Try using a different original Nintendo DSi AC adapter with the unit if possible. If the unit does function with a different AC adapter, only the AC adapter will be required for inspection by a Nintendo technician. Alternatively, should the product be outside of the warranty period a new official AC Adapter can be purchased by contacting Nintendo Customer Service.
If this does not correct your issue, the unit and AC Adapter will both need to be returned to Nintendo Australia for inspection/repair.
NOTE: Only the AC Adapter from the same model Nintendo DSi can be used, eg. If testing for an issue with a Nintendo DSi make sure to test ONLY with parts from another Nintendo DSi, not a Nintendo DS Lite or Nintendo DS. The same also applies to the Nintendo DSi XL.