Does anyone know who can apply and how to apply for DJI Research discount ??

This is too confusing. my sister is grad student, she wants to get a DJI  drone but as we know DJI drones are so expensive, she figured out there is a DJI research discount 40% Off Exist.

Have you looked here?
DJI Educational Discounts

I believe that the 40% discount to which your sister is referring is for certain type of drones that are selected by DJI and you have to have SDK developer account. Whereas the education discount is 10% for several products.

"6- What discount does the Educational Discount offer?

10% off for eligible DJI products: Mavic Series, Phantom Series, Inspire Series, Ronin Series, Spark Series, Osmo Series, Tello Series, DJI Goggles Series, Developer Tools (not including Osmo Pocket, DJI Smart Controller, Mavic 2 Pro (DJI Smart Controller), Mavic 2 Zoom (DJI Smart Controller) and other newly launched or low stock products)."

Here's the source: https://store.dji.com/education (under the FAQ section).

Here's the Research Discount information