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

Tianxiang Gou; Vicenţiu D. Rădulescu

doi:10.5802/crmath.716

Article de recherche - Équations aux dérivées partielles

Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential
[Unicité des solutions positives aux équations elliptiques non linéaires fractionnaires à potentiel harmonique]

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 (VO)
Résumé

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

(- Δ)^{s} u + (ω + | x |^{2}) u = | u |^{p - 2} u in R^{n},

where $n \geq 1$ , $0 < s < 1$ , $ω > - λ_{1, s}$ , $2 < p < \frac{2 n}{(n - 2 s)^{+}}$ , and $λ_{1, s} > 0$ is the lowest eigenvalue of the operator $(- Δ)^{s} + | x |^{2}$ . 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 :

(- Δ)^{s} u + (ω + | x |^{2}) u = | u |^{p - 2} u in R^{n},

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

Reçu le : 2024-09-21
Accepté le : 2024-12-28
Publié le : 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

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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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