Existence of ground state solutions for the general Choquard equation with Riesz fractional derivative operator

Sarah Abdullah Qadha; Muneera Abdullah Qadha; Haibo Chen; Mohamed Abdalla; Mohammed Z. Alqarni; Sarah Abdullah Qadha; Muneera Abdullah Qadha; Haibo Chen; Mohamed Abdalla; Mohammed Z. Alqarni

doi:10.3934/math.20251129

AIMS Mathematics

2025, Volume 10, Issue 11: 25489-25503. doi: 10.3934/math.20251129

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Research article

Existence of ground state solutions for the general Choquard equation with Riesz fractional derivative operator

1.
School of Mathematics and Statistics, Central South University, Changsha 410083, China
2.
Department of Mathematics, Faculty of Education at Al-Mahweet, Sana'a University, Al-Mahweet, Yemen
3.
Mathematics Department, Faculty of Science South Valley University, Qena 83523, Egypt
4.
Mathematics Department, Faculty of Science King Khalid University, Abha, Saudi Arabia

Received: 18 September 2025 Revised: 26 October 2025 Accepted: 31 October 2025 Published: 06 November 2025
MSC : 35J60, 35J20

This work examines the existence of a ground state solution for the following general nonlinear Choquard equation:
$-\Delta u+u = \left({(-\Delta )}_{E}^{\frac{-\alpha }{2}}\mathrm{*}\mathcal{Q}\left(u\right)\right)\mathcal{q}\left(u\right), in\;{\mathbb{R}}^{N}, $
where $ N\ge 3, $ $ \mathcal{Q} $ is the primitive function of $ \mathcal{q}, $ $ \mathcal{Q}\in {C}^{1}\left(\mathbb{R}; \mathbb{R}\right) $ fulfils the general Berestycki–Lions conditions, and $ {(-\Delta)}_{E}^{\frac{-\alpha }{2}} $ is the equivalent Riesz fractional operator of order $ \alpha \in \left(\mathrm{0, 2}\right) $. In this case, the Riesz potential has not previously been investigated. The existence of a solution is established through the application of variational techniques. This modification not only expands the theoretical understanding of such equations but also opens up new avenues for practical applications, particularly in fields such as quantum mechanics and astrophysics.
- existence,
- general Choquard equation,
- variational methods,
- ground state solution,
- equivalent Riesz Potential
Citation: Sarah Abdullah Qadha, Muneera Abdullah Qadha, Haibo Chen, Mohamed Abdalla, Mohammed Z. Alqarni. Existence of ground state solutions for the general Choquard equation with Riesz fractional derivative operator[J]. AIMS Mathematics, 2025, 10(11): 25489-25503. doi: 10.3934/math.20251129

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Abstract

This work examines the existence of a ground state solution for the following general nonlinear Choquard equation:

$-\Delta u+u = \left({(-\Delta )}_{E}^{\frac{-\alpha }{2}}\mathrm{*}\mathcal{Q}\left(u\right)\right)\mathcal{q}\left(u\right), in\;{\mathbb{R}}^{N}, $

where $ N\ge 3, $ $ \mathcal{Q} $ is the primitive function of $ \mathcal{q}, $ $ \mathcal{Q}\in {C}^{1}\left(\mathbb{R}; \mathbb{R}\right) $ fulfils the general Berestycki–Lions conditions, and $ {(-\Delta)}_{E}^{\frac{-\alpha }{2}} $ is the equivalent Riesz fractional operator of order $ \alpha \in \left(\mathrm{0, 2}\right) $. In this case, the Riesz potential has not previously been investigated. The existence of a solution is established through the application of variational techniques. This modification not only expands the theoretical understanding of such equations but also opens up new avenues for practical applications, particularly in fields such as quantum mechanics and astrophysics.

References

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