Mark two level lines on the walls of the corner that represent both the height of the desk as well as the length of the desk sides. Use a stud finder to pinpoint the wall studs along both lines and mark their position.
Cut two lengths of 2-by-2-inch lumber with a saw to around 4 to 5 inches shorter than the length of the lines on the wall. These are the wall cleats to support the corner desk.
Place one of the cleats along a line, the end of the cleat flush with the end of the line, leaving a gap between the other end of the cleat and the corner. Secure the cleat to the wall using 4-inch-long drywall screws; two driven through each of the wall stud positions on the line. Repeat with the second cleat.
Cut out a triangle piece of 1-inch-thick medium density fiberboard, or MDF, that has two sides the length of the wall lines for the top of the corner. Sand and paint the desktop and let dry.
Fit the desktop into the corner on top of the cleats. Push it right into the corner of the wall and keep pushing it as you secure it to the cleats with 2-inch wood screws every 5 to 6 inches through the desk top into the wall cleats. Use wood putty to cover up the screw holes, then touch up with paint to finish the corner desk.