check your battery. may be KONG> also the comutator <==(not sure how this spell)

alternator is like dynamo.. it is charge your battery when your engine car was switch on

my just exp same thing like this. he replace the battery but still got the same problem. then send formen. he said alternator... now it is ok.
try lok for formen. maybe another problem

for engine starting problems, there are only few major components to check:

- check your starter, perhaps can add starter relay, or even check if the fuses are burning out (though they typically either work or they don't) or even the wiring around the fuse box area.

- check your spark plugs, see if they are fouled/covered in carbon, or even melted/thinned our spark tip/grooves

- battery, basic checking would be it's voltage, minimum also 11.5, ideally you'd want it to be between 12-12.5 volts. if you know workshops with more advanced meters, can also check it's cold crank ampere (CCA) to determine if the battery is in good condition or not. (doesn't matter if it is new battery, some batteries are just bad and their CCA drops very fast in just few months of usage)
alternator should not be a problem during cranking, as your battery is the one supplying power, and after engine is started (along with the alternator) only then the alternator takes over in supplying power as well as charging the battery for it's lost current due to the cranking.