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

TL;DR

This paper classifies all positive solutions to a critical elliptic PDE with boundary conditions on the half-space, showing they are explicit, symmetric functions parametrized by points in the lower half-space.

Contribution

It provides a complete classification and explicit form of solutions for a critical elliptic equation with boundary conditions on the half-space, extending known results.

Findings

All positive solutions are of a specific explicit form.

Solutions are parametrized by points in the lower half-space.

The results establish uniqueness and explicit solution formulas.

Abstract

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