WP Ajax check is user logged in - http://wordpress.stackexchange.com/questions/69814/check-if-user-is-logged-in-using-jquery/69819#69819

ajax_is_user_logged_in.js

var data = {
    action: 'is_user_logged_in'
};

jQuery.post(ajaxurl, data, function(response) {
    if(response == 'yes') {
        // user is logged in, do your stuff here
    } else {
        // user is not logged in, show login form here
    }
});

functions.php

<?php
function ajax_check_user_logged_in() {
    echo is_user_logged_in()?'yes':'no';
    die();
}
add_action('wp_ajax_is_user_logged_in', 'ajax_check_user_logged_in');
add_action('wp_ajax_nopriv_is_user_logged_in', 'ajax_check_user_logged_in');
?>