Nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order: Two and three solutions

Weichun Bu; Tianqing An; Guoju Ye; Yating Guo; Weichun Bu; Tianqing An; Guoju Ye; Yating Guo

doi:10.3934/math.2021801

AIMS Mathematics

2021, Volume 6, Issue 12: 13797-13823. doi: 10.3934/math.2021801

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Research article

Nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order: Two and three solutions

1.
College of Science, Hohai University, Nanjing 210098, China
2.
College of Science, Zhongyuan University Of Technology, Zhengzhou 450007, China

Received: 19 July 2021 Accepted: 23 September 2021 Published: 26 September 2021
MSC : 35J91, 35A15, 35R11, 35J67

In this article, we consider the following nonlocal fractional Kirchhoff-type elliptic systems

$ \begin{equation*} \left\{\begin{array}{l} -M_{1}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\eta(x)-\eta(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}} \ \ \ \ \ dxdy +\int_{\Omega}\frac{|\eta|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\eta-|\eta|^{\overline{p}(x)}\eta\right)\\ \; \; \; = \lambda F_{\eta}(x, \eta, \xi)+\mu G_{\eta}(x, \eta, \xi), \; \; x \in \Omega, \\ -M_{2}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\xi(x)-\xi(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}} \ \ \ \ \ dxdy +\int_{\Omega}\frac{|\xi|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\xi-|\xi|^{\overline{p}(x)}\xi\right)\\ \; \; \; = \lambda F_{\xi}(x, \eta, \xi)+\mu G_{\xi}(x, \eta, \xi), \; \; x \in \Omega, \\ \; \eta = \xi = 0, \; \; x \in \mathbb{R}^{N}\backslash \Omega, \end{array} \right. \end{equation*} $

where $ M_{1}(t), M_{2}(t) $ are the models of Kirchhoff coefficient, $ \Omega $ is a bounded smooth domain in $ \mathbb R^{N} $, $ (-\Delta)_{p(\cdot)}^{s(\cdot)} $ is a fractional Laplace operator, $ \lambda, \mu $ are two real parameters, $ F, G $ are continuous differentiable functions, whose partial derivatives are $ F_{\eta}, F_{\xi}, G_{\eta}, G_{\xi} $. With the help of direct variational methods, we study the existence of solutions for nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order, and obtain at least two and three weak solutions based on Bonanno's and Ricceri's critical points theorem. The outstanding feature is the case that the Palais-Smale condition is not requested. The major difficulties and innovations are nonlocal Kirchhoff functions with the presence of the Laplace operator involving two variable parameters.
- Kirchhoff systems,
- $ p(\cdot) $-Laplace operator,
- variational methods,
- Bonanno's critical points theorem,
- Ricceri's critical points theorem
Citation: Weichun Bu, Tianqing An, Guoju Ye, Yating Guo. Nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order: Two and three solutions[J]. AIMS Mathematics, 2021, 6(12): 13797-13823. doi: 10.3934/math.2021801

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Abstract

In this article, we consider the following nonlocal fractional Kirchhoff-type elliptic systems

$ \begin{equation*} \left\{\begin{array}{l} -M_{1}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\eta(x)-\eta(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}} \ \ \ \ \ dxdy +\int_{\Omega}\frac{|\eta|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\eta-|\eta|^{\overline{p}(x)}\eta\right)\\ \; \; \; = \lambda F_{\eta}(x, \eta, \xi)+\mu G_{\eta}(x, \eta, \xi), \; \; x \in \Omega, \\ -M_{2}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\xi(x)-\xi(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}} \ \ \ \ \ dxdy +\int_{\Omega}\frac{|\xi|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\xi-|\xi|^{\overline{p}(x)}\xi\right)\\ \; \; \; = \lambda F_{\xi}(x, \eta, \xi)+\mu G_{\xi}(x, \eta, \xi), \; \; x \in \Omega, \\ \; \eta = \xi = 0, \; \; x \in \mathbb{R}^{N}\backslash \Omega, \end{array} \right. \end{equation*} $

where $ M_{1}(t), M_{2}(t) $ are the models of Kirchhoff coefficient, $ \Omega $ is a bounded smooth domain in $ \mathbb R^{N} $, $ (-\Delta)_{p(\cdot)}^{s(\cdot)} $ is a fractional Laplace operator, $ \lambda, \mu $ are two real parameters, $ F, G $ are continuous differentiable functions, whose partial derivatives are $ F_{\eta}, F_{\xi}, G_{\eta}, G_{\xi} $. With the help of direct variational methods, we study the existence of solutions for nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order, and obtain at least two and three weak solutions based on Bonanno's and Ricceri's critical points theorem. The outstanding feature is the case that the Palais-Smale condition is not requested. The major difficulties and innovations are nonlocal Kirchhoff functions with the presence of the Laplace operator involving two variable parameters.

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