Explicit forms for extremals of sharp Sobolev trace inequalities on the unit balls

We show explicit forms for extremals of some fourth-order sharp Sobolev trace inequalities on the unit balls recently proved by Ache-Chang and Case. We also give a classification result of the bi-harmonic equation on \({\mathbb {R}}^4_+\) ${\mathbb{R}}_{+}^4$ with some conformally covariant boundary conditions. Moreover, we show a classification result for an associated integral equation.