Positive eigenfunctions of a class of fractional Schrödinger operator with a potential well

Differential Integral Equations 35 (2022), no. 1-2, 123–150

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 Schr"{o}dinger equations to the nonlocal fractional setting.