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On the Critical Kirchhoff Problems with Super-linear Nonlinearities and Variable Potentials

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Abstract

In this paper, we consider the following critical Kirchhoff equation:

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2{\mathrm{d}}x\right) \Delta u +V(x)u=\mu f(u)+u^5, \quad x \in {\mathbb {R}}^3, \end{aligned}$$

where \(\mu \) is a positive parameter, a, \(b>0\) are real constants, \(f \in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\) and \(V \in \) \({\mathcal {C}}{^1}({\mathbb {R}}^3,{\mathbb {R}})\). Under some suitable conditions, the existence of ground state solution to the above problem is established without the monotonicity condition on \(f(u)/u^3\). Moreover, it is worth noting that we consider the above problem with variable potentials. Our results improve and extend related ones in the existing literature to some extent.

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Acknowledgements

The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.

Funding

This work is supported by the National Natural Science Foundation of China (no. 11961014) and Guangxi Natural Science Foundation (2021GXNSFAA196040). The authors would like to thank the Editors and referees for their useful suggestions and comments.

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  1. College of Science, Guilin University of Technology, Guilin, 541004, Guangxi, People’s Republic of China

    Ting Xiao & Qiongfen Zhang

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  1. Ting Xiao
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  2. Qiongfen Zhang
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The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

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Correspondence to Qiongfen Zhang.

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Communicated by Amin Esfahani.

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Xiao, T., Zhang, Q. On the Critical Kirchhoff Problems with Super-linear Nonlinearities and Variable Potentials. Bull. Iran. Math. Soc. 48, 3351–3380 (2022). https://doi.org/10.1007/s41980-022-00699-8

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  • DOI: https://doi.org/10.1007/s41980-022-00699-8

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