论文标题
Coron的关键车道系统问题
Coron's problem for the critical Lane-Emden system
论文作者
论文摘要
在本文中，我们解决了关键车道填充系统\ [\ [case}-ΔU= | V |^{p-1} v＆\ mbox {in}ω_________________________________v= | u | ω_ε，\ end {cases} \] 其中$ n \ ge 4 $，$ p \ in（1，\ frac {n-1} {n-2}）$，$ \ frac {1} {p+1}+\ frac {1} {q+1} = \ frac {n-2} {n-2} {n-2} {n} {n} $ and $ a $ aby $ a $ aind y Inup aind and and and a ranuius dombius。我们证明该系统承认一个积极解决方案家族，将孔周围集中在孔中心为$ε\至0 $，也获得了解决方案的具体定性描述。据我们所知，这是关键的车道系统在有限域上的第一个存在结果，而自1990年代初期，由于Mitidieri（1993）[30]和van der der vorst（1991），不存在星形有界域的不存在结果[30] [36]。
In this paper, we address the solvability of the critical Lane-Emden system \[\begin{cases} -Δu=|v|^{p-1}v &\mbox{in } Ω_ε,\\ -Δv=|u|^{q-1}u &\mbox{in } Ω_ε,\\ u=v=0 &\mbox{on } \partial Ω_ε, \end{cases}\] where $N \ge 4$, $p \in (1,\frac{N-1}{N-2})$, $\frac{1}{p+1} + \frac{1}{q+1}=\frac{N-2}{N}$, and $Ω_ε$ is a smooth bounded domain with a small hole of radius $ε> 0$. We prove that the system admits a family of positive solutions that concentrate around the center of the hole as $ε\to 0$, obtaining a concrete qualitative description of the solutions as well. To the best of our knowledge, this is the first existence result for the critical Lane-Emden system on a bounded domain, while the non-existence result on star-shaped bounded domains has been known since the early 1990s due to Mitidieri (1993) [30] and van der Vorst (1991) [36].
