{"title":"Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential","authors":"Tianxiang Gou","doi":"10.1017/prm.2024.44","DOIUrl":null,"url":null,"abstract":"In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, <jats:disp-formula> <jats:alternatives> <jats:tex-math>\\[ (-\\Delta)^s u+ \\left(\\omega+|x|^2\\right) u=|u|^{p-2}u \\quad \\mbox{in}\\ \\mathbb{R}^n, \\]</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210524000441_eqnU1.png\" /> </jats:alternatives> </jats:disp-formula>where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$n \\geq 1$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline1.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$0< s<1$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline2.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\\omega >-\\lambda _{1,s}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline3.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$2< p< {2n}/{(n-2s)^+}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline4.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\\lambda _{1,s}>0$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline5.png\" /> </jats:alternatives> </jats:inline-formula> is the lowest eigenvalue of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(-\\Delta )^s + |x|^2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline6.png\" /> </jats:alternatives> </jats:inline-formula>. The fractional Laplacian <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(-\\Delta )^s$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline7.png\" /> </jats:alternatives> </jats:inline-formula> is characterized as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\\mathcal {F}((-\\Delta )^{s}u)(\\xi )=|\\xi |^{2s} \\mathcal {F}(u)(\\xi )$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline8.png\" /> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\\xi \\in \\mathbb {R}^n$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline9.png\" /> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\\mathcal {F}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline10.png\" /> </jats:alternatives> </jats:inline-formula> 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.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"124 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/prm.2024.44","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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