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

Abstract

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

arXiv comment13 pages. All comments are welcome