"What have they to say?" vs "What do they have to say?"

Is "what have they to say?" grammatical? If so, what is the difference with the phrase "what do they have to say?"? Do they mean the same thing?

Solution 1:

Yes, it's grammatical. Though it's uncommon in American English, and therefore sounds better to an American ear if delivered in an RP British accent, which makes it falute much higher. The difference is strictly syntactic -- that is, it makes no difference in meaning.

The verb have has two usages in English, both common:

1. It can be an irregular verb meaning 'possess', as in
   He has a green house.
2. It can be an auxiliary verb with no meaning for the perfect and many other constructions
   He has converted it to a duplex.
   He had to pay a lot.

In American English, the two usages have different syntactic affordances. In particular, the first, meaningful, sense of have is treated like a lexical verb and undergoes Do-Support in questions and negatives.

• Do you have the time/a minute/a car/a college degree?
• I don't have the time/a minute/a car/a college degree.

The second, meaningless, auxiliary usage is treated like an auxiliary:
inverted with the subject in questions, and contracted with negatives

• Has he converted it to a duplex?
• He hasn't converted it yet.

The converses are strange. If you treat lexical have like an auxiliary, you get the British-sounding variant.

• %Have you the time/a minute/a car/a college degree? ("%" marks dialectal variants)
• %I haven't the time/a minute/a car/a college degree.

and sometimes you violate idiom constraints

• *I haven't to go tomorrow; they cancelled it,
  instead of
• I don't have to go tomorrow; they cancelled it.

But if you treat auxiliary have like a lexical verb, you get garbage

• *Does he have converted it to a duplex?
• *He doesn't have converted it yet

Solution 2:

Yes, "what have they to say" is "grammatical," that is, it is a correct grammatical construction. It is an independent clause. It could stand as a complete sentence with the capitalization of "What" and with an ending question mark as punctuation. "what have they to say?" and "what do they have to say?" would have the same meaning to most readers. "what have they to say?" likely will sound formal and a bit archaic to most American English speakers.