I have seen many uses, even in books, of the three words "does there exist" and of the three words "does there exists". Examples:
And even in the same paragraph in another book:
Does there exists a uniform way of specifying languages ...? That is, does there exist a uniform construction such that ...? 
I would guess that the form does there exist is the right one, but a confirmation would be nice, or even better -- an explanation/reference.