Christoffer Solbrekke, as with any iphone, I strongly recommend that you clean it. Disassemble it by using [http://www.ifixit.com/Device/iPhone_5#guideList|these guides.] Clean it properly using [guide|5825|this guide.] It was written for a 3G but all the points still apply to your phone as well. Once it is properly cleaned '''replace the battery" reassemble and re-evaluate. With a new battery and properly cleaned, you will have a good starting point should any further trouble shooting be necessary. Hope this helps, good luck.