Counterexamples to Wang's Conjecture

Abstract

In this paper, we give a negative answer to Wang's conjecture. For every $n\geq 3$, there exist $\varepsilon>0$ and $δ\in (0,1)$ such that, for all $q$ and $λ$ satisfying \[ \frac{n}{n-2}-\varepsilon<q\le \frac{n}{n-2}, \qquad \frac{1}{q-1}\cdot δ<λ\le \frac{1}{q-1}, \] there exists a bounded Euclidean domain \(Ω\subset\mathbb{R}^n\) 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*}