Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential

IF 1.3 3区 数学 Q1 MATHEMATICS
Tianxiang Gou
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引用次数: 0

Abstract

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, \[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \] where $n \geq 1$ , $0< s<1$ , $\omega >-\lambda _{1,s}$ , $2< p< {2n}/{(n-2s)^+}$ , $\lambda _{1,s}>0$ is the lowest eigenvalue of $(-\Delta )^s + |x|^2$ . The fractional Laplacian $(-\Delta )^s$ is characterized as $\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} \mathcal {F}(u)(\xi )$ for $\xi \in \mathbb {R}^n$ , where $\mathcal {F}$ denotes the Fourier transform. This solves an open question in [M. Stanislavova and A. G. Stefanov. J. Evol. Equ. 21 (2021), 671–697.] concerning the uniqueness of ground states.
带谐波势的分数非线性椭圆方程基态的唯一性
在本文中,我们证明了以下带谐波势的分数非线性椭圆方程基态的唯一性:[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \] 其中 $n \geq 1$ , $0<;s<1$ , $\omega >-\lambda _{1,s}$ , $2< p< {2n}/{(n-2s)^+}$ , $\lambda _{1,s}>0$ 是 $(-\Delta )^s + |x|^2$ 的最小特征值。分数拉普拉斯函数 $(-\Delta )^s$ 的特征为 $\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} 。\对于 $\xi \in \mathbb {R}^n$ 来说,这里的 $\mathcal {F}$ 表示傅立叶变换。这解决了[M. Stanislavova and A. G. Stefanov.
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来源期刊
CiteScore
3.00
自引率
0.00%
发文量
72
审稿时长
6-12 weeks
期刊介绍: A flagship publication of The Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics journal publishing peer-reviewed papers of international standard across the whole spectrum of mathematics, but with the emphasis on applied analysis and differential equations. An international journal, publishing six issues per year, Proceedings A has been publishing the highest-quality mathematical research since 1884. Recent issues have included a wealth of key contributors and considered research papers.
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