When I sing high notes, I sound breathy, and the notes don't hold very long. How can I maintain my high notes?

High notes require more strength in your vocal fold adductor muscles (the cricothyroids) to keep them together, as well as a larger amount of breath pressure (note: not breath flow...)

You need to excercise your vocal muscles by singing high notes EVERY OTHER DAY -- not every day because they need 48 hrs to heal between practice sessions. Its just like building strength in your biceps through lifting weights -- you can't start out to heavy or you'll tear muscles. You can't do what's comfortable, or you'll never build muscles. And, it takes time, but becomes easier the longer you practice.

The folds must be closed firmly -- not tightly, with the pressure from the lungs increased on the folds. Increased breath pressure is a MUST. You have to attack the high notes, you don't sing them as easily as the low notes. Nobody does, no matter what they may tell you. It takes courage and athleticism.

This will all result in a resonant high note.

There's alot more to it than just that, such as vowel covering/modification and soft palate placement. It would take a book to explain it all. The best way is to find a voice teacher that shares your voice type and that has performed professionally. This is really the only way to learn how to do it right.

Learn to moderate how much air you release (a teacher can help you with this). Learning correct breathing techniques will help you to maintain higher notes longer and to control the sound and make it richer. It is vital to use deep breathing techniques (a good resource is The Art of Breathing by Nancy Zi).

Breathy high notes mean that your vocal chords aren't closed all they way. Make sure to lift your soft palate when you sing. This feels kind of like a yawn. Practice in front of the mirror. This will help create a purer tone. Also, don't forget to push with your diaphragm and OPEN YOUR MOUTH!!

Don't push so hard, if you push less for the high note it will come out brighter, then go from there.