临界零质量Kirchhoff型方程的解及其渐近行为

数学物理学报 ›› 2022, Vol. 42 ›› Issue (6): 1729-1743.

临界零质量Kirchhoff型方程的解及其渐近行为

李安然*(),樊丹丹,魏重庆

^{山西大学数学科学学院, 太原 030006}

收稿日期:2021-09-10 出版日期:2022-12-26 发布日期:2022-12-16
通讯作者: 李安然 E-mail:lianran@sxu.edu.cn
基金资助:
国家自然科学基金(12071266);国家自然科学基金(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);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

Anran Li*(),Dandan Fan,Chongqing Wei

^{School of Mathematical Sciences, Shanxi University, Taiyuan 030006}

Received:2021-09-10 Online:2022-12-26 Published:2022-12-16
Contact: Anran Li E-mail:lianran@sxu.edu.cn
Supported by:
the NSFC(12071266);the NSFC(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);the Shanxi Basic Research Program (Free Exploration) Project(202103021224013)

摘要/Abstract

摘要：

该文主要利用变分法研究了$\mathbb{R}^{3}$上一类带临界项的零质量Kirchhoff型方程非平凡解的存在性和渐近行为.首先在非线性项满足一些适当的条件下, 验证方程对应的泛函具有山路结构并给出了相应山路能量水平的估计.然后利用第二集中紧性引理验证方程对应的泛函满足Palais-Smale局部紧性条件, 进而由山路定理得到方程山路型非平凡解的存在性, 进一步利用基态解的定义得到方程基态解的存在性.最后该文研究了上述山路型非平凡解当参数趋于0时的渐近行为: 它们会收敛到相应零质量Schrödinger方程的一个山路型非平凡解.

关键词: 带临界项的零质量Kirchhoff型方程, 变分法, 山路定理, 第二集中紧性引理, 基态解

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

中图分类号:

O177.91

李安然,樊丹丹,魏重庆. 临界零质量Kirchhoff型方程的解及其渐近行为[J]. 数学物理学报, 2022, 42(6): 1729-1743.

Anran Li,Dandan Fan,Chongqing Wei. Existence and Asymptotic Behaviour of Solutions for Kirchhoff Type Equations with Zero Mass and Critical[J]. Acta mathematica scientia,Series A, 2022, 42(6): 1729-1743.

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