Engine vibrates strong at morning

I have a Peugeot 307 automatic model 2004 gasoline, and my problem is when you start the car in the morning you can press the gas pedal full down the car don’t move or it moves slightly from he’s position it has a power loss . At the next 20 seconds it gets better and better until after one minute turns to auto gas ( Icom) then it performs like new !
You hear it misfirering and the engine vibrates strong like the engine has a not adjusted carburetor. When the car is hot the problem appears the same but not so extreme. When you drive it with gasoline the engine vibrates slightly in stand only ! I think it is the gasoline pump but I am not sure because of it strange behavior of the engine! Can some one give me a TIP?