No search results when External user accessing Ananymous sharePoint 2010 site

     When Ananymous user login in to the sharepoint site and search for something.... He will get no information. But the internal user can able to fetch the search results. And in some cases site will ask for Ananymous user's credentials.

For the first one:

Go to site actions--site settings-->Under 'Site Administration'-->Search and Offline Availability and select
-Always index all webparts on this site.

Asking user credentials:

The reason is OSSSearchResults.aspx. To disable that we can select an option from Site Collection Search Dropdown Mode i.e, Do not show scope description, and default to target results page.

Due to the above selction the search results will gets displayed on   /Search/Results.aspx and this page will not ask any user credentials.