带有临界指数的Kirchhoff型方程正解的存在性

Existence of Positive Solution for Kirchhoff Type Problem Involving Critical Exponent

研究了如下一类带临界非线性项的Kirchhoff型方程:{-(a+b∫_a|▽u|~2dx)Δu=λf(x,u)+u^5 u=0 x∈Ω其中a,b,λ>0,Ω是R^3中的一个有界且带光滑边界的区域.在f没有(AR)条件的假设下,运用Brézis-Lieb引理和山路引理证明了方程至少存在1个正解.

This paper considers the following Kirchhoff type problem involving a critical non-linearity：{-（a＋b∫_a｜▽u｜-2dx）Δu=λf（x,u）＋u-5 u=0 x∈Ω where a,b,λ0 and Ω is a smooth bounded domain with smooth boundary Ω.Under f not satisfying（AR）condition,it shows that the above problem has at least one positive solution via the variation method of Brézis-Lieb lemma and mountain pass lemma.