期刊论文

中文

数学理论与应用

1006-8074

2021

41

01

71-90

中南大学研究生科研创新项目（1053320211512）资助；

本校为第一机构

本文研究一类涉及临界非线性的Kirchhoff型方程:-(a+b∫_(Ω)|▽u|^(2)dx)△u=λu+g(x)|u|^(x*-2)u,u∈H0^(1)(Ω)其中Ω是RN(N≥3)中光滑有界区域,参数a,b,λ>0,g(x)是一变号函数,且满足g∈L^(∞)(Ω)与集合{x∈Ω:g(x)>0}具有正测度.当N=3时,λ在λ1a的右侧小邻域得到方程的正基态解。对于N=4时,g在不同条件下,得到了方程正解的存在与不存在性.对于N≥5时,通过变分方法,得到方程的正基态解.

In this paper,we investigate a class of Kirchhoff type problem involving a critical nonlinearity-(a+b∫Ω|▽u|2dx)Δu=λu+g(x)|u|2*-2u,u∈H10(Ω),where Ω is a smooth bounded domain in RN(N≥3),a,b,λ＞0 and g(x)is a sign-changing potential,g ∈ L∞(Ω)with the set{x ∈ Ω:g(x)＞0}of positive measure.For N=3,we obtain a positive ground state solution to the problem with λ in a small right neighborhood of λ1a.For N=4,under some assumptions on g,we get the existence and non-existence of the positive solutions.For N≥5,by using the variational met...