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

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 $u\in C^{\mathrm{\infty }}(\bar{B})$ to the equation $$\{\begin{array}{ll}\mathrm{\Delta }u=0& in{B}^n,\\ u_{\nu }+\lambda u=u^q& on{S}^{n-1},\end{array}$$ must be constant. In this paper, we give a proof of this conjecture.