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

  • Jing ZhangEmail author


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.


Kirchhoff type problems Critical nonlinearity Variational method Critical point 



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.

Author’s Contributions

All authors contributed equally to the manuscript.


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.

Compliance with Ethical Standards

This study conforms to ethical standards.

Competing Interests

The author declares that there is no competing interests.


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Authors and Affiliations

  1. 1.School of Mathematical SciencesInner Mongolia UniversityHohhotChina
  2. 2.Mathematics Sciences CollegeInner Mongolia Normal UniversityHohhotChina

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