Hi there i have the same problem with mine too, do you flash any rom in it or kernel?
I guess its a problem with the kernel or rom you have flashed, mine have the same problem, but you can put it charging in the usb cable in the computer and let it stay there for a whyle, then you have to try to boot it up by using the 3 keys (power+home+vol up) it will boot in recovery mode then you ask it to reeboot and install an updated kernel or reflash the rom, while the device is on, do not put it off, because if you do it you will have the same problem to put it on again, but when its on it charges normally.
I have changed mine for a new one but the problem i am sure it was the things i flashed it. I was using another kernel to install and root the device and again put the original stock kernel in the device to get rid of the yellow triangle, but that came up with that problem. I even dowloaded root explorer in order to reset the battery status bin (you have to be rooted to do this) and reboot the phone but with no avail. The other you have to do this is install CWM MANAGER V1.80 MY KERNEL CF-ROOT 4.0 MY ANDROID VERSION 2.3.3 BASEBAND i9100UHKE2 and now note this the stock kernel was the uhke4 but the kernel i put it it is 18.104.22.168-I9100XWKE7-CL215725root@DELL103#2 COMPILATION NUMBER: GINGERBREAD.UHKE4
Note: i dont know if it will work for you, i am using a stock rom from samsung. If you do not flash anything its better to return the device or even you have flashed it buy a usb jig and reset the counter and find for the repair, they have to fix that for you.
I hope that have provided you some help.