Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain
Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain
复制标题
DOI：
10.1007/s00526-023-02551-1
复制
发表时间：
2022-06
影响因子：
2.1
通讯作者：
M. Squassina;Minbo Yang;Shunneng Zhao
中科院分区：
文献类型：
--
作者：
M. Squassina;Minbo Yang;Shunneng Zhao
本文研究了下列临界Hartree方程-Δ u=（Ω u2 μ Ω（Ω））|X射线衍射|μ d <$）u2 μ n-1 + ε u，in Ω，u= 0，on <$Ω，其中N≥ 4，0< μ≤ 4，ε> 0是小参数，Ω是RN中的有界区域，2 μ n = 2 N-μ N-2是Hardy-Littlewood-Sobolev不等式意义下的临界指数.通过建立各种形式的局部Pohozaev恒等式并应用爆破分析，我们首先研究了Hartree方程上的单泡解的爆破点的位置.其次，我们证明了当ε很小时，爆破解集中在Robin函数的非退化临界点处的局部唯一性。
In this paper we are interested in the following critical Hartree equation-Δ u=(∫ Ω u 2 μ∗(ξ)| x-ξ| μ d ξ) u 2 μ∗-1+ ε u, in Ω, u= 0, on∂ Ω, where N≥ 4, 0< μ≤ 4, ε> 0 is a small parameter, Ω is a bounded domain in RN, and 2 μ∗= 2 N-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 ε small.