A customer called saying his door stop wouldn’t stay in the wall. I assumed he meant the door stop wouldn’t stay in the baseboard, but you have to ask these questions. Yes, he did mean the wall.
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Installing a door stop in drywall is generally not a great idea unless the force of the impact is spread out among a larger area. Drywall will puncture easily, so placing a door stop there is asking for a hole, which defeats the idea of a door stop anyway. It is probably a better idea to move the door stop down to the baseboard, which will weather an impact better.
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Another option is to place the door stop on the back of the door itself so that when it contacts the wall, it will hit the baseboard. Some people install them at the top of the door, but you have the same problem with the force being focused on a small area (the tip of the stopper) and will puncture the drywall.
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You might also consider installing a hinge-pin door stopper. These sit at the top of the hinge pin and the force is placed at the door and the jamb. You will need to pop out the hinge pin, thread the stopper onto the pin, and tap the hinge pin back into the hinge.
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