One way that you can do to get internet connection on the Samsung Galaxy SIII is using Wi-Fi to connect to Wi-Fi network.
See also: How to Connect to Wi-Fi Network on Samsung Galaxy S3
Here you can also keep your phone's Wi-Fi on during it is asleep. But, you should note that when you turn this option on then you must pay it with more battery consumption.
How to turn on keep Wi-Fi on during
See also: How to Connect to Wi-Fi Network on Samsung Galaxy S3
Here you can also keep your phone's Wi-Fi on during it is asleep. But, you should note that when you turn this option on then you must pay it with more battery consumption.
How to turn on keep Wi-Fi on during