We had one that did that, we had to change the power connector, and surprisingly that helped. If that fails, they've backed up their products up with us every time. Give them a call.
Definitely call amazon. They are very good about supporting kindles. When I sat on my kindle and broke the screen Best Buy replaced it no questions asked and from what I've read Amazon would have done the same thing if I had gone to them.
Amazon makes their money selling the books and they can't sell you a book if your Kindle is kaput. They must hate me. I only buy old out-of-copyright classics like Dickens and Austen stuff.