0, they do not compete in the Premier League.
Barcelona are a spanish team so they're in the spanish league.
Barcelona, the current greatest club in the world has won champions league three (3) times.
Liverpool have never won the title in the Premier League era.
Man Utd have won the English league (first division) 7 times, not including the English Premier League, which the have won 11 times.
i think three times
As of the end of the 2014/2015 Premier League, Burnley have never won it.
As of the end of the 2014/2015 Premier League, Swansea have never won it.
Manchester United have won the Premier League 13 times.
Until the 2011 Champions League, FC Barcelona has won their 4th trophy.