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It depends upon whether this is a light pollution gradient (you have extra light that needs to be subtracted) or vignetting (you're missing light in areas that need to be multiplied to compensate). I'm going to assume that it's the latter - vignetting.
Since there isn't any nebulosity in the image, you can just remove the stars from the image and use it as a flat. Here are instructions for PS (I assume they'll work in Elements - but I've never used it before):
1. Open the image in PS.
2. Convert to 16-bit mode
3. Duplicate layer.
3. Gaussian blur the new layer 100 px (depends upon image size - you want to remove all stars and other interesting features, and keep the image gradients smooth).
4. Open Levels and stretch the white point left until it reaches right-most edge of the histogram (53 with your web sized image).
5. Change the blend mode of the layer to Divide.
6. Flatten the two layers.
Your vignetting is extremely severe - there was no detail around the edges in your image, so this won't help with that. You can see that the central region becomes more evenly illuminated (this will work much better with full sized images - convert from RAW directly to a 16-bit image for editing if you can).
Hope this helps.
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