Existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems

Zhiwei Hao; Huiqin Zheng; Zhiwei Hao; Huiqin Zheng

doi:10.3934/era.2023167

Electronic Research Archive

2023, Volume 31, Issue 6: 3309-3321. doi: 10.3934/era.2023167

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Existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems

Zhiwei Hao ^,,
Huiqin Zheng

School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China

Received: 10 February 2023 Revised: 21 March 2023 Accepted: 27 March 2023 Published: 10 April 2023

In this paper, we deal with the existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems as follows:

$ \left\{ \begin{array}{l}M\Big(\int_Q\frac{1}{p(x, y)}\frac{| v(x)-v(y)|^{p(x, y)}}{| x-y|^{d+sp(x, y)}}dxdy\Big)(-\Delta_{p(x)})^s v(x)\ \, \, \, \, \, \, \,\\ = \lambda| v(x)|^{r(x)-2}v(x), \;\;\;\;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\;\;\;\;\;\;\, \, \, \, \, \, \, \, \, \, \, \;\; \;\;\;\, \, \, \, \, \, \, \, \, \text{in}\;\;\Omega, \\ v = 0, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\;\;\, \, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \, \, \, \, \, \text{in}\;\mathbb{R}^d\backslash\Omega, \end{array}\right. $

where $ (-\triangle_{p(x)})^s $ is the fractional $ p(x) $-Laplacian. Different from the previous ones which have recently appeared, we weaken the condition of $ M $ and obtain the existence and multiplicity of solutions via the symmetric mountain pass theorem and the theory of the fractional Sobolev space with variable exponents.
Citation: Zhiwei Hao, Huiqin Zheng. Existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems[J]. Electronic Research Archive, 2023, 31(6): 3309-3321. doi: 10.3934/era.2023167

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Abstract

In this paper, we deal with the existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems as follows:

$ \left\{ \begin{array}{l}M\Big(\int_Q\frac{1}{p(x, y)}\frac{| v(x)-v(y)|^{p(x, y)}}{| x-y|^{d+sp(x, y)}}dxdy\Big)(-\Delta_{p(x)})^s v(x)\ \, \, \, \, \, \, \,\\ = \lambda| v(x)|^{r(x)-2}v(x), \;\;\;\;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\;\;\;\;\;\;\, \, \, \, \, \, \, \, \, \, \, \;\; \;\;\;\, \, \, \, \, \, \, \, \, \text{in}\;\;\Omega, \\ v = 0, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\;\;\, \, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \, \, \, \, \, \text{in}\;\mathbb{R}^d\backslash\Omega, \end{array}\right. $

where $ (-\triangle_{p(x)})^s $ is the fractional $ p(x) $-Laplacian. Different from the previous ones which have recently appeared, we weaken the condition of $ M $ and obtain the existence and multiplicity of solutions via the symmetric mountain pass theorem and the theory of the fractional Sobolev space with variable exponents.

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