In areas that my HTC Rezound had a strong/near strong LTE signal my S4 won't stay connected. It will acquire the LTE signal, but then kick back to 3G after a few seconds. I have noticed several others on other Android sites having this problem as well. Is this a software issue with the phone? Is there a fix in the works?
No real expert here, but I can tell you that I also went from Rezound to S4 and have not experienced this issue at all. Sounds like a problem with your particular S4, I'd pay a visit to the Verizon store.
The phone typically works for me without issues you can try switching from the default of GLOBAL to LTE/CDMA and see if that helps
Settings | Connectivity Tab | More Networks | Network Mode | and select LTE / CDMA and see if that helps It kinda locks the phone to what it will use 99.99% of the time anyways. other then that I am not sure what to suggest other then attempt to goto a VZW store and explain your issue and ask for a new SIM....