• 论文 •

### 临界零质量Kirchhoff型方程的解及其渐近行为

1. 山西大学数学科学学院, 太原 030006
• 收稿日期:2021-09-10 修回日期:2022-08-22 发布日期:2022-12-16
• 通讯作者: 李安然,E-mail:lianran@sxu.edu.cn E-mail:lianran@sxu.edu.cn
• 基金资助:
国家自然科学基金(12071266,11701346)、山西省高等学校科技创新项目(2019L0024)、山西省回国留学人员科研教研资助项目(2020-005)和山西省基础研究计划(自由探索类)资助项目(202103021224013)

### 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.

• O177.91