穿孔区域中临界Hartree方程的泡泡解
Bubble solution for the critical Hartree equation in pierced domain
AI总结：
本文在包含原点的有界光滑区域中挖去半径为ε的小球形成穿孔区域，针对N≥5、μ充分接近0的临界Hartree方程，证明了当ε趋于零时存在在原点处爆破的泡泡解。
摘要：
本文研究如下临界Hartree方程解的存在性：在穿孔区域Ω_ε中，-Δu等于（对Ω_ε上u^{2_μ^*}与|x-y|^{-μ}乘积的积分）乘以u^{2_μ^*-1}，且在边界∂Ω_ε上u=0。其中2_μ^*=(2N-μ)/(N-2)是Hardy-Littlewood-Sobolev不等式意义下的上临界指数，N≥5，0<μ<4且μ充分接近0，Ω_ε:=Ω\B(0,ε)，Ω是ℝ^N中包含原点的有界光滑区域，ε为正参数。当ε趋于零时，我们构造出在原点处爆破的泡泡解。
英文摘要：
In this article, we establish the existence of solutions to the following critical Hartree equation \begin{align*} \begin{cases} -Δu=\left(\int_{Ω_\varepsilon}\frac{u^{2_μ^*}}{|x-y|^μ}dy\right)u^{2_μ^*-1}, &\text{ in } Ω_\varepsilon, \\ u=0, &\text{ on } \partialΩ_\varepsilon, \end{cases} \end{align*} where $2_μ^*=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $N\geq 5$, $0<μ<4$ with $μ$ sufficiently close to $0$, $Ω_\varepsilon:=Ω\backslash B(0,\varepsilon)$ and $Ω$ is a bounded smooth domain in $\mathbb{R}^N$, which contains the origin, and $\varepsilon$ is a positive parameter. As $\varepsilon$ goes to zero, we construct bubble solution which blows up at the origin.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
