What version of Premiere Elements on Windows 10 (assumed 64 bit)?
When you get the dialog with the Continue Trial button, are you then clicking on License this product and doing that using your purchased serial number? Have you verified that you have the correct purchased serial number for the Premiere Elements that you purchased? By any chance, is your Premiere Elements the pre-installed type that came with the Sony computer?
The usual drill
a. Do you have the latest version of QuickTime installed (7.7.6 since Microsoft has had a problem with 7.7.8 latest version)?
b.Are you running the programs from a user account with administrative privileges. And, also have you right clicked the computer desktop icon and selected and applied Run As Administrator?
c. As you getting any error message except the pop of the the trial or buy message?
d. What was the last activity in the program before it stopped working with the trial or buy pop up?
Let us start here and then decide what next. And questions or need clarification on the above, please ask.
Madget, what version of premiere elements are you using? please send logs for investigation and snapshot of error at email@example.com tool for collecting logs could be downloaded from https://helpx.adobe.com/creative-cloud/kb/cc-log-collector.html