摘要
该文致力于研究R^(4)中一类带有陡峭位势的临界Kirchhoff型方程{-(a+b∫_(R^(4))|▽u|^(2)dx)Δu+λV(x)u=|u|^(2)u+f(u),x∈R^(4),u∈H^(1)(R^(4)),其中a,b>0是常数且参数λ>0.在4维空间中,|u|^(2)u的非线性增长在2^(*)=4时达到Sobolev临界指数.假设非负连续位势V是底部为V^(-1)(0)的陡峭位势且f∈C(R,R)满足一定的条件.利用变分方法,获得了方程至少存在一个基态解.此外,还研究了当|x|→∞时,基态解的集中行为和当b→0,λ→∞时,基态解的渐近行为.
In this paper,we focus on dealing with a class of critical Kirchhoff type equation {-(a+b∫_(R^(4))|▽u|^(2)dx)Δu+λV(x)u=|u|^(2)u+f(u),x∈R^(4),u∈H^(1)(R^(4)),where a,b>0 are constants andλ>0.The nonlinear growth of|u|^(2)u reaches the Sobolev critical exponent since 2^(*)=4 in dimension 4.Assume that V is the nonnegative continuous potential,which represents a potential well with the bottom V^(-1)(0)and f∈C(R,R)satisfies suitable conditions.By the variational methods,the existence of at least a ground state solution is obtained.Moreover,we study the concentration behavior of the ground state solutions as λ→∞ and their asymptotic behavior as b→and λ→∞,respectively.
作者
陈征艳
张家锋
Chen Zhengyan;Zhang Jiafeng(School of Data Science and Information Engineering,Guizhou Minzu University,Guiyang 550025)
出处
《数学物理学报(A辑)》
北大核心
2025年第2期450-464,共15页
Acta Mathematica Scientia
基金
国家自然科学基金(11861021)
贵州省教育厅自然科学研究项目(QJJ2023012,QJJ2023061,QJJ2023062)
贵州民族大学自然科学研究项目(GZMUZK[2022]YB06)。
关键词
Kirchhoff型方程
临界增长
变分方法
陡峭位势
基态解
Kirchhoff type equation
critical growth
variational methods
steep potential well
ground state solutions
