论文标题
在有限域中，关键哈特里方程的爆炸解决方案的局部唯一性
Local Uniqueness of blow-up solutions for critical Hartree equations in bounded domain
论文作者
论文摘要
在本文中，我们对以下关键hartree方程\ begin {equation*} \ begin {case}-ΔU= \ displayStyle {\ big（\int_Ω\ frac {u^u^{2_μ^ast} （ξ）} {| x-ξ|^μ}dξ\ big） u = 0，~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~参数，$ω$是$ \ mathbb {r}^n $中的一个有界域，而$2_μ^\ ast = \ frac {2n-μ} {n-2} $是Hardy-Little Wood-sobolev intorate的关键指数。通过建立各种版本的局部pohozaev身份并应用爆炸分析，我们首先研究了爆炸点的位置，用于单个起泡解决方案上方的爆炸解决方案。接下来，我们证明了爆炸解决方案的局部唯一性，该解决方案集中在$ \ varepsilon $ small的罗宾功能的非分类临界点上。
In this paper we are interested in the following critical Hartree equation \begin{equation*} \begin{cases} -Δu =\displaystyle{\Big(\int_Ω\frac{u^{2_μ^\ast} (ξ)}{|x-ξ|^μ}dξ\Big)u^{2_μ^\ast-1}}+\varepsilon u ,~~~\text{in}~Ω,\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}~\partialΩ, \end{cases} \end{equation*} where $N\geq4$, $0<μ\leq4$, $\varepsilon>0$ is a small parameter, $Ω$ is a bounded domain in $\mathbb{R}^N$, and $2_μ^\ast=\frac{2N-μ}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for $\varepsilon$ small.
