发表机构
School of Mathematics, Hunan University; School of Mathematics and Computational Science, Xiangtan University（湖南大学数学学院; 湘潭大学数学与计算科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了反例，否定了Wang猜想：对任意维数n≥3，在特定参数范围内存在主曲率大于1的有界域，使得非线性Robin问题有非平凡正解。
AI 中文摘要
本文对Wang猜想给出了否定的回答。对于每个$n\geq 3$，存在$\varepsilon>0$和$\delta \in (0,1)$，使得对所有满足\\[ \frac{n}{n-2}-\varepsilon<q\le \frac{n}{n-2}, \qquad \frac{1}{q-1}\cdot \delta<\lambda\le \frac{1}{q-1} \\]的$q$和$\lambda$，存在一个有界欧几里得域$\Omega\subset\mathbb{R}^n$，其主曲率大于$1$，使得以下非线性Robin问题 admits a nonconstant positive solution: \begin{align*} \begin{cases} \Delta u=0, & \text{in}\\ \Omega,\\\\[2mm] \dfrac{\partial u}{\partial\nu}+\lambda u=u^q, & \text{on}\\ \partial\Omega. \end{cases} \end{align*}
英文摘要
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*}
