论文标题
一个奇异的扰动分数Kirchhoff问题
A singularly perturbed fractional Kirchhoff problem
论文作者
论文摘要
在本文中，我们首先建立了对分数Kirchhoff问题的肯定解决方案的独特性和非分类性， \ big（a+b {\ int _ {\ mathbb {r}^{n}}}}}} |（ - δ）^{\ frac {s} {2}} {2}}} \ Mathbb {r}^{n}，\ end {equation*}其中$ a，b，m> 0 $，$ 0 <\ frac {n} {4} <s <s <1 $，$ 2 <p <p <2^*_ s = \ frac {2n} {2n} {2n} {n-2s} {n-2s} $ and $（$（n）$（$（$）lap and frac fraf fraf fraf。然后，将这种非分类结果与Lyapunov-Schmidt还原方法结合在一起，我们将半经典溶液的存在引导到奇异的扰动问题\ begin {equation*} \ big（\ varepsilon^{2S} b {\ int _ {\ Mathbb {r}^{n}}}} |（-Δ）^{\ frac {s} {2} {2}} u |^2dx \ big）（ - δ）^su+v（x）^su+v（x） \ Mathbb {r}^{n}，\ end {equation*} for $ \ varepsilon> 0 $足够小，并且潜在函数$ v $。
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 $V$.
