论文标题
一类Kirchhoff方程的多峰值积极解决方案的局部唯一性
Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations
论文作者
论文摘要
本文是双重的。在第一部分中，将非排效结果与lyapunov-schmidt减少方法相结合，我们得出了多峰值的阳性解决方案的存在，以奇异扰动问题\ begin \ begin {equation*} \ big（\ varepsilon^{2S} B {\ int _ {\ MathBb {r}^{n}}}}} |（-Δ）^{\ frac {s} {2} {2}} {2}} u |^2dx \ big）（ - δ）^su+v（x）^su+v（x） \ end {equation*}对于$ \ varepsilon> 0 $很小，$ 2S <n <4s $，$ 1 <p <2^*_ s-s-os $，以及在函数$ v $上进行一些温和的假设。主要困难来自非局部运算符，混合了非局部术语，这导致相应的不受干扰的问题被证明是偏微分方程的系统，而不是单个分数Kirchhoff方程。在第二部分中，根据$ v $的一些假设，我们通过使用局部PohozǎEV身份来展示正面多峰解决方案的局部唯一性。
This paper is twofold. In the first part, combining the nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of multi-peak positive 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,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} for $\varepsilon> 0$ sufficiently small, $2s<N<4s$, $1<p<2^*_s-1$ and some mild assumptions on the function $V$. The main difficulties are from the nonlocal operator mixed the nonlocal term, which cause the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single fractional Kirchhoff equation. In the second part, under some assumptions on $V$, we show the local uniqueness of positive multi-peak solutions by using the local Pohozǎev identity.
