Opuscula Math. 44, no. 5 (2024), 749-765
https://doi.org/10.7494/OpMath.2024.44.5.749
Opuscula Mathematica
Properties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential
César E. Torres Ledesma
Hernán C. Gutierrez
Jesús A. Rodríguez
Manuel M. Bonilla
Abstract. In this article we consider the following fractional semilinear elliptic equation \[(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,\] where \(s\in (0,1)\), \(N\gt 2s\), \(\sigma\in (0,\frac{2s}{N-2s})\) and \(\omega\in (0, \lambda_1)\). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result.
Keywords: harmonic potential, fractional Sobolev space, ground state solution, bifurcation result, variational method.
Mathematics Subject Classification: 45G05, 35J60, 35B25.
- C. Alves, C. Torres Ledesma, Existence and concentration of solution for a non local regional Schrödinger equation with competing potentials, Glasg. Math. J. 61 (2019), no. 2, 441-460. https://doi.org/10.1017/S0017089518000289
- F. Almgren, E. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), 683-773. https://doi.org/10.1090/S0894-0347-1989-1002633-4
- A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), no. 4, 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
- X. Chang, Ground state solutions of asymptotically linear fractional Schrödinger equation, J. Math. Phys. 54 (2013), no. 6, 061504. https://doi.org/10.1063/1.4809933
- M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys. 53 (2012), no. 4, 043507. https://doi.org/10.1063/1.3701574
- E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521-573.
- Z. Ding, H. Hajaiej, On a fractional Schrödinger equation in the presence of harmonic potential, Electron. Res. Arch. 29 (2021), no. 5, 3449-3469. https://doi.org/10.3934/era.2021047
- S. Dovetta, E. Serra, P. Tilli, Action versus energy ground states in nonlinear Schrödinger equations, Math. Ann. 385 (2023), 1545-1576. https://doi.org/10.1007/s00208-022-02382-z
- P. Felmer, C. Torres Ledesma, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 75-98. https://doi.org/10.1007/s00526-014-0778-x
- G. Fibich, X.-P. Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Phys. D 175 (2003), no. 1-2, 96-108. https://doi.org/10.1016/S0167-2789(02)00626-7
- T. Gou, Uniqueness of ground states for fractional nonlinear elliptic problem with harmonic potential, arXiv:2208.12068v2,(2022). https://doi.org/10.48550/arXiv.2208.12068
- H. Hajaiej, L. Song, Comment on the uniqueness of the ground state solutions of a fractional NLS with a harmonic potential, arXiv:2209.05389v1, (2022). https://doi.org/10.48550/arXiv.2209.05389
- H. Hajaiej, P. Markowich, S. Trabelsi, Minimizers of a class of constrained vectorial variational problems: Part I, Milan J. Math. 82 (2014), no. 1, 81-98. https://doi.org/10.1007/s00032-014-0218-6
- L. Jeanjean, Some continuation properties via minimax arguments, Electron. J. Differential Equations 2011 (2011), Article no. 48. https://doi.org/10.48550/arXiv.1104.0767
- S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^N\), J. Math. Phys. 54 (2013), no. 3, 031501. https://doi.org/10.1063/1.4793990
- L. Song, Properties of the least action level, bifurcation phenomena and the existence of normalized solutions for a family of semi-linear elliptic equations without the hypothesis of autonomy, J. Differential Equations 315 (2022), 179-199. https://doi.org/10.1016/j.jde.2022.01.035
- C. Torres Ledesma, Existence and symmetry result for fractional \(p\)-Laplacian in \(\mathbb{R}^n\), Commun. Pure Appl. Anal. 16 (2017), no. 1, 99-113. https://doi.org/10.3934/cpaa.2017004
- X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997), no. 3, 633-655. https://doi.org/10.1137/S0036141095290240
- J. Zhang, X. Bao, J. Zhang, Existence and concentration of solutions to Kirchhoff-type equations in \(\mathbb{R}^2\) with steep potential well vanishing at infinity and exponential critical nonlinearities, Adv. Nonlinear Anal. 12 (2023), no. 1, 20220317. https://doi.org/10.1515/anona-2022-0317
- W. Zhang, S. Yuan, L. Wen, Existence and concentration of ground-states for fractional Choquard equation with indefinite potential, Adv. Nonlinear Anal. 11 (2022), no. 1, 1552-1578. https://doi.org/10.1515/anona-2022-0255
- César E. Torres Ledesma (corresponding author)
https://orcid.org/0000-0002-6889-4982- FCA Research Group, Departamento de Matemáticas, Instituto de Investigación en Matemáticas, Faculta de Ciencias Físicas y Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n. Trujillo-Perú
- Hernán C. Gutierrez
- https://orcid.org/0000-0001-8947-5368
- FCA Research Group, Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n. Trujillo-Perú
- Jesús A. Rodríguez
- https://orcid.org/0000-0001-7070-4781
- FCA Research Group, Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n. Trujillo-Perú
- Manuel M. Bonilla
- https://orcid.org/0000-0002-5437-7802
- FCA Research Group, Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n. Trujillo-Perú
- Communicated by Vicenţiu D. Rădulescu.
- Received: 2023-07-17.
- Revised: 2024-03-19.
- Accepted: 2024-03-21.
- Published online: 2024-07-01.
