In this paper, we consider the following Kirchhoff type equation: $$ \textstyle\begin{cases} - (a+b \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= f(x,u) &\text{in } \varOmega , \\ u=0 &\text{on } \partial \varOmega , \end{cases} $$ where $a,b>0$ are constants and $\varOmega \subset \mathbb{R}^{N}$ ( $N=1,2,3$ ) is a bounded domain with smooth boundary ∂Ω. By applying Morse theory, we obtain some existence and multiplicity results of nontrivial solutions for either a or b being sufficiently small.

中文翻译：

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基于莫尔斯理论的基尔霍夫型方程的一些摄动结果

在本文中，我们考虑以下Kirchhoff类型方程：$$ \ textstyle \ begin {cases}-（a + b \ int _ {\ varOmega} \ vert \ nabla u \ vert ^ {2} \，dx）\ Delta u = f（x，u）＆\ text {in} \ varOmega，\\ u = 0＆\ text {on} \ partial \ varOmega，\ end {cases} $$其中$ a，b> 0 $是常量而$ \ varOmega \ subset \ mathbb {R} ^ {N} $（$ N = 1,2,3 $）是具有光滑边界∂Ω的有界域。通过应用摩尔斯理论，我们获得了对于a或b足够小的非平凡解的存在性和多重性结果。