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Sign-Changing Solutions for Fractional Kirchhoff-Type Equations with Critical and Supercritical Nonlinearities

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Abstract

This paper concerns the existence of sign-changing solutions for the following fractional Kirchhoff-type equation with critical and supercritical nonlinearities

$$\begin{aligned} \left( a+b[u]^{2}\right) (-\Delta )^{\alpha }u+V(x)u=f(x,u)+\lambda |u| ^{r-2}u,\,\, \text {in}\,\,\mathbb {R}^3, \end{aligned}$$

where \(a,b>0\) are constants, \(\alpha \in (\frac{3}{4}, 1)\), \(\lambda >0\) is a real parameter, \(r\ge 2_{\alpha }^{*}=\frac{6}{3-2\alpha }\), \((-\Delta )^{\alpha }\) is the fractional Laplace operator, the potential function V and the nonlinearity f satisfy some suitable conditions. By combining an appropriate truncation argument with Moser iteration method, we prove that the existence of sign-changing solutions for the above equation when the parameter \(\lambda \) is sufficiently small. Our results enrich and improve the previous ones in the literature.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant nos. 11661053, 11771198, 11901276 and 11961045) and supported by the Provincial Natural Science Foundation of Jiangxi, China (nos. 20181BAB201003, 20202BAB201001 and 20202BAB211004).

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  1. Department of Mathematics, Nanchang University, Nanchang, 330031, Jiangxi, People’s Republic of China

    Liu Gao, Chunfang Chen, Jianhua Chen & Chuanxi Zhu

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  1. Liu Gao
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Correspondence to Chunfang Chen.

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Gao, L., Chen, C., Chen, J. et al. Sign-Changing Solutions for Fractional Kirchhoff-Type Equations with Critical and Supercritical Nonlinearities. Mediterr. J. Math. 18, 78 (2021). https://doi.org/10.1007/s00009-021-01733-5

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  • DOI: https://doi.org/10.1007/s00009-021-01733-5

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