临界零质量Kirchhoff型方程的解及其渐近行为  

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

作　　者：李安然[1] 樊丹丹 魏重庆 Li Anran;Fan Dandan;Wei Chongqing(School of Mathematical Sciences,Shanxi University,Taiyuan 030006)

机构地区：[1]山西大学数学科学学院,太原030006

出　　处：《数学物理学报（A辑）》2022年第6期1729-1743,共15页Acta Mathematica Scientia

基　　金：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);山西省基础研究计划(自由探索类)资助项目(202103021224013)。

摘　　要：该文主要利用变分法研究了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.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.

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 

分 类 号：O177.91[理学—数学]