Where did you get your serial number from?
Also ensure that the serial number you're using is for Premiere Elements and not for Photoshop Elements.
If you have confirmed that you have the correct purchased serial number for Premiere Elements 13, you should contact Adobe Chat Activation to help you. It is the only one to handle this type of problem.
If something has been lost in the translation and your problem is really that you keep getting the License this Product or Buy message when you try to open Premiere Elements, then you should be able to resolve the issue by clicking on License this Product and giving the purchased serial number that you have for the specific product that you have purchased.
Also, make sure that you have a serial number and are not trying to use a redemption code or other code.
Please let us know if you have any questions.