Lightroom uses the Melissa RGB color space.
Melissa RGB is the ProPhoto RGB color space with the same gamma curve as sRGB.
You can't change this color space.
While Photoshop supports the ProPhoto RGB color space, Photoshop Elements doesn't.
The closest color space is sRGB.
Check each program as follows.
Make sure Photoshop Elements is set to sRGB.
Do the following in Photoshop Elements.
1) Go to Edit > Color Settings.
2) Select Always Optimize Colors for Computer Screens.
Do the following in Lightroom.
1) Go to Edit > Preferences (Windows) or Lightroom > Preferences (Mac).
Or, press Ctrl + ,.
2) Click the External Editor tab.
3) Set Color Space to sRGB, the closest match to the ProPhoto RGB color space used in Lightroom.
The gamma of ProPhoto RGB is 1.8
In Lightroom, the gamma for ProPhoto RGB is 1.0.
This is the same gamma as raw files.
Go to Gamma Explained.