Why does my phone keep cutting on and off?

My phone keeps cutting off and on , showing the Apple sign . Then after a while it dies… I have tried new batteries . Rebooting it. And

I even got a new one . It had the same problem .

we took it to Verizon and they said it’s not the batterie , it’s the phone . Turns out they just wanted us to buy a new one from them … but I still have the same problem with it on a day to day basis . Please help me!