Abstract
In this paper, we are concerned with the following eigenvalue problem \begin{equation*} (-\Delta)^su+\lambda g(x)u=\alpha u,\ \ u\in H^s(\mathbb{R}^N),\ N\geq3, \end{equation*} where $s\in(0,1),\,\alpha,\lambda\in\mathbb{R}$ and \begin{equation*} g(x)\equiv0\ \text{on}\ \bar{\Omega},\ \ g(x)\in(0,1]\ \text{on}\ \mathbb{R}^N\backslash\bar{\Omega} \ \text{and}\ \lim_{|x|\rightarrow\infty}g(x)=1 \end{equation*} for some bounded open set $\Omega\subset\mathbb{R}^N$. We discuss the existence and some properties of the first two eigenvalues for this problem, which extend some classical results for semilinear Schrödinger equations to the nonlocal fractional setting.
Citation
Guangze Gu. Zhipeng Yang. "Positive eigenfunctions of a class of fractional Schrödinger operator with a potential well." Differential Integral Equations 35 (1/2) 123 - 150, January/February 2022. https://doi.org/10.57262/die035-0102-123
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