There exists some explanation for this phrase. You have come to the right place to find it! "There exists" is basically an alternative version of "there is." "exist" is basically "is." "There" here doesn't have its normal meaning (like it would in here is my car, there is yours). "There exists" sounds more academic, more formal. Look at some examples here: I am certain there exists some logical explanation for paranormal events. There exist some frogs with three eyes. I don't recommend using this phrase in standard, casual conversation. If you are teaching something or giving a lecture, it would be perfect, though! Personally, I only use it when I write something like an essay.