I have a 11hp briggs and stratton mower motor model 258707 and it wont start...what is the problem

check for the small wire to the carby that makes contact when the throttle is in the stop position is not shorting out

if its a briggs and stratton engine with the plastic carb fixed to top of the fuel tank then you will need to replace the diaphragm and blow out the fuel pick up pipe

Check the battery and battery lead terminals are clean (wire brush or a scotchbrite) and tight.
Still not working: - then check the battery lead connections to earth (chassis), starter solenoid and starter motor.

It could be the battery is just no good. I had mine do that and I took the battery to have it checked and they said it was ok and charged it for me came home started right away but after I turned it off it wouldn't start again so I went and got a new battery and have not had that happen again.

what do you mean by overheating make sure all the fins in engine are clean and free of debris and also the top of flywheel is clear as well it must be to keep it running cool

check the drive belt for having oil on it and missing pieces and enough tension

if the motor has been on its side the oil has runs out thru the muffler and air filter. briggs are famous for doing that. let it sit overnight and remove the air cleaner and you see is fill of oil. next day take the spark plug and clean it and check the gap then put it back, take carburator cleaner and spray some inside the carburator, check the oil level and start.if it wont start then is time to clean carburator. the air filter if is wet in oil and is the dry type. get a new one.

check gas tank with light is there dirt in tank? did you change fuel filter? is there spark at the spark plug? try a 2 second spurt of starting fluid in carburetor and crank does it fire?if it does then you most likely need to clean carburetor with carburetor cleaning spray, flush fuel lines with spray and swap fuel filter.

I would check the stator which is located on the top or side of the motor, this is what produced spark and if there is rust on it it will lose spark. so basically remove the fan housing the top of the motor and you should see it either on top or directly on the side i believe it has one wire going to it green in color and is held down by two bolts. good like look out for the rust

With your model number I could see if it has a low oil safety switch I have seen them go bad (check your oil level )
With all wires un plugged from the coil you should have spark
if not coil is bad .
Use another spark plug to test for spark, ground the side of the plug by holding it against some-ting metal on the engine , spark
plug wire hooked to the plug. leave a spark plug in the cylinder
head so gas wont fly out and hit the test spark plug catching fire (seen it done).
Good luck