Multiplicity of nontrivial solutions for a class of fractional Kirchhoff equations

Liuyang Shao; Haibo Chen; Yicheng Pang; Yingmin Wang; Liuyang Shao; Haibo Chen; Yicheng Pang; Yingmin Wang

doi:10.3934/math.2024203

AIMS Mathematics

2024, Volume 9, Issue 2: 4135-4160. doi: 10.3934/math.2024203

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Research article

Multiplicity of nontrivial solutions for a class of fractional Kirchhoff equations

1.
School of Mathematics and Statistics, GuiZhou University of Finance and Economics, Guiyang, Guizhou, 550025, China
2.
School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, China

Received: 12 November 2023 Revised: 11 December 2023 Accepted: 19 December 2023 Published: 15 January 2024
MSC : 35B38, 35D30, 35J10, 35J20, 35J62

In this article, we study a class of fractional Kirchhoff with a superlinear nonlinearity:

$ \begin{equation} \begin{cases} M(\int_{\mathbb{R}^{N}}|(-\triangle)^{\frac{\alpha}{2}}u|^{2}dx)(-\triangle)^{\alpha}u+\lambda V(x)u = f(x, u)\; \; \mbox{in}\; \; \mathbb{R}^{N}, \\ u\in H^{\alpha}(\mathbb{R}^{N}), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; N\geq1, \; \; \; \; \; \; \; \; (1.1)\notag \end{cases} \end{equation} $

where $ \lambda > 0 $ is a parameter, $ a $ and $ b $ are positive numbers satisfying $ M(t) = am(t)+b $, $ m:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+} $ is continuous. $ V: \mathbb{R}^{N}\times\mathbb{R}\rightarrow \mathbb{R} $ is continuous. $ f $ satisfies $ \lim\limits_{|t|\rightarrow \infty}f(x, t)/|t|^{k-1} = Q(x) $ uniformly in $ x\in\mathbb{R}^{N} $ for each $ 2 < k < 2_{\alpha}^{\ast}, (2_{\alpha}^{\ast} = \frac{2N}{N-2\alpha}) $. We investigated the effects of functions $ m $ and $ Q $ on the solution. By applying the variational method, we obtain the existence of multiple solutions. Furthermore, it is worth mentioning that the ground state solution has also been obtained.
- fractional Kirchhoff,
- varied method,
- multiplicity,
- nontrivial solution
Citation: Liuyang Shao, Haibo Chen, Yicheng Pang, Yingmin Wang. Multiplicity of nontrivial solutions for a class of fractional Kirchhoff equations[J]. AIMS Mathematics, 2024, 9(2): 4135-4160. doi: 10.3934/math.2024203

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Abstract

In this article, we study a class of fractional Kirchhoff with a superlinear nonlinearity:

$ \begin{equation} \begin{cases} M(\int_{\mathbb{R}^{N}}|(-\triangle)^{\frac{\alpha}{2}}u|^{2}dx)(-\triangle)^{\alpha}u+\lambda V(x)u = f(x, u)\; \; \mbox{in}\; \; \mathbb{R}^{N}, \\ u\in H^{\alpha}(\mathbb{R}^{N}), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; N\geq1, \; \; \; \; \; \; \; \; (1.1)\notag \end{cases} \end{equation} $

where $ \lambda > 0 $ is a parameter, $ a $ and $ b $ are positive numbers satisfying $ M(t) = am(t)+b $, $ m:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+} $ is continuous. $ V: \mathbb{R}^{N}\times\mathbb{R}\rightarrow \mathbb{R} $ is continuous. $ f $ satisfies $ \lim\limits_{|t|\rightarrow \infty}f(x, t)/|t|^{k-1} = Q(x) $ uniformly in $ x\in\mathbb{R}^{N} $ for each $ 2 < k < 2_{\alpha}^{\ast}, (2_{\alpha}^{\ast} = \frac{2N}{N-2\alpha}) $. We investigated the effects of functions $ m $ and $ Q $ on the solution. By applying the variational method, we obtain the existence of multiple solutions. Furthermore, it is worth mentioning that the ground state solution has also been obtained.

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