Counterexamples to Wang's Conjecture

In this paper, we give a negative answer to Wang’s conjecture. For every $n\ge 3$, there exist $\epsilon >0$ and $\delta \in (0,1)$ such that, for all $q$ and $\lambda$ satisfying $$\frac{n}{n-2}-\epsilon <q\le \frac{n}{n-2},\frac{1}{q-1}\cdot \delta <\lambda \le \frac{1}{q-1},$$ there exists a bounded Euclidean domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with principal curvatures bigger than $1$ for which the following nonlinear Robin problem admits a nonconstant positive solution: \begin{align*} \begin{cases} Δu=0, & $\text{in}$\ Ω,\[2mm] \dfrac{\partial u}{\partialν}+λu=u^q, & $\text{on}$\ \partialΩ. \end{cases} \end{align*}