I found an iphone on the floor and it won't turn on?

It won't go past the apple logo, it just shuts off. Its a gray iphone 5

Is it your iPhone, or someone you know? If it's yours, just charge it overnight, then try pressing both the home button and lock button at the same time for three seconds.

if it's not yours, return it to the appropriate person. Otherwise, you can be charged with stealing.

you can return it to the police station for them to keep for lost and found.

Return it to apple

Take it to the Apple store. I'm sure they can get it back to its owner.

May as well take it to be seen about at an Apple Store, in terms of finding the owner.

Activation Lock would prevent unauthorised use anyway.