If the car starts but won't stay ON there are several reasons that are causing it, but it will depend on the condition of the engine when the problem occurs.
The reason why car starts but won't stay on
(1)Bad electronic sensor.
(2) Problem with fuel injection system.
(3) Bad electric fuel pump.
Car troubleshooting
If starting happens only when the engine is cold. This starting occurs when the engine is cold but not when the engine has warmed up, then the problem is bad electronic sensors, but this will only apply for fuel injected cars.As indicated that the starting problem only happens when the engine is cold. It is reasonable to suspect that one of the electronic sensors in the car's fuel injection system has gone bad. A bad sensor can throw the computer off in its ability to calculate/regulate the amount of fuel the engine gets. In particular, a sensor is possibly not sensing the cold outside temperature and therefore the engine is not getting the rich air/fuel mixture it must have to stay running (when it's cold).
What to do?
Take the car to a reputable repair garage. The engine will most likely have to be hooked up to an engine analyzer which will then be used to determine if one of the sensors is indeed faulty. Most garages charge around $40 - $70 for this analysis. In addition to this, there will, of course, be the charges for the replacement of any sensor(s) which the mechanic found to be faulty. In most cases though, it is rarely more than one sensor which is the culprit.
If the starting happens not only when the engine is cold but also after the engine has warmed up, here's what you do. Locate the fuel pump (refer to this vehicle's manual if you need assistance). Now, turn the ignition key to the point just before the engine starts to turn over and leave the key in that position. If the fuel pump is located inside the gas tank, remove the gas cap and put your ear up to the gas tank opening and listen for a "whirring" sound. If the fuel pump is located under the hood, listen for that same "whirring" sound coming directly from the fuel pump. In either case, this sound means the electronic fuel pump is working.
If you hear the "whirring sound", then there is a problem with the fuel injection system, the problem could be that the fuel injection system is not operating normally.
What to do?
Fuel injection systems are complex and hard to diagnose. If the car is not under warranty, take it to a mechanic that has expertise in this particular make/model. We highly recommend you get a second opinion if the first mechanic suggests that you need a new fuel injection system. Very often, the problem is not the fuel injection system itself but rather one of its sensors. A skilled mechanic will be able to determine if all the sensors are working properly.
However, if you did not hear the "whirring sound", then this could be a bad electric fuel pump. Any vehicle with fuel injection has an electric fuel pump. A properly working electric fuel pump makes a sort of "whirring" type sound. Since there is no "whirring sound" , it could be the electric fuel pump must be bad. A bad fuel pump will not provide the engine with the fuel it needs to stay running after it starts.
What to do?
Replacing the fuel pump can be a simple or complex job depending on where it is located. If you feel so inclined, a manual for this vehicle will provide detailed instructions on removing and replacing the fuel pump. If you're not up to it, shop around for the best price.