Check the fuel pump relay to see if the contacts are pitted. (If you don't want to open the relay then borrow one from, say, the headlights, and see if it gets better.)
When I read this I thought right away that the start circuit is energizing the fuel pump to prime the system, but once started the system doesn't keep it energized. You may have a bad processor! But with the cost of those do other things first to make sure if it is or isn't a bad one.
Also, see about getting/borrowing a pressure gauge that can be used on gas. Check it immeditately after turning the key on. Then while it's running check it again. If it stays where it should (guessing here - ~50-70 PSI is commong on FI) then the pump is working and something else is the issue. If it drops then check the fuel pressure regulator.
You can get quite inventive in diagnosing this. The relay simply provides power to the pump. Pull it and then find the 12V supply and the contact going to the pump. Jump them (caution: do this at your own risk!) and see if that cures your problem.
Find the wire leading to the pump and put a test light on it. When the key is turned on it should light for anywhere from 1/2 a second to several seconds and then turn off. Once it's running it shoud turn on continuously.
When did you last change the fuel filter?