Q:

live or living in some city

They ....................... in Atlanta for more than ten years.

Should it be live or living?
Which one is correct and why?

A:

Neither is idiomatic.
If they are still living in Atlanta, you should use have lived or have been living.

The present perfect construction with for TIMESPAN and a stative verb is ordinarily a continuative perfect, signifying that the state described persists in the present.

If they live somewhere else now, you should use lived.

The simple past with for TIMESPAN signifies that the state described ended at some time in the past.

Have lived may work here, too, as an existential perfect, signifying that the past state described is a currently relevant episode in their history. However, additional information must be provided to distinguish this reading from the continuative perfect; for instance:

They have lived in Atlanta before, for ten years, so they should have many friends there.

For more about continuative and existential perfects, see this, especially §3.2.

