Datamining of the game files following Patch v4.2.0 has revealed that Fortnite might be getting a competitive mode in the near future. A new ‘Comp’ folder has been added to the Playlists section within the files which also contains past Limited Time Modes:

The lack of ‘Playlist_Comp_Duos’ and ‘Playlist_Comp_Squads’ files suggests that it’d only be available for Solo play, though this may be done as a way to test things out before rolling it out to Duos and Squads at a later date.

Note: None of this information is official or confirmed, it is purely speculation based on files added to the game. Do not take it as fact.