具有变号位势的临界Kirchhoff型方程正解的存在性

Existence of Positive Solutions for a Critical Kirchhoff Type Equation with a Sign Changing Potential

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|^(2)dx)△u=λu+g(x)|u|^(x*-2)u,u∈H0^(1)(Ω)whereΩis 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 withλin a small right neighborhood ofλ1 a.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 method,we obtain that the equation has a ground state solution.