Avoid dropdown menu close on click inside

This should help as well

$(document).on('click', 'someyourContainer .dropdown-menu', function (e) {
  e.stopPropagation();
});

Bootstrap provides the following function:

                 | This event is fired immediately when the hide instance method 
hide.bs.dropdown | has been called. The toggling anchor element is available as the 
                 | relatedTarget property of the event.

Therefore, implementing this function should be able to disable the dropdown from closing.

$('#myDropdown').on('hide.bs.dropdown', function (e) {
    var target = $(e.target);
    if(target.hasClass("keepopen") || target.parents(".keepopen").length){
        return false; // returning false should stop the dropdown from hiding.
    }else{
        return true;
    }
});

Removing the data attribute data-toggle="dropdown" and implementing the open/close of the dropdown can be a solution.

First by handling the click on the link to open/close the dropdown like this :

$('li.dropdown.mega-dropdown a').on('click', function (event) {
    $(this).parent().toggleClass('open');
});

and then listening the clicks outside of the dropdown to close it like this :

$('body').on('click', function (e) {
    if (!$('li.dropdown.mega-dropdown').is(e.target) 
        && $('li.dropdown.mega-dropdown').has(e.target).length === 0 
        && $('.open').has(e.target).length === 0
    ) {
        $('li.dropdown.mega-dropdown').removeClass('open');
    }
});

Here is the demo : http://jsfiddle.net/RomaLefrancois/hh81rhcm/2/

The absolute best answer is to put a form tag after the class dropdown-menu

so your code is

<ul class="dropdown-menu">
  <form>
    <li>
      <div class="menu-item">bla bla bla</div>
    </li>
  </form>
</ul>