Hartree型Brézis-Nirenberg问题特征值问题的定性分析
Qualitative analysis to an eigenvalue problem of the Hartree type Brézis-Nirenberg problem
AI总结：
本文研究临界Hartree型Brézis-Nirenberg方程，在Robin函数临界点非退化条件下，证明小ε时爆破解的Morse指标可由Hessian矩阵负特征值个数确定，首次将相关定性分析推广到非局部情形。
摘要：
本文研究临界Hartree方程：在有界光滑区域Ω⊂ℝ^N（N≥5）中，-Δu=(∫_Ω u^{2*_μ}(y)/|x-y|^μ dy)u^{2*_μ-1}+εu，u>0，且在∂Ω上u=0，其中μ∈(0,4)，2*_μ=(2N-μ)/(N-2)是Hardy-Littlewood-Sobolev不等式意义下的上临界指数。在Robin函数R(x)的临界点x_0∈Ω满足非退化条件下，我们证明当ε>0充分小时，集中在x_0处的爆破解u_ε的Morse指标可以用Hessian矩阵D²R(x)在x_0处的负特征值个数来计算。与通常的局部情形相比，由于Hartree型非线性，我们的问题是非局部的，由此产生若干困难，需要引入对u_ε处相应线性化问题的特征对{(λ_{i,ε},v_{i,ε})}的新估计。据我们所知，本文似乎是第一篇考虑Hartree型Brézis-Nirenberg问题定性分析的工作，我们的结果将M. Grossi等人在[GP]中以及F. Takahashi在[Ta3]中建立的工作推广到非局部情形。
英文摘要：
In this paper, we are concerned with the critical Hartree equation \begin{equation*} \begin{cases} -Δu=\left(\displaystyle{\displaystyle{\int_Ω}}\frac{u^{2^{*}_μ}(y)}{|x-y|^μ}dy\right)u^{2^{*}_μ-1}+\varepsilon u,\quad u>0,\quad &\text{in $Ω$,}\\ u=0,\quad &\text{on $\partialΩ$,} \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ ($N\geq 5$) is a smooth bounded domain, $μ\in (0,4)$ and $2^{*}_μ=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under a non-degeneracy condition on the critical point $x_0\inΩ$ of the Robin function $R(x)$, we perform that for $\varepsilon>0$ sufficiently small, the Morse index of the blow-up solutions $u_\varepsilon$ concentrating at $x_0$ can be computed in terms of the negative eigenvalues of the Hessian matrix $D^{2}R(x)$ at $x_0$. Compared with the usual local cases, our problem is non-local due to the nonlinearity with Hartree-type, and several difficulties arise and new estimates of the eigenpairs $\{\left(λ_{i,\varepsilon},v_{i,\varepsilon}\right)\}$ to the associated linearized problem at $u_{\varepsilon}$ should be introduced. To our knowledge, this seems to be the first paper to consider the qualitative analysis of a Hartree type Brézis-Nirenberg problem and our results extend the works established by M. Grossi et al in \cite{GP} and F. Takahashi in \cite{Ta3} to the non-local case.
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