论文标题
Sobolev CriticalSchrödinger系统的标准化基态的存在和渐近行为
Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems
论文作者
论文摘要
该论文涉及以下非线性schrödinger系统的归一化基态的存在和渐近特性： \ left \ {\ begin {aligned} ＆-ΔU+λ_1u = | u |^{2^* - 2} u+{να} | u |^{α-2} | v |^βu，\ quad \ quad \ text {in} \ mathbb {r}^n， ＆-ΔV+λ_2v = | v |^{2^* - 2} v+{νβ} | u |^α| ＆\ int u^2 = a^2，\; \; \; \ int v^2 = b^2， \ end {Aligned} \ right。 \ end {equation*}其中$ n = 3,4 $，$α，β> 1 $，$ 2 <α+β<2^*= \ frac {2n} {n-2} $。我们证明，对于$ν<0 $，不存在正常化的基态。当$ν> 0 $和$α+β\ le 2+ \ frac {4} {n} $时，我们表明该系统具有$ 0 <ν<ν_0$的归一化基态解决方案，将明确给出常数$ν_0$。在这种情况下，在$α+β> 2+ \ frac {4} {n} $的情况下，我们证明存在阈值$ν_1\ ge 0 $，因此对于$ν>ν_1$而存在的归一化基态解决方案，并且不存在$ν<ν_1$。我们还提供$ν_1= 0 $的条件。最后，我们将最小化器的渐近行为视为$ν\ to0^+$或$ν\ to+\ infty $。
The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrödinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-δu+λ_1 u=|u|^{2^*-2}u+{να} |u|^{α-2}|v|^βu,\quad \text{in }\mathbb{R}^N, &-δv+λ_2 v=|v|^{2^*-2}v+{νβ} |u|^α|v|^{β-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $α,β>1$, $2<α+β<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $ν<0$. When $ν>0$ and $α+β\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<ν<ν_0$, the constant $ν_0$ will be explicitly given. In the case $α+β>2+\frac{4}{N}$ we prove the existence of a threshold $ν_1\ge 0$ such that a normalized ground state solution exists for $ν>ν_1$, and does not exist for $ν<ν_1$. We also give conditions for $ν_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $ν\to0^+$ or $ν\to+\infty$.
