You have to have a clean rug and a clean floor before you tackle your problem corners. Sweep your rug thoroughly to remove any surface dust, lint and other debris. Next, roll up the rug and thoroughly clean and dry the floor beneath it. If you have a rug that's small enough for you to handle on your own, take it outside and shake it out. If you steam-clean your carpets, make sure they're completely dry before continuing.
Your next task is to fix those curling corners. Set an iron to its hottest setting and let it heat up. Fold a clean, white towel in half and place it over the rolled-up corner. If you skip the towel, you may melt your rug. Iron on top of the towel for 15 to 20 seconds, applying light, downward pressure. Lift up the iron and the towel and check for curling and damage. Repeat until the rug lies flat then move onto the next corner.
Rug gripper tape is specially designed to prevent carpets from curling. It has an adhesive coating on both sides, so it sticks to your rug and then sticks to the floor. Apply at least 6 inches of gripper tape along both edges of your rug at the corner. The tape should overlap at the point of the corner. Once you have the tape in place, press your rug to the floor. Walk on the rug or place a heavy book on top for a few minutes to help the gripper tape make a strong seal. Repeat on all four corners.
Keeping your rug from moving will help prevent further curling and other damage. A non-slip rug pad is a simple solution to this problem. Measure your rug and the rug pad. Cut the rug pad so it's 1 to 2 inches smaller than your rug on all sides. Lay it down on the floor then lay your rug on top of it. Smooth the rug down with your hands or walk on it to make sure the non-slip pad hasn't bunched or folded anywhere.