I think my iphone 5's battery have some problem.
Apple has a battery replacement program for iPhone 5. You can check eligibility on this Apple page


  Apple has determined that a very small percentage of iPhone 5 devices may suddenly experience shorter battery life or need to be charged more frequently. The affected iPhone 5 devices were sold between September 2012 and January 2013 and fall within a limited serial number range. (...)
  
  The program covers affected iPhone 5 batteries for two years after the first retail sale of the unit or until March 1, 2015, whichever provides longer coverage.