There is and there are are used for both singular and plural. There exist gives the same meaning as them but When you say there is , are ,,, it is about something more specific ,concrete ,nearer to the speaker . There are 9 pupils in the class , There are 11 players in the team .( The team is one he might be joining or playing in or watching its match.) While there exists , about something a bit unknown ,unexperienced by and far from the speaker .The speaker knows them because they exist not because he experienced their existence . There exist 9 planets in the solar system . There exist 11 players in a football team ( as an information about it , while the speaker is not taking part in it , far from his own experience).