It was already duplicate like this :

location ^~ /myapp/index.php {
auth_basic "Private Section";
auth_basic_user_file $document_root/.htpasswd;
location ~ \.php$ {
include /etc/nginx/fastcgi_params;
fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
fastcgi_pass 127.0.0.1:9000;
}
}

The inside location ~ \.php$ is silly because so I remove it like you said, but no change.

By the way the problem is not when I use the complete url (http://www.domain.fr/myapp/index.php) but when I don't add the index.php page (http://www.domain.fr/myapp/)