Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (6): 1729-1743.

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Existence and Asymptotic Behaviour of Solutions for Kirchhoff Type Equations with Zero Mass and Critical

Li Anran, Fan Dandan, Wei Chongqing   

  1. School of Mathematical Sciences, Shanxi University, Taiyuan 030006
  • Received:2021-09-10 Revised:2022-08-22 Published:2022-12-16
  • Supported by:
    Supported by the NSFC(12071266, 11701346), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(2019L0024), the Research Project Supported by Shanxi Scholarship Council of China(2020-005) and the Shanxi Basic Research Program (Free Exploration) Project(202103021224013)

Abstract: In this paper, the existence and asymptotic behaviour of solutions for Kirchhoff type equations with zero mass and critical term are studied. First, under some appropriate assumptions of the nonlinearity, it is proved that the functional associated to the equation enjoys mountain pass structure and the estimate of mountain pass energy level is also given. Then the second concentration compactness lemma is used to verify that the corresponding functional satisfies Palais-Smale local compactness condition, thus the existence of nontrivial solutions is obtained by mountain pass theorem, furthermore, the existence of ground state solutions is obtained by using the definition of the ground state solution. Finally, the asymptotic behaviour of these mountain pass type solutions is studied whenever the parameter tends to zero. It can be proved that they converge to a mountain pass type solution of our problem when the parameter equals to 0.

Key words: Kirchhoff type equations with zero mass and critical term, Variational methods, Mountain Pass theorem, The second concentration compactness lemma, Ground state solution

CLC Number: 

  • O177.91