Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential

Tianxiang Gou; Vicenţiu D. Rădulescu

doi:10.5802/crmath.716

Research article - Partial differential equations

Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential

Tianxiang Gou ¹; Vicenţiu D. Rădulescu ^2,^3,^4,⁵

¹ School of Mathematics and Statistics, Xi’an Jiaotong University, 710049 Xi’an, Shaanxi, China
² Faculty of Applied Mathematics, AGH University of Kraków, 30-059 Kraków, Poland
³ Department of Mathematics, University of Craiova, 200585 Craiova, Romania
⁴ Brno University of Technology, Technická 3058/10, 61600 Brno, Czech Republic
⁵ Simion Stoilow Institute of Mathematics of the Romanian Academy, 010702 Bucharest, Romania

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 353-363.

Abstract (Original language)
Résumé

In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential:

\begin{equation*} (-\Delta )^s u+ \bigl (\omega +\vert {x}\vert ^2\bigr ) u=\vert u\vert ^{p-2}u \quad \text{ in } \mathbb{R}^n, \end{equation*}

where $n \ge 1$, $0<s<1$, $\omega >-\lambda _{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, and $\lambda _{1,s}>0$ is the lowest eigenvalue of the operator $(-\Delta )^s + \vert x\vert ^2$. This solves an open question raised in [15] concerning the uniqueness of solutions to the equation.

Dans cet article, nous établissons l’unicité des solutions positives pour l’équation elliptique non linéaire fractionnaire suivante avec potentiel harmonique :

\begin{equation*} (-\Delta )^s u+ \bigl (\omega +\vert {x}\vert ^2\bigr ) u=\vert u\vert ^{p-2}u \quad \text{ in } \mathbb{R}^n, \end{equation*}

où $n \ge 1$, $0<s<1$, $\omega >-\lambda _{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$ et $\lambda _{1,s}>0$ est la plus petite valeur propre de l’opérateur $(-\Delta )^s + \vert x\vert ^2$. Cela résout une question ouverte soulevée dans [15] concernant l’unicité des solutions de cette équation.

Received: 2024-09-21
Accepted: 2024-12-28
Published online: 2025-05-20

DOI: 10.5802/crmath.716

Classification: 35A02, 35R11
Keywords: Uniqueness, positive solutions, harmonic potential, fractional elliptic equations
Mots-clés : Unicité, solutions positives, potentiel harmonique, équations elliptiques fractionnaires

Author's affiliations:

Tianxiang Gou ¹; Vicenţiu D. Rădulescu ^{2, 3, 4, 5}

¹ School of Mathematics and Statistics, Xi’an Jiaotong University, 710049 Xi’an, Shaanxi, China
² Faculty of Applied Mathematics, AGH University of Kraków, 30-059 Kraków, Poland
³ Department of Mathematics, University of Craiova, 200585 Craiova, Romania
⁴ Brno University of Technology, Technická 3058/10, 61600 Brno, Czech Republic
⁵ Simion Stoilow Institute of Mathematics of the Romanian Academy, 010702 Bucharest, Romania

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2025__363_G4_353_0,
     author = {Tianxiang Gou and Vicen\c{t}iu D. R\u{a}dulescu},
     title = {Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {353--363},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.716},
     language = {en},
}

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Tianxiang Gou; Vicenţiu D. Rădulescu. Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 353-363. doi : 10.5802/crmath.716. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.716/

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