New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity

Gao, Yuan; Liu, Lishan; Wei, Na; Gu, Haibo; Wu, Yonghong

doi:10.3390/fractalfract9100646

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New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity

by

Yuan Gao

¹,

Lishan Liu

^2,3,*

,

Na Wei

^4,5

,

Haibo Gu

³

and

Yonghong Wu

⁵

¹

School of Statistics and Data Science, Qufu Normal University, Qufu 273165, China

²

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

³

School of Mathematical Sciences, Xinjiang Normal University, Urumqi 830017, China

⁴

School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

⁵

Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia

^*

Author to whom correspondence should be addressed.

Fractal Fract. 2025, 9(10), 646; https://doi.org/10.3390/fractalfract9100646

Submission received: 11 September 2025 / Revised: 28 September 2025 / Accepted: 30 September 2025 / Published: 4 October 2025

(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)

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Abstract

By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations:

M ({[u]}_{s, p}^{p} + \int_{R^{N}} V (x) {| u |}^{p} d x) [{(- Δ)}_{p}^{s} u +

{V (x) | u |}^{p - 2} u] = {λ | u |}^{q - 2} u ln | u |, x \in R^{N}, (P)

. In these equations,

λ \in R ∖ {0},

p \in (1, + \infty)

,

s \in (0, 1), s p < N,

p_{s}^{*} = \frac{N p}{N - s p}

,

M (τ) = a + b τ^{θ - 1}

,

a, b \in R^{+},

1 < θ < \frac{p_{s}^{*}}{p}

,

V (x) \in C (R^{N}, R)

is a potential function and

{(- Δ)}_{p}^{s}

is the fractional p-Laplacian operator. The existence of solutions is deeply influenced by the positive and negative signs of

λ

. More precisely, (i) Equation (P) has one ground state solution for

λ > 0

and

p θ < q < p_{s}^{*}

, with a positive corresponding energy value; and (ii) Equation (P) has at least two nontrivial solutions for

λ < 0

and

p < q < p_{s}^{*}

, with positive and negative corresponding energy values, respectively.

Keywords: fractional Kirchhoff problem; logarithmic nonlinearity; Nehari manifold; Ekeland variational principle; multiple solutions

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MDPI and ACS Style

Gao, Y.; Liu, L.; Wei, N.; Gu, H.; Wu, Y. New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity. Fractal Fract. 2025, 9, 646. https://doi.org/10.3390/fractalfract9100646

AMA Style

Gao Y, Liu L, Wei N, Gu H, Wu Y. New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity. Fractal and Fractional. 2025; 9(10):646. https://doi.org/10.3390/fractalfract9100646

Chicago/Turabian Style

Gao, Yuan, Lishan Liu, Na Wei, Haibo Gu, and Yonghong Wu. 2025. "New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity" Fractal and Fractional 9, no. 10: 646. https://doi.org/10.3390/fractalfract9100646

APA Style

Gao, Y., Liu, L., Wei, N., Gu, H., & Wu, Y. (2025). New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity. Fractal and Fractional, 9(10), 646. https://doi.org/10.3390/fractalfract9100646

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New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity

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