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

Huiling Wu; Liping Wu; Huiling Wu; Liping Wu

doi:10.3934/era.2026368

Electronic Research Archive

2026, Volume 34, Issue 12: 8740-8755. doi: 10.3934/era.2026368

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Research article

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

Huiling Wu ^,,
Liping Wu

School of Cyber Science and Engineering, Minjiang University, Fuzhou 350108, China

Received: 17 July 2026 Revised: 28 August 2026 Accepted: 22 September 2026 Published: 23 October 2026

This work investigated normalized solutions for the Choquard equation with external potential and mixed local-nonlocal nonlinearities
$ \left\{ \begin{array}{lcl} -\Delta {u}+(V(x)+\lambda)u = \gamma(I_{\alpha}*|u|^{\frac{N+\alpha}{N}})|u|^{\frac{\alpha}{N}-1}u+\mu |u|^{p-2}u,\,\,x\in { \mathbb{R}}^{N}, \\ \int_{{ \mathbb{R}}^{N}}|u|^{2}dx = \rho^{2}, \end{array}\right. $
where $ N\ge 3 $, $ \gamma, \, \mu > 0 $, $ \lambda\in \mathbb{R} $, $ p\in(2+\frac{4}{N}, 2^{*}) $, $ 2^{*} = \frac{2N}{N-2} $, and $ I_{\alpha} $ stood for the Riesz potential when $ \alpha\in(0, N). $ Under finite potential conditions, normalized ground states were verified. For trapping potentials, normalized ground states paired with bound states were obtained.
- Choquard equation,
- lower critical exponent,
- normalized solutions,
- external potential,
- variational methods
Citation: Huiling Wu, Liping Wu. Normalized solutions to a Choquard equation with potential and mixed local-nonlocal nonlinearities[J]. Electronic Research Archive, 2026, 34(12): 8740-8755. doi: 10.3934/era.2026368

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Abstract

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

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

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

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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 10.1016/s0294-1449(16)30422-x
[7]	G. Li, H. Ye, The existence of positive solutions with prescribed $L^2$-norm for nonlinear Choquard equations, J. Math. Phys., 55 (2014), 121501. https://doi.org/10.1063/1.4902386 doi: 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 doi: 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 doi: 10.1007/s11464-020-0189-6
[10]	P. Jin, X. Tang, D. Hu, The existence of normalized solutions of the $L^{2}$-supercritical Choquard equations, Milan J. Math., 93 (2025), 149–171. https://doi.org/10.1007/s00032-025-00415-1 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 10.1016/S0362-546X(96)00021-1

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