Multiple solutions for a singular fractional Kirchhoff problem with variable exponents

Najla Alghamdi; Abdeljabbar Ghanmi; Najla Alghamdi; Abdeljabbar Ghanmi

doi:10.3934/math.2025039

AIMS Mathematics

2025, Volume 10, Issue 1: 826-838. doi: 10.3934/math.2025039

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Multiple solutions for a singular fractional Kirchhoff problem with variable exponents

Najla Alghamdi ^{1
,
,},
Abdeljabbar Ghanmi ²

1.
Department of Mathematics and Statistics, Faculty of Sciences, University of Jeddah, Jeddah, Saudi Arabia
2.
ENIT-LAMSIN, BP. 37, 1002 Tunis-Belvédère, Tunis El Manar university, Tunis, Tunisia

Received: 05 November 2024 Revised: 23 December 2024 Accepted: 06 January 2025 Published: 14 January 2025
MSC : 31B30, 35J35, 35J60

In this work, we studied the multiplicity of solutions for a Kirchhoff problem involving the $ \kappa(\xi) $-fractional derivative and critical exponent. More precisely, we transformed the studied problem into an integral equation that lead to the study of the critical point for the energy functional; after that, we presented and proved some properties related to this functional and demonstrated that the energy functional satisfied the geometry of the mountain pass geometry. Finally, by applying the mountain pass theorem for the even functional, we proved that this functional admitted infinitely many critical points, which means that the studied problem has infinitely many solutions.
- $ \psi $-Hilifer fractional derivative,
- mountain pass geometry,
- concentration-compactness principle,
- variational methods
Citation: Najla Alghamdi, Abdeljabbar Ghanmi. Multiple solutions for a singular fractional Kirchhoff problem with variable exponents[J]. AIMS Mathematics, 2025, 10(1): 826-838. doi: 10.3934/math.2025039

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Abstract

In this work, we studied the multiplicity of solutions for a Kirchhoff problem involving the $ \kappa(\xi) $-fractional derivative and critical exponent. More precisely, we transformed the studied problem into an integral equation that lead to the study of the critical point for the energy functional; after that, we presented and proved some properties related to this functional and demonstrated that the energy functional satisfied the geometry of the mountain pass geometry. Finally, by applying the mountain pass theorem for the even functional, we proved that this functional admitted infinitely many critical points, which means that the studied problem has infinitely many solutions.

References

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