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

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.