Samsung Galaxy S III update for Verizon has some issues with LTE connectivity
According to those who have already installed the update, it apparently interferes with signal strength. Some users who had been getting 4 bars of 4G LTE service are now getting just 1 or 2 bars. Others are saying that data speeds have slowed considerably while some aren't receiving any 4G LTE signal at all. And even the usual fixes, like turning on and off airplane mode, or doing a factory restart, aren't solving the problem.
One unofficial solution is to change your settings from Global to LTE/CDMA. Go to Settings > More Networks > Mobile Networks and change the network mode to LTE/CDMA instead of Global. Let us know if that helps you out by dropping a comment in the box below. This is a pretty big update for Samsung Galaxy S III owners on Verizon with some cool features that have been long awaited. Let's hope Samsung can take care of this quickly.
source: AndroidCommunity via AndroidAuthority