Normalized solutions to a Choquard equation with potential and mixed local-nonlocal nonlinearities

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This work investigated normalized solutions for the Choquard equation with external potential and mixed local-nonlocal nonlinearities

$\left\{\begin{array}{l} -\Delta u+(V(x)+\lambda) u=\gamma\left(I_{\alpha} *|u|^{\frac{N+\alpha}{N}}\right)|u|^{\frac{\alpha}{N}-1} u+\mu|u|^{p-2} u, x \in \mathbb{R}^{N}, \\ \int_{\mathbb{R}^{N}}|u|^{2} d x=\rho^{2}, \end{array}\right.$

where $N \geq 3, \gamma, \mu>0, \lambda \in \mathbb{R}, p \in\left(2+\frac{4}{N}, 2^{*}\right), 2^{*}=\frac{2 N}{N-2}$, and Iα stood for the Riesz potential when α ∈ (0,N). Under finite potential conditions, normalized ground states were verified. For trapping potentials, normalized ground states paired with bound states were obtained.

Keywords

Choquard equation / lower critical exponent / normalized solutions / external potential / variational methods

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105. https://doi.org/10.1002/sapm197757293

[2] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. https://doi.org/10.1016/0362-546X(80)90016-4

[3] S. I. Pekar, Untersuchung über die Elektronentheorie der Kristalle , De Gruyter Brill, 1954. https://doi.org/10.1515/9783112649305

[4] F. Gao, M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242. https://doi.org/10.1007/s11425-016-9067-5

[5] V. Moroz, J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. https://doi.org/10.1016/j.jfa.2013.04.007

[6] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 1 (1984), 223-283. https://doi.org/10.1016/s0294-1449(16)30422-x

[7] G. Li, H. Ye, The existence of positive solutions with prescribed L²-norm for nonlinear Choquard equations , J. Math. Phys., 55 (2014), 121501. https://doi.org/10.1063/1.4902386

[8] X. Luo, Normalized standing waves for the Hartree equations, J. Differ. Equations, 267 (2019), 4493-4524. https://doi.org/10.1016/j.jde.2019.05.009

[9] H. Luo, L. Wang, Normalized ground state solutions for nonautonomous Choquard equations, Front. Math., 18 (2023), 1269-1294. https://doi.org/10.1007/s11464-020-0189-6

[10] P. Jin, X. Tang, D. Hu, The existence of normalized solutions of the L²-supercritical Choquard equations , Milan J. Math., 93 (2025), 149-171. https://doi.org/10.1007/s00032-025-00415-1

[11] Y. Meng, B. Wang, Multiplicity of normalized solutions to a class of non-autonomous Choquard equations, J. Geom. Anal., 35 (2025), 12. https://doi.org/10.1007/s12220-024-01844-x

[12] X. Li, J. Bao, W. Tang, Normalized solutions to lower critical Choquard equation with a local perturbation, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 3216-3232. https://doi.org/10.3934/dcdsb.2022213

[13] S. Mo, S. Ma, Normalized solutions to lower critical Choquard equation in mass-supercritical setting, J. Geom. Anal., 36 (2026), 82. https://doi.org/10.1007/s12220-026-02330-2

[14] S. Yao, H. Chen, V. D. Radulescu, J. Sun, Normalized solutions for lower critical Choquard equations with critical Sobolev perturbation, SIAM J. Math. Anal., 54 (2022), 3696-3723. https://doi.org/10.1137/21m1463136

[15] J. Chen, Z. Chen, Normalized solutions to Brezis-Nirenberg-type problem for a Schrödinger equation with van der Waals type potentials, II: existence and multiplicity, Z. Angew. Math. Phys., 76 (2025), 73. https://doi.org/10.1007/s00033-025-02455-4

[16] Y. Ding, H. Wang, Normalized solutions to Schrödinger equations with critical exponent and mixed nonlocal nonlinearities, J. Geom. Anal., 34 (2024), 215. https://doi.org/10.1007/s12220-024-01667-w

[17] P. Cao, J. Wang, W. Zou, On the standing waves for nonlinear Hartree equation with confining potential, J. Math. Phys., 53 (2012), 033702. https://doi.org/10.1063/1.3691190

[18] Y. Ao, X. Zhao, W. Zou, Normalized solutions for nonlinear Choquard equations with general nonlocal term, J. Fixed Point Theory Appl., 25 (2023), 17. https://doi.org/10.1007/s11784-022-01027-7

[19] W. Xing, H. Suo, C. Lei, Normalized solutions for the Choquard equation with potential and combined nonlinearities, Adv. Nonlinear Anal., 15 (2026), 20250150. https://doi.org/10.1515/anona-2025-0150

[20] J. Chen, Z. Chen, Normalized solutions for a Schrödinger equation with van der Waals type potentials and external potential: Mass supercritical, Math. Methods Appl. Sci., 48 (2025), 5781-5789. https://doi.org/10.1002/mma.10636

[21] Y. Li, L. Zhao, Normalized solutions for Schrödinger equations with potentials and general nonlinearities, Cal. Var., 63 (2024), 99. https://doi.org/10.1007/s00526-024-02699-4

[22] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. https://doi.org/10.1016/S0362-546X(96)00021-1