论文标题
$ d $耦合的schrödinger系统的最少能量积极的灵魂在维度三
Least energy positive soultions for $d$-coupled Schrödinger systems with critical exponent in dimension three
论文作者
论文摘要
在本文中，我们考虑具有关键指数的耦合schrödinger系统：\ begin {equination*} \ begin {case}-ΔU_I+λ_{i} u_i = \ sum \ sum \ sum \ limits_ {j = 1} in}ω，\\ u_i \ in H_0^1（ω），\ Quad I = 1,2，...，d。 \ end {cases} \ end {equation*}在这里，$ω\ subset \ mathbb {r}^{3} $是一个平稳的界面域，$ d \ geq 2 $，$β_{ii}> 0 $ $ i $，$β_{ij {ij} =β_= j $ i ne，我们研究Brézis-nirenberg类型问题：$-λ_{1}（ω）<λ_{1}，\ cdots，λ_{d} <-λ^*（ω）$，其中$λ_{1}（1}（ω）$是$-Δ$的第一个eigenvalue of $-Δ λ_1（ω））$。我们通过各种参数获得了针对弱合作的情况（$β_{ij}> 0 $ small）的最低能量积极解决方案（$β_{ij}> 0 $ small）（$β_{ij {ij} \ leq 0 $）。该证明是通过数学诱导对方程数进行的，并且需要对该系统进行更多精致的能量估计。此外，我们提出了一个新的不存在的结果，与较高的案例$ n \ geq 5 $相比揭示了一些不同的现象。看来，这是第一篇论文为第三维度中的关键Schrödinger系统存在最少的能量积极解决方案提供了相当完整的图片。
In the present paper, we consider the coupled Schrödinger systems with critical exponent: \begin{equation*} \begin{cases} -Δu_i+λ_{i}u_i=\sum\limits_{j=1}^{d} β_{ij}|u_j|^{3}|u_i|u_i \quad ~\text{ in } Ω,\\ u_i \in H_0^1(Ω) ,\quad i= 1,2,...,d. \end{cases} \end{equation*} Here, $Ω\subset \mathbb{R}^{3}$ is a smooth bounded domain, $d \geq 2$, $β_{ii}>0$ for every $i$, and $β_{ij}=β_{ji}$ for $i \neq j$. We study a Brézis-Nirenberg type problem: $-λ_{1}(Ω)<λ_{1},\cdots,λ_{d}<-λ^*(Ω)$, where $λ_{1}(Ω)$ is the first eigenvalue of $-Δ$ with Dirichlet boundary conditions and $λ^*(Ω)\in (0, λ_1(Ω))$. We acquire the existence of least energy positive solutions to this system for weakly cooperative case ($β_{ij}>0$ small) and for purely competitive case ($β_{ij}\leq 0$) by variational arguments. The proof is performed by mathematical induction on the number of equations, and requires more refined energy estimates for this system. Besides, we present a new nonexistence result, revealing some different phenomena comparing with the higher-dimensional case $N\geq 5$. It seems that this is the first paper to give a rather complete picture for the existence of least energy positive solutions to critical Schrödinger system in dimension three.
