Existence, uniqueness, and localization of positive solutions to nonlocal problems of the Kirchhoff type via the global minimum principle of Ricceri

In Hyoun Kim; Yun-Ho Kim; In Hyoun Kim; Yun-Ho Kim

doi:10.3934/math.2025210

AIMS Mathematics

2025, Volume 10, Issue 3: 4540-4557. doi: 10.3934/math.2025210

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Existence, uniqueness, and localization of positive solutions to nonlocal problems of the Kirchhoff type via the global minimum principle of Ricceri

In Hyoun Kim ¹,
Yun-Ho Kim ^{2
,
,}

1.
Department of Mathematics, Incheon National University, Incheon 22012, Korea, Republic of Korea
2.
Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea

Received: 11 October 2024 Revised: 13 January 2025 Accepted: 27 January 2025 Published: 03 March 2025
MSC : 35B33, 35D30, 35J20, 35J60, 35J66

The purpose of this paper is to demonstrate the existence and uniqueness of positive solutions to fractional $ p $-Laplacian problems with discontinuous Kirchhoff-type functions. The crucial tools for getting these results are the uniqueness result of the Brézis–Oswald–type problem and the abstract global minimum principle. The primary features of this paper are the discontinuity of the Kirchhoff coefficient in $ [0, \infty) $ and the localization of solutions.
- fractional $ p $-Laplacian,
- Kirchhoff-type function,
- weak solution,
- uniqueness,
- global minima
Citation: In Hyoun Kim, Yun-Ho Kim. Existence, uniqueness, and localization of positive solutions to nonlocal problems of the Kirchhoff type via the global minimum principle of Ricceri[J]. AIMS Mathematics, 2025, 10(3): 4540-4557. doi: 10.3934/math.2025210

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Abstract

The purpose of this paper is to demonstrate the existence and uniqueness of positive solutions to fractional $ p $-Laplacian problems with discontinuous Kirchhoff-type functions. The crucial tools for getting these results are the uniqueness result of the Brézis–Oswald–type problem and the abstract global minimum principle. The primary features of this paper are the discontinuity of the Kirchhoff coefficient in $ [0, \infty) $ and the localization of solutions.

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