Hello Ivaylo,
After doing some testing, the issue is caused by insufficient memory_limit. This is because, the way Joomla deal with that search field is by inserting all strings from all language files into database table. Since Easysocial has almost 10k of language strings, it hits the memory limit error. There is nothing we can do about this other than reducing language string in Easysocial.
However, what you can do to make it work again, edit this file :
../administrator/components/com_languages/models/strings.php to exclude easysocial language
http://take.ms/0QRCO . In the event where you need to search for string in Easysocial, you can open the Easysocial language files in any editor and search from there.