Critical Exponent Elliptic Equations on the Half-Space: Uniqueness and Explicit Solutions

We prove that all positive solutions of $-\mathrm{\Delta }u=u^{\frac{2n}{n-2}}$ on the upper half space ${\mathbb{R}}_{+}^n$ (for $n\ge 3$) satisfying the boundary condition $D_{x_n}u=-u^{\frac{n}{n-2}}$ are of the form $u(x)=a{\left(\frac{\lambda }{{\lambda }^2+\mathrm{\mid }x-y{\mathrm{\mid }}^2}\right)}^{\frac{n-2}{2}}$, where $a=a(n)$, $\lambda >0$, and $y=(y_1,\dots ,y_n)$ is a point in the lower half-space with $y_n<0$.