Procedure for Drooping Eye Corners?

Correcting a drooping outer eye is a challenging position. It often combines the lower eyelid, the cheek, the cannthus, the upper eyelid and the lateral brow. Unless the lower eyelid alone has a problem (looseness or prior surgery pulling it down), all five of these measures are usually necessary to achieve a haronious, balanced result.
It can make a dramatic difference in a patient's appearance, particularly in reducing the "sad" appearance many patients have.
Below two articles on the topic, including some extreme examples:

There are many potential reasons for "drooping eye corners"
You must look at the brow, upper eyelid, canthal support mechanisms, lower eyelid, and cheek.
Any and all of these anatomic structures can contribute to your problem. Therefore, I would highly suggest that you see a plastic surgeon to discuss your specific problem and recommended treatment options.