What version of Premiere Elements and on what computer operating system?
Did you purchase Premiere Elements as a download from Adobe and have it previously installed on your computer?
If so, you should be able to go to your account at Adobe web site(My Products section), find your version listed in your products, and click on the program's listing there to reveal the purchased serial number.
If you purchase the box packaged product from a retailer, the serial number should be on a label in the packaging. If the box with the serial number is gone, no where to go.