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Contact your local ASUS support to confirm, but if you can provide the serial number of the notebook, it should include some kind of manufacturer warranty from the date the unit was produced at the ASUS factory.
http://www.asus.com/support/CallUsSince the systems aren't normally sold right away, we normally cover the warranty from the customer's invoice date (as long as a copy can be provided of course) from an authorized ASUS reseller, so if you purchased it on ebay you may not have this extra warranty if it was not purchased from an authorized ASUS reseller as a brand new unit.
The GL553VE is a fairly new model though, so expect you may still be covered under the manufacturer warranty from the serial number if you can confirm with ASUS support.
Also, do you know if anything was upgraded in the unit like the memory and are you sure you have the correct and working AC adapter for the unit?
Normally the battery charge light should turn on once you plug in the AC adapter while the unit is turned off.
If you send back the unit for repair, it's better to remove any upgrades so parts don't get lost and you can also make sure they aren't related to the problem while you're at it.
Thanks and good luck!