You can play competitively on the PS4 through websites like gamebattles. I'm not sure if that's what you mean by esports. If you are referring to the big LAN events like MLG and UMG, then those are played on the Xbox One. Those large CoD events were played on the PS3 back in Black Ops 1 and MW2 (to a lesser extent) so it is possible that future CoD LAN events may be played on the PS4, but I believe Advanced Warfare LAN events will be mostly held on the Xbox One
I am thinking that Xbox will have COD and PS will have Destiny(if it goes into esports). If not then COD will probably be on the PS4 if Halo 5 is any good for esports.
Just speculation so don't take it as facts.
you can play ps3/ps4 on MLG GBs or UMG but the major LAN tournaments are on xbox
Major LAN events: Cod champs, MLG, Gfinity, UMG, X Games (not sure if it there's any for AW), ESWC
LAN Events will be held on Xbox for as long as the Pro players play on that console.
90% or more top players play on the Xbox, so MLG etc.. will continue to support that console. The Playstation eSport community just isn't big enough.