I recently purchased a used ESP EC-401 that is equipped with Grover tuners and it's having difficulty staying in tune. The low 3 strings are usually fine, but the high 3 go out of tune from a simple bend.

I recently changed the strings on it to Ernie Ball Cobalts.

I really don't know what to do.
Sounds like the nut, not the tuners.

Grover makes very nice tuners and they very rarely move once tuned.

Try some Big bends nut sauce or take a #2 pencil and rub it on the nut slots. the graphite from the pencil will hopefully lube the strings so they don't bind.

Having a tech widen the string slots is the best answer
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I'm hardly a tech but those nut slots look too tight for the top three strings. What gauge are they?
Well, since the guitar comes with 10-46 gauge strings (or even 9-42) the slots are cut for those. There is probably nothing wrong with the tuners at all, it's just the strings getting caught in the nut.
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How you know you have too many guitars...

Apparently once also known as PonyFan #834553.
nut slots need to be bigger-- unless you know what you're doing, take it to a reputable shop and have someone take care of it.