Hello Julxu,
you haven't provided enough details to understand what you are seeing
Describe the topology and provide configuration of BGP and EIGRP processes on the two routers
>> I was expecting when the main link down the backup link will up immediately.
With EIGRP you could use dialer watch as a way to trigger a DDR call on an backup link but it is not clear what type of backup link you have.
see
http://www.cisco.com/en/US/docs/ios/dial/configuration/guide/dia_backup_dia_watch_ps6441_TSD_Products_Configuration_Guide_Chapter.html
this is useful if the backup link is on a different router then the primary link. With dialer watch you can monitor the presence in the IP routing table or one or more monitored (watched prefixes) and to use this to trigger the backup DDR call.
>> can I put some ACL on EIGRP to let the routers know only the special ip can update the route, not every equipments
This is a different matter something can be done but without knowing your scenario it is difficult to say anything useful.
Hope to help
Giuseppe
