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January/February 2022 Positive eigenfunctions of a class of fractional Schrödinger operator with a potential well
Guangze Gu, Zhipeng Yang
Differential Integral Equations 35(1/2): 123-150 (January/February 2022).

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

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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.

Information

Published: January/February 2022
First available in Project Euclid: 6 January 2022

Subjects:
Primary: 35J10 , 35J50 , 35Q40

Rights: Copyright © 2022 Khayyam Publishing, Inc.

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Vol.35 • No. 1/2 • January/February 2022
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