Reinstall Photoshop Elements 15 on Win 10 , i have a serial number , no disk.

If your original purchase was a DVD version, then there is no link available for you to download and you will have to purchase a new copy.

If your original purchase was an electronic download from Adobe, then you should have a link in your Adobe.com account.  Go to this link.  You will have a sign in page (if you are not automatically signed in).  Once you are signed in, you should see all of your purchased products with serial number and download links to your Elements 15. 

If that is not available, start an online chat session with Adobe support by going to Adobe.com and clicking on the blue Contact Us button under the Support menu.  I'm not sure if they will give you a download link or not.  They keep changing their policies and I don't know what the current one is.

If you are successful, please report back.