Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (3): 666-685.

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Positive Ground State Solutions for Nonlinear Critical Kirchhoff Type Problem

Yiqun Cheng(),Kaimin Teng*()   

  1. School of Mathematical Sciences, Taiyuan University of Technology, Taiyuan 030024
  • Received:2020-04-17 Online:2021-06-01 Published:2021-06-09
  • Contact: Kaimin Teng;
  • Supported by:
    the NSFC(11501403);the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province(2018);the NSF of Shanxi Province(201901D111085)


In this paper, we consider the following Kirchhoff type problem $ \begin{equation}\left\{ \begin{array}{l}-\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\Big)\triangle u+V(x)u=|u|^{p-2}u+\varepsilon|u|^4u, \, \, \, x\in\mathbb{R}^3, \\ u\in H^{1}(\mathbb{R}^3), \end{array} \right. \end{equation}$ where $a>0$, $b>0$, $4< p < 6$ and $V (x)\in L_{\rm loc}^{\frac{3}{2}}(\mathbb{R}^3)$ is a given nonnegative function such that $\lim\limits_{|x|\rightarrow\infty}V(x): =V_{\infty}$. Under suitable conditions on $V(x)$, we prove that the existence of ground state solutions for small $\varepsilon$.

Key words: Kirchhoff type problem, Critical nonlinearity, Ground state

CLC Number: 

  • O175.2