Learn English – “There exists some people…” or “There exist some people…”

The dreaded Elements of Style have this to say on this matter:

The word people is not to be used with words of number, in place of persons. If of "six people" five went away, how many "people" would be left?

So at least in formal or technical discourse you might want to prefer "persons".

I have also observed that "persons" is generally and widely used in official documents, highway signs, etc. in Britain, Canada, and the U.S.

In informal contexts, however, "people" is probably preferable, especially as it will allow you to steer clear of the, er, people from all walks of life who will try to tell you that "persons" is not a valid word at all.

But as both your question and Strunk point out, you should avoid mixing them too much, because there is no exact correspondence between "person"/"persons" and "people"/"peoples".

The word people is predominantly plural (see Merriam-Webster's top definitions), although it can be singular.

According to this source (a Pearson Education Q&A),

In the PLURAL sense, people is used as the plural of person very frequently. It is a plural count noun and takes a plural verb. It never has an -s ending; it is already plural.

Most of the time, people will be plural; and as far as I can tell, you will never see the phrase people is--at least, not in a grammatically correct sentence.

Here is what that same source has to say of the singular people:

the SINGULAR sense of people is used to refer to ALL the men, women, and children of a particular tribe, nation, country or ethnic group, speaking of them as a UNIT, and so the phrase a great people is indeed singular. It is a singular count noun.

(Okay, people is appears as a phrase in that quotation, but is refers to phrase and not people.)

People, even when singular, should always take are.

For more on whether to use is or are, please visit the Online Writing Lab.