How to host my website created in Laravel on the Hosting Server? [HTTP ERROR 500]

Question

I’m trying to host my site developed in Laravel on the website hosting (Locaweb), but an error (HTTP ERROR 500) is preventing the operation.

File structure:

http://i.imgur.com/Ncg8bsl.png

Version of php: PHP Version 5.6.14

I modified the public/index.php

//require __DIR__.'/../bootstrap/autoload.php';
require __DIR__.'/../core/bootstrap/autoload.php';

//$app = require_once __DIR__.'/../bootstrap/app.php';
$app = require_once __DIR__.'/../core/bootstrap/app.php';

.htaccess (is the standard of Laravel)

<IfModule mod_rewrite.c>

    <IfModule mod_negotiation.c>
        Options -MultiViews
    </IfModule>

    RewriteEngine On

    # Redirect Trailing Slashes If Not A Folder...
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteRule ^(.*)/$ /$1 [L,R=301]

    # Handle Front Controller...
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteCond %{REQUEST_FILENAME} !-f
    RewriteRule ^ index.php [L]

    # Handle Authorization Header
    RewriteCond %{HTTP:Authorization} .
    RewriteRule .* - [E=HTTP_AUTHORIZATION:%{HTTP:Authorization}]

</IfModule>

After these changes, this error appears to me:

A página de xxx.com.br não está funcionando

xxx.com.br não consegue atender a esta solicitação no momento.

HTTP ERROR 500

Use Locaweb as hosting, and also set up core folder and subfolders as 777 permission.

ps. Locally everything works.