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

Part of: Miscellaneous topics - Partial differential equations Partial differential equations

Published online by Cambridge University Press:  11 April 2024

Tianxiang Gou
Tianxiang Gou*
Affiliation:
School of Mathematics and Statistics, Xi'an Jiaotong University, 710049 Xi'an, Shaanxi, China ([email protected])
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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.

Keywords

Uniquenessground statesharmonic potentialfractional elliptic equations

MSC classification

Primary: 35A02: Uniqueness problems
Secondary: 35R11: Fractional partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 14
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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