In order to make sure that your custom list is visible to the SharePoint search results for the anonymous users, you need to set the AnonmyousPermMask64 property for that particular list.
The whole list of the powershell commands is given as follows.
$web = get-spweb “http://yoursiteurl/subweb/subsubweb”
$mylist = $web.Lists["ListName"]
$mylist.BreakRoleInheritance($true)
$mylist.AnonymousPermMask64= $list.AnonymousPermMask64 -bor ([int][Microsoft.SharePoint.SPBasePermissions]::AnonymousSearchAccessList)
$list.Update()
Happy SharePointing :-)