I have a Galaxy Note 8. As you know the side of the screen is curved. So when i stick the screen protector, the side is not fully sealed. The side will tend to "come up"

Is there any screen protector brand that can fully cover the entire screen?

are you looking for tempered glass? if yes, whitedome is good, but pricey. my suggestion is to look for those fully glued protectors (like UV glue type)

If you just want to protect the surface from scratches, you can try those silicone type screen protector, it is quite soft and can cover the edges (the material is like those stickers you paste on your windscreen)