I have seen many uses, even in books, of the three words "does there exist" and of the three words "does there exists". Examples:

  • Does there exist a political business cycle? [1]
  • Does there exists a general procedure for justifying ... [2]

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 ...? [3]

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.

  • 2
    The presence of the dummy auxiliary verb "do" means that the main verb must be in the plain (infinitival) form, i.e. "exist". – BillJ Apr 30 '16 at 15:24
  • 2
    Except for a handful of books by non-native English speakers, this mistake seems to have come into existence only after the invention of word-processing software. See Ngram. – Peter Shor Apr 30 '16 at 16:06
  • I think that Ronald V. Book is a native English speaker... but thanks for the Ngram! – Bach Apr 30 '16 at 18:59

You can have singular and plural, but like this:

Does there exist a polical business cycle?
Do there exist political business cycles?

But (as BillJ said) both keep the infinitive form "exist".

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