What is the right way to spell "else's"?

“Else’s” is correct.

@Kropotkin has it right.

The problem belongs to someone else. And the way that we show possession in American English is with the apostrophe – S. British English may have different customs.

Exactly the same as “It is somebody’s problem.”

@gailcalled That has a different nuance to me. If it’s merely “someone’s problem”, that someone could still be you!

Not referring to nuanced meaning but only the issue of the apostrophe.

“It’s someone else’s problem” is correct in UK English also