A student whose self-esteem in performnace is evident stands out among the crowd. You need to prepare them psychologically as well as musically. Is their memory of their work complete or do they only rely on muscle memory? As a good test, , give them score paper and ask them to write down the first 4 measures of their piece. If hey can't do it or forget certain things, such as dynamics or slurrs, they need to secure their memory. They do this by studying the score in their head away from the piano. Every work they play should be studied in this manner along with regular keyboard practice. It also helps if they have knowledge of basic theory so they can understand the harmonic progression of the works, it gives them one more tool to hold on to if they have a memory lapse. Knowing that you have several backups handy in case of a performance messup goes a very long way to performing with confidence.
-Set a date 1 or two weeks before the competition date as the "competition date". Prepare them to be fully ready for this earlier date. On that date hold a small recital at your or their home, they must dress with what they will wear to the competition, perform the works and see what happens. Like this they still have over a week to fix the problem spots and the competition becomes just another performance as opposed to this stressful one-time thing.
-From about 4 weeks before the competition ask your students to do non-piano homework, ask them to take 10-20 minutes every day to close their eyes and visualize clearly the competition day. Where it is (if they donīt know what it looks like, any performance space will do), who is there, what they will be wearing. They visualize it all, from the moment they walk onto the stage to the moment they walk off, they hear the coughs the sighs, the occasional whispers, and throughout it all, they see themselves playing confidently, every note sounding like they want it to sound. If they begin feeling the nerves in their stomach as they picture this, have them stop and remember a time when something they did made them feel happy and fulfilled, gave them a huge sense of accomplishment, then transfer this feeling back to the competition stage in their head. Try to keep that feeling of accomplishment in their performance.
This type of excercise does require mental effort on their part but by the time they reach the actual competition/performance day, they will have performed the work successfully in their head dozens of times, and had the chance to perform it for real at least a week before. Although there will still be nerves on the day, this type of excercise gets them to focus them positively and this type of confidence is evident to all the judges. I have been to many competitions, you can tell.
For more ideas like this I recommend compser Michael Colgrass's book "My Lessons with Kumi" it is available on Amazon.