The only other plugin I use is Polylang for multilingual content, and the language is part of the permalinks.
For example : http://mywebsite.com/fr/docs

I see 404 errors on search requests, so I suppose it is related to my permalinks configuration, but this configuration is exactly what I need.
For example, this AJAX request of the live search results with 404 : http://mywebsite.com/fr/docs/?ajax=true&livesearch=true&s=test

Here is the link to my website for more details : http://goo.gl/mOYqA5

Am I doing something wrong?

Thanks for writing. I’ve been through your site and everything is working as it should. For example, this is what I get when using the live search: https://cloudup.com/cqWkuYaKMye

Which browser are you using?

Best,
Juan.

The problem was that all search request were done on http://mywebsite.com/fr/docs (or http://mywebsite.com/en/docs-en for english language).

I just replaced the call to home_url() in header.php and scripts.php by a custom function which retrieve the home page url without /docs (or /docs-en) of the request path.

So now search requests are done on http://mywebsite.com/fr or http://mywebsite.com/en and it works as expected.

I have another limitation for now, it would have been great if it was possible to have multiple languages for the info boxes. This would allow to show different version of the homepage with info boxes depending on language.

Rudy

Thanks for writing back. Great news then.

Well, it is possible to have multiple languages in all contents. That depends on the plugin you use to translate your site, and its configuration. Check if the plugin has a section where you can set what contents should be translatable.

Best,
Juan.