ABSTRACT

In this paper, we consider a type of Kirchhoff problem as follows $-(a+b{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2)\mathrm{\Delta }u=(1+\epsilon K(x\left)\right)u^{2^{\ast }-1},u>0,\mathrm{i}\mathrm{n}{\mathbb{R}}^N,$ where a, b>0 are given constants, $\epsilon >0$ is a small parameter and $2^{\ast }=2N/(N-2),(N\ge 3)$. We show that if $K\left(x\right)$ has k critical points near which $K\left(x\right)$ satisfies some expansion assumption, then by Lyapunov–Schmidt reduction method, we construct multi-peak solutions for $\epsilon >0$ small, which concentrate at the k critical points of $K\left(x\right)$.

中文翻译：

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一类具有临界指数的基尔霍夫方程的多峰解

摘要

在本文中，我们考虑一种基尔霍夫问题如下 $-(\mathrm{一种}+乙{\int }_{{\mathbb{电阻}}^N}|\mathrm{\nabla }你{|}^2)\mathrm{\Delta }你=(1+\epsilon 钾(X\left)\right){你}^{2^{＊}-1},你>0,\mathrm{一世}\mathrm{n}{\mathbb{电阻}}^N,$其中a , b >0 是常数，$\epsilon >0$ 是一个小参数并且 $2^{＊}=2N/(N-2),(N\ge 3)$. 我们证明如果$钾\left(X\right)$附近有k 个临界点$钾\left(X\right)$ 满足一些展开假设，然后通过 Lyapunov-Schmidt 约简方法，我们构造多峰解 $\epsilon >0$小，集中在k 个临界点$钾\left(X\right)$.