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

In a recent work published in Calc.Var.Partial Differential Equations, 59 (2020), Guo and Wang proposed a conjecture that for any \(1<q<\frac{n}{n-2}\) $1<q<\frac{n}{n-2}$ and \(0<\lambda \le \frac{1}{q-1}\) $0<\lambda \le \frac{1}{q-1}$ , a positive solution \(u\in C^{\infty }({\bar{B}})\) $u\in C^{\infty }(\overline{B})$ to the equation \(\begin{aligned} \left\{ \begin{array}{ll} \Delta u=0 & in\ B^n,\\ u_{\nu }+\lambda u=u^q& on\ S^{n-1}, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }u=0\hfill & in{B}^n,\hfill \\ u_{\nu }+\lambda u=u^q\hfill & on{S}^{n-1},\hfill \end{array}\right)\end{array}$$ must be constant. In this paper, we give a complete proof of this conjecture.