Multiplicity of nontrivial solutions for a class of fractional Kirchhoff equations

• Published in: AIMS mathematics Vol. 9; no. 2; pp. 4135 - 4160
• Main Authors: Shao, Liuyang, Chen, Haibo, Pang, Yicheng, Wang, Yingmin
• Format: Journal Article
• Language: English
• Published: AIMS Press 2024
• Subjects: fractional kirchhoff; multiplicity; nontrivial solution; varied method
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2024203

In this article, we study a class of fractional Kirchhoff with a superlinear nonlinearity: <disp-formula> <tex-math id="FE1"> \begin{document}$ \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} $\end{document} </tex-math></disp-formula> 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.