https://www.adobe.com/account.html for serial numbers on your Adobe page... or
For preinstalled, you SHOULD have received a piece of paper with the computer with the serial number
Please try the following possible solution
The serial number part
1. Go to the Adobe web site http://www.adobe.com, sign in with your Adobe ID and Password, go to the Search field at the top right of that web page and type in My Products.
2. Follow the My Products link to find your Adobe Premiere Elements 10 product. If you find it, chick on Adobe Premiere Elements 10 to reveal its serial number.
The software part
1. Go to the following web site and download and install the tryout of Premiere Elements 10. During installation insert your purchased serial number.
Be sure to read and carry out carefully the web page's Note: Very Important Instructions" and its click "here" before all else in order to avoid Access Denied.
Pre-installed programs do not always come with serial numbers. This type of matter needs to be resolved at the time of purchase of the computer.
You need both installation files as well as the purchased serial number for those files to move forward. Adobe cannot help you on this type of situation.
Your last resort is Sony who probably will not be able to help you unless you have a sales agreement stating your rights.
We will be watching for further developments. I am hoping that the route that I have suggested works.
What video card/graphics card does your computer use? NVIDIA GeForce?