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Existence of Solutions for Kirchhoff Type Problems with Critical Nonlinearity in \(\mathbb {R}^{3}\)

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Abstract

In this paper, the existence and multiplicity of solutions of Kirchhoff type problems with critical nonlinearity is considered in \(\mathbb {R}^{3}: -\varepsilon ^{2}\left (a+b\displaystyle {\int }_{\mathbb {R}^{3}}|\nabla u|^{2}dx\right ){\Delta } u + V(x)u -\varepsilon ^{2}{\Delta }(u^{2})u = K(x)|u|^{22^{\ast }-2}u + h(x,u)\), \((t, x) \in \mathbb {R} \times \mathbb {R}^{3}\). Under suitable assumptions, we prove that it has at least one solution and for any \(m \in \mathbb {N}\), it has at least m pairs of solutions.

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Acknowledgments

I would like to thank the referee for his/her valuable comments and helpful suggestions which have led to an improvement of the presentation of this paper.

Funding

The author is supported by The Inner Mongolia Autonomous Region university scientific research project (NJZY18021) and Postdoctoral research project of Inner Mongolia University (21100-5175504) and Inner Mongolia Normal University introduces high-level scientific research projects (2016YJRC005) and Research project of Inner Mongolia Normal University (2016ZRYB001). All sources of funding for the research reported played the same role in the manuscript.

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Authors and Affiliations

  1. Mathematics Sciences College, Inner Mongolia Normal University, Hohhot, 010022, China

    Jing Zhang

  2. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Jing Zhang

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  1. Jing Zhang
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All authors contributed equally to the manuscript.

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Correspondence to Jing Zhang.

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Research supported by The Inner Mongolia Autonomous Region university scientific research project (NJZY18021) and Postdoctoral research project of Inner Mongolia University (21100-5175504) and Inner Mongolia Normal University introduces high-level scientific research projects (2016YJRC005) and Research project of Inner Mongolia Normal University (2016ZRYB001)

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Zhang, J. Existence of Solutions for Kirchhoff Type Problems with Critical Nonlinearity in \(\mathbb {R}^{3}\). J Dyn Control Syst 26, 363–381 (2020). https://doi.org/10.1007/s10883-019-09439-4

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  • DOI: https://doi.org/10.1007/s10883-019-09439-4

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