility("hidden");
            });
        };
        this.elementsToHide = [];
        this.setElementsVisibility = function (visibility) {
            $.each($this.elementsToHide, function (i) {
                $("#" + this).css("visibility", visibility);
            });
        };
    });
} ());