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arXiv 2203.07464math.APmath.CA
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奇摄动分数阶Kirchhoff问题

A singularly perturbed fractional Kirchhoff problem

Vicentiu D. Rădulescu, Zhipeng Yang

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23 pages, comments are welcome
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摘要：

本文首先建立了分数阶Kirchhoff问题\begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su+mu=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*}正解的唯一性和非退化性，其中$a,b,m>0$，$0<\frac{N}{4}<s<1$，$2<p<2^*_s=\frac{2N}{N-2s}$和$(-Δ)^s$是分数阶Laplacian算子.然后，结合这个非退化结果和Lyapunov-Schmidt约化方法，我们得到了奇异摄动问题\begin{equation*} \Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su+V(x)u=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*}的半经典解的存在性，其中$\varepsilon> 0$足够小，且具有一个势函数$V$。

英文摘要：

In this paper, we first establish the uniqueness and non-degeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su+mu=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b,m>0$, $0<\frac{N}{4}<s<1$, $2<p<2^*_s=\frac{2N}{N-2s}$ and $(-Δ)^s$ is the fractional Laplacian. Then, combining this non-degeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of semiclassical solutions to the singularly perturbation problem \begin{equation*} \Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su+V(x)u=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} for $\varepsilon> 0$ sufficiently small and a potential function $V$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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