Uhm, Is PvP the answer for "competitive aspect" ?
Diablo leaderboards for example are all about just PvE.
Or the guilds in WoW trying to get first world kills on mythic bosses.
There's bound to be a bunch of crazies in this game who're gonna try and solo kaiju, or last as long as they can anyway and compete with the other ones as to who can do it first.
Or who can mine the most, frag the most enemies, find new enemies/locations the first, etc etc ...