Should I buy a Korean Iphone 5S?

I'm from Mumbai and iphone here costs to much! Also, is there any risk of fake copy of the phone or some carrier issues?

You will have some difficulty, though no technical difficulties. The iPhone 5S has a "universal" communications chipset so it is capable of working with all major mobile carriers (GSM & CDMA frequencies) in India (and the world). So no problem technically.

Whether the Korean device you buy is a fake, that is for you to determine. Technically, it is very difficult to determine unless you have it enabled in Korea (and you have a Korean data plan). If you buy it from an Apple store within Korea (not an Apple-authorized re-seller or an authorized distributor), Apple stores will sell you real 5S models - no fakes. But other stores/sellers can have fakes.

That's the good news.

HOWEVER, any electronic device bought in another country carries a more limited warranty and service. If anything was wrong with your iPhone, you might have to go back to Korea to have it fixed or replaced (and even that might not be possible since your legal address is not in Korea). Some Asian countries have warranty/service available pan-Asia. You need to make sure a Korean phone can be serviced within India. Since there's no direct Apple presence in India yet, it is unlikely you will get warranty/service in India for it.

Also, you will have to pay import duties (taxes) to bring the phone into India. And if you try to smuggle it in without paying these but you get caught, the phone can be confiscated and additional fines/penalties can be imposed on you as well.

If the difference in price is less than 30%, it is usually not worth the risks.

I think amazon buy origine 5s, you can buy any fear, but amazon is profesional company and singifut.
and price is less than 30%