After you upgrade Microsoft SharePoint Foundation 2010 (which is configured for multi-tenancy) to Microsoft SharePoint Foundation 2013, the search function does not work. When a user tries to search on the tenant sites, no search results are returned.
This behavior is by design. SharePoint Foundation 2013 does not support searching on tenant sites. Creating partitioned SharePoint Server applications requires an enterprise license, and therefore is not possible in SharePoint Foundation 2013.