带临界指数的Kirchhoff型线性耦合方程组正解的多重性  

Multiple Positive Solutions of Kirchhoff Type Linearly Coupled System with Critical Exponent

作　　者：段雪亮 吴晓凡 魏公明[2] 杨海涛[3] Duan Xueliang;Wu Xiaofan;Wei Gongming;Yang Haitao(School of Mathematics and Statistics,Zhengzhou Normal University,Zhengzhou 450044;College of Science,University of Shanghai for Science and Technology,Shanghai 200093;School of Mathematical Sciences,Zhejiang University,Hangzhou 310027)

机构地区：[1]郑州师范学院数学与统计学院,郑州450044 [2]上海理工大学理学院,上海200093 [3]浙江大学数学科学学院,杭州310027

出　　处：《数学物理学报（A辑）》2024年第3期699-716,共18页Acta Mathematica Scientia

基　　金：河南省高等学校重点科研项目(23A110018)。

摘　　要：该文研究了如下带Sobolev临界指数的Kirchhoff型线性耦合方程组{−(1+b_(1)∥u∥^(2))Δu+λ_(1)u=u5+βv,x∈Ω,−(1+b_(2)∥v∥^(2))Δv+λ_(2)v=v^(5)+βu,x∈Ω,u=v=0在∂Ω上,其中Ω⊂R^(3)是一个开球,∥⋅∥表示H_(0)^(1)(Ω)的范数,β∈R是一个耦合参数.常数b_(i)≥0和λ_(i)∈(−λ_(1)(Ω),−1/4λ_(1)(Ω)),i=1,2,这里λ_(1)(Ω)是(−Δ,H_(0)^(1)(Ω))的第一特征值.在含有Kirchhoff项的情形下,利用变分法证明了方程组有一个正基态解和一个高能量的正解,并研究了当β→0时这两个解的渐近行为.This paper deals with the following Kirchhoff type linearly coupled system with Sobolev critical exponent{−(1+b_(1)∥u∥^(2))Δu+λ_(1)u=u5+βv,x∈Ω,−(1+b_(2)∥v∥^(2))Δv+λ_(2)v=v^(5)+β u,x∈Ω,u=v=0on∂Ω,whereΩ⊂R^(3)is an open ball,∥⋅∥is the standard norm of H_(0)^(1)(Ω) and β∈R is a coupling parameter.Constants b_(i)≥0 andλ_(i)∈(−λ_(1)(Ω),−1/4λ_(1)(Ω)),i=1,2,whereλ_(1)(Ω)is the first eigenvalue of(−Δ,H_(0)^(1)(Ω)).Under the effects of Kirchhoff terms,we prove that the system has a positive ground state solution and a positive higher energy solution for someβ>0 by using variational method.Moreover,we study the asymptotic behaviours of these solutions asβ→0.

关 键 词：KIRCHHOFF 型方程 线性耦合方程组 SOBOLEV 临界指数 变分法 

分 类 号：O175.29[理学—数学]