A proof of Guo-Wang's conjecture on the uniqueness of positive harmonic functions in the unit ball

TL;DR

This paper proves Guo-Wang's conjecture that under certain conditions, the only positive harmonic functions in the unit ball with specified boundary behavior are constant.

Contribution

The paper provides a rigorous proof confirming Guo-Wang's conjecture on the uniqueness of positive harmonic functions with boundary conditions in the unit ball.

Findings

Positive harmonic functions with given boundary conditions are constant under specified parameters.

The conjecture by Guo-Wang is validated through the proof.

The result extends understanding of boundary value problems for harmonic functions.

Abstract

Guo-Wang [Calc.Var.Partial Differential Equations,59(2020)] conjectured that for $1<q<\frac{n}{n-2}$ and $0<\lambda\leq \frac{1}{q-1}$, the positive solution $u\in C^{\infty}(\bar B)$ to the equation \[ \left\{ \begin{array}{ll} \Delta u=0 &in\ B^n,\\ u_{\nu}+\lambda u=u^q&on\ S^{n-1}, \end{array} \right. \] must be constant. In this paper, we give a proof of this conjecture.