Creating a Desktop shortcut in Linux Mint

In ~/Desktop, create a file named Eclipse.desktop with content:

[Desktop Entry]
Version=x.y
Name=ProgramName
Comment=This is my comment
Exec=/home/peter/Documents/exec.sh
Icon=/home/peter/Pictures/icon.png
Terminal=false
Type=Application
Categories=Utility;Application;

if you need to run the command as root, change:

Exec=gksu /usr/local/something/AppRequiringRootPriv