My question is about the sentence
A few opportunities exist to get a better education in the U.S.(1)
Some people said: "It is a correct sentence."
However, I don't think so and will explain why.
The part "to get a better education in the U.S" refers to the main verb "exist", as you know, because, it doesn't refer to "a few opportunities".
(If they said:"A few opportunities to get a better education in the U.S exist", it would be OK, but right now, it has to refer to "exist")
Therefore, the infinitive clause "to get a better..." has to function to specify a purpose in that sentence.
My opinion is that this sentence grammatically means:
A few opportunities exist because the reason why they exist is to get a better education. Their aim to exist is to get a better education.
Adding "for someone" doesn't even solve that problem as well.
A few opportunities exist for me to get a better education in the U.S.
It also seems like they exist because their aim to exist is my getting a better education..
Finally, It is easy to read and understand them but the sentence(1) is not a natural sentence, it has no meaning in English grammar. It also proves that the questions "What opportunities are there to get a better education" and "What opportunities exist to get a better education" are also not natural.
Do you agree with me? If you don't, could you please explain why? At what point I am wrong?