I had this problem and called Verizon (big mistake) to see if they had a solution. The tech asked me a few simple questions like did the battery look like it was bulging and some other crazy nonsense. They are sending me a new battery but in the meantime my phone did it again. I finally got it fully charged overnight and by 10am it was in the yellow zone and by noon it was in the red and about to shut off. Here's what I did to solve my own problem:
Go to settings: battery manager: tap the battery icon and then you will see what programs and such and the percentage of battery use for each running item on your phone. I discovered that an app I had downloaded for a dating site (I know, I know) was running in the background and using 90% of my battery power. I unloaded this app, put my phone back on the charger and haven't had any problems since.
