Re: why do the galaxy note 2 can't be charged by another charger ?
A bad battery won't charge with any charger. But these days, since the industry standardized on the microUSB charging port, almost any charger will charge almost any phone. If the charger doesn't put out enough current it'll take longer to reach full charge, but it'll still work. (If the charger can put out more than the phone needs, the phone will only draw as much as it needs.)
If you try different chargers and different cables, and the phone still doesn't show that the battery is charging, I'd suspect one of two things - either a bad battery or one of the pins on the charging port became disconnected. (Considering the strain that connector gets if it's charged by cable every day, it's amazing that they last more than a few weeks.) That would require a new microUSB board or resoldering the connector to the existing board.