Source : Webster's Revised Unabridged Dictionary (1913)

Outstay \Out*stay"\, v. t.
   To stay beyond or longer than.

         She concluded to outstay him.            --Mad. D'
                                                  Arblay.

Source : WordNet®

outstay
     v 1: stay too long; "overstay or outstay one's welcome" [syn: {overstay}]
     2: surpass in staying power; "They outstayed their competitors"